%%%@@@ input from paper
\documentstyle[amssymb,12pt]{amsart}

\textwidth15.6truecm
\textheight22.8truecm
%\oddsidemargin0cm
%\topmargin-1cm
\hoffset-0.5truecm
%\voffset-0.5truecm

 

%%%@@@ input from makros
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%        Makros zum Formatieren des Entwurfs fuer Christian:               %
%        Begonnen am 3. November 1993                                      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%% \theoremstyle{plain} %% This is the default|
\catcode`\@=11
\def\section{\@startsection{\@string\section}%
        1\z@{6\p@\@plus4\p@}{1\p@}%
        {\defaultfont\bf}}
\catcode`\@=12
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Christians Original-Figure-environment: Keine LaTeX-figures!             %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \newcounter{Figure} % Brauch' ma des?
 \newenvironment{Figure}%
    {\refstepcounter{Figurenum}%
         \vskip10pt minus 5pt \vbox\bgroup\hsize\textwidth\begin{center}}%
    {\vskip6pt\centerline{\hfil\small Figure~%
         \arabic{Figurenum}\hfil}\end{center}\egroup\vskip10pt minus 5pt}
 \newcounter{Figurenum}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Figure-environment ist nun als LaTeX-figure definiert!                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newenvironment{Figure}{\begin{figure}\caption{}}{\end{figure}}

\def\la{\lambda}
\def\ep{\epsilon}
\def\si{\sigma}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
 
\theoremstyle{definition}
\newtheorem{dfn}{Definition}[section]
 
\theoremstyle{remark}
\newtheorem{rem}{Remark}[section]
\newtheorem{ex}[rem]{Example}
 
\numberwithin{equation}{section}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 					Diverse Kleinigkeiten, Abkuerzungen etc.:							%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\let\epsilon\varepsilon
\newcommand{\esf}{{e}}			% elementary symmetric function
\newcommand{\hsf}{{h}}			% homogeneous symmetric function
\newcommand{\ksf}{{k}}			% Delta^2 of homogeneous symmetric function
\newcommand{\schf}[2]{{s}_{#1}\of{#2,\bold{x}}}
												% Schur function
\newcommand{\defeq}{:=}					% define equal...
\newcommand{\tableau}{T}				% semistandard tableau
\newcommand{\Ferrer}{B}				   % Ferrers board
\newcommand{\digraph}{D}				% directed graph
\newcommand{\sdigraph}{S}				% Stembridge's directed graph
\newcommand{\horedge}{a}				% horizontal edge
\newcommand{\veredge}{\hat{a}}		% vertical edge
\newcommand{\path}{P}					% lattice path
\newcommand{\pathset}{M}				% lattice path's adjoined edge set
\newcommand{\fpath}{{\cal P}}			% family of lattice paths
\newcommand{\nfpath}{{\cal N}}		% number of lattice paths
\newcommand{\nifpath}{{\cal P}_o}	% nonintersecting family of lattice paths
\newcommand{\SG}[1]{\boldsymbol{S}_{#1}}
												% Symmetric group
\newcommand{\GF}[1]{\bold{GF}\of{#1}}
												% Generating function
\newcommand{\sgn}{\operatorname{sgn}}			% Signum
\newcommand{\inv}{\operatorname{inv}}			% Inversionen
\newcommand{\sschf}[2]{{sp}_{#1}\of{#2,\bold{x}}}
\newcommand{\isschf}[2]{{sp}_{n,m}\of{#1,#2;\bold{x},\bold z}}
\newcommand{\osschf}[2]{{sp}_{n,1}\of{#1,#2,\bold{x},z}}
												% symplectic Schur function
\newcommand{\reflect}{{\bold{R}}}		% reflection
\newcommand{\mreflect}{\tilde{\bold{R}}}		% modified reflection
\newcommand{\oschf}[2]{{o}_{#1}\of{#2,\bold{x}}}
												% orthogonal Schur function
\newcommand{\oschftriv}[2]{{o}_{#1}\of{#2,\seqof{1,1,\dots,1}}}
												% "coarse" orthogonal Schur function
\newcommand{\liff}{\text{ iff }}		% logical: if and only if
\newcommand{\limp}{\text{ implies }}
												% logical: implies
\renewcommand{\land}{\text{ and }}	% logical: and
 
\newcommand{\thmref}[1]{Theorem~\ref{#1}}
\newcommand{\secref}[1]{\S\ref{#1}}
\newcommand{\lemref}[1]{Lemma~\ref{#1}}
\newcommand{\figref}[1]{Figure~\ref{#1}}
 
\newcommand{\proof}{{\par\noindent\bf Proof: }} 
\newcommand{\QED}{\nobreak\strut\hfill\qedsymbol\par\vskip3mm}
\newcommand{\noproof}{\nobreak\hfill\qedsymbol\par\vskip3mm}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%                    Diverse Klammerungen:                                 %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\seqof}[1]{\left\langle#1\right\rangle}
\newcommand{\setof}[1]{\left\{#1\right\}}
\newcommand{\of}[1]{\left(#1\right)}
\newcommand{\parof}[1]{\left(#1\right)}
\newcommand{\numof}[1]{\left|#1\right|}
\newcommand{\detof}[2]{\left|{#2}\right|_{#1\times #1}}
\newcommand{\firstcolumn}[1]{%\underbrace{
#1%}_{\text{first column}}
\quad \vdots\quad }
\newcommand{\firstrow}[1]{%\underbrace{
#1%}_{\text{first row}}
\quad \vdots\quad }
\newcommand{\brkof}[1]{\left[#1\right]}
\newcommand{\intof}[1]{\lfloor #1\rfloor}
\newcommand{\absof}[1]{\left|#1\right|}
\newcommand{\sgnof}[1]{\operatorname{sgn}\of{#1}}
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Spezielle Makros fuer symporth.                                           %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\basicop}{basic reflected crossing-over}
\newcommand{\rgvi}{reflection-Gessel-Viennot involution}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Spezielle Makros fuer Proctor-Sundaram-Bijektion.                         %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\proctor}{{\Bbb P}}		% All Proctor objects
\newcommand{\varproctor}{{\Bbb RP}}	% Variant of (restricted) Proctor objects
\newcommand{\pcore}{\proctor^{C}}	% Core of Proctor objects: P-Tableaux
\newcommand{\phull}{\proctor^{H}}	% All \setminus Core
\newcommand{\varphull}{\varproctor^{H}}	% All \setminus Core

\newcommand{\sundaram}{{\Bbb S}}		% All Sundaram objects
\newcommand{\score}{\sundaram^{C}}	% Core of Sundaram objects: S-Tableaux
\newcommand{\shull}{\sundaram^{H}}	% All \setminus Core

\newcommand{\inftiesincolumn}[2]{\numof{\infty\in\setof{#1_{.,#2}}}}
\newcommand{\numofentries}[2]{\numof{#1\in\setof{#2}}}
\newcommand{\leqincol}[3]{\setof{#3_{.,#2}\leq #1}}
\newcommand{\morethanincol}[3]{\setof{#3_{.,#2}> #1}}
\newcommand{\pairsincol}[3]{\operatorname{pairs}\of{
	\lessthanincol{#1}{#2}{#3}}}
\newcommand{\singsincol}[3]{\operatorname{singles}\of{
	\lessthanincol{#1}{#2}{#3}}}
\newcommand{\inverted}[1]{\overline{#1}}
\newcommand{\nivsecpath}[2]{\brkof{N_{#1}\cap P_{#2}}}

     % Das ganze AMS-LaTeX-Zeug ausgelagert...                   %
%%%@@@ end input from makros

%%%@@@ input from paths
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%               PATHS.TEX: Begonnen am 16. Juli 1993                       %
% Dieses kleine Macropaket soll einfache Befehle zum Zeichnen von Gittern  %
% und Gitterpunktwegen einfuehren. (Fuer AMSLaTeX!)                        %
% Wesentlich ueberarbeitet und gestrafft am 1. Mai 1994.                   %
% Ergaenzt am 24. August 1994: Autmoatisches labelling von Wegen,          %
% Giambelli-Wege ...                                                       %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fonts:                                                                   %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\font\scalefont = cmti8
 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Basic parameters:                                                        %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\unitlength = 5mm					% instead of 4 mm
\thicklines

\newcount\xmin                % coordinates of extremal points of the grid
\newcount\xmax
\newcount\ymin
\newcount\ymax

\newcount\gridwidth
\newcount\gridheight

\newcount\nofxpoints          % number of points in horizontal direction
\newcount\nofypoints          % number of points in vertical direction

\newcount\dcnta               % dummy-counter A
\newcount\dcntb               % dummy-counter B
\newcount\dcntc					% dummy-counter C

% Special macros for drawing a grid with lower left corner (xmin, ymin)
% and upper right corner (xmax, ymax):
\def\begingrid#1#2#3#4{			% arguments: xmin, ymin, xmax, ymax
	\global\xmin = #1
	\global\ymin = #2
	\global\xmax = #3
	\global\ymax = #4
	% simple test:
	\ifnum\xmin > \xmax\errmessage{PATHS: \xmin > \xmax|}\fi
	\ifnum\ymin > \ymax\errmessage{PATHS: \ymin > \ymax|}\fi
	% computation of heigth and width
	\global\gridwidth = \xmax
	\global\gridheight = \ymax
	\global\advance\gridwidth by -\xmin
	\global\advance\gridheight by -\ymin
	\nofxpoints = \gridwidth
	\advance\nofxpoints by 1
	\nofypoints = \gridheight
	\advance\nofypoints by 1
		% adjust some space around the picture
		% \advance\gridwidth by 4
		% \advance\gridheight by 4
		% \advance\xmin by -2
		% \advance\ymin by -2
	% LaTeX's picture command
	\begin{picture}(\gridwidth, \gridheight)(\xmin, \ymin)
		% restore correct the lengths
		% \advance\gridwidth by -4
		% \advance\gridheight by -4
		% \advance\xmin by 2
		% \advance\ymin by 2
	% draw gridpoints:
	\global\dcnta = \ymin
	\loop\ifnum\dcnta<\ymax
		\makegridline\dcnta	% do inner loop in a separate macro
		\global\advance\dcnta by 1
	\repeat
	\makegridline\ymax
}
 
% inner loop of grid-drawing procedure
\def\makegridline#1{
   \begingroup
   \global\dcntb = \xmin
   \loop\ifnum\dcntb<\xmax
      \put(\dcntb, #1){\circle*{0.1}}
      \global\advance\dcntb by 1
   \repeat
   \put(\xmax, #1){\circle*{0.1}}
   \endgroup
}
 
% draw axes
\def\makexaxis{\thinlines
        \dcnta = \gridwidth
        \advance\dcnta by 1
        \put(\xmin, 0){\vector(1,0){\dcnta}}
        \thicklines
}
\def\makeyaxis{\thinlines
        \dcntb = \gridheight
        \advance\dcntb by 1
        \put(0, \ymin){\vector(0,1){\dcntb}}
        \thicklines
}
\def\makeaxes{\thinlines
        \dcnta = \gridwidth
        \advance\dcnta by 1
        \put(\xmin, 0){\vector(1,0){\dcnta}}
        \dcntb = \gridheight
        \advance\dcntb by 1
        \put(0, \ymin){\vector(0,1){\dcntb}}
        \thicklines
}
 
% scaling on axes
\def\makexscale{
   \dcnta = \xmin
	\ifnum\dcnta<-9\dcnta=-9\fi
   \loop\ifnum\dcnta<0
		\put(\dcnta,0){\line(0,-1){0.1}}
      \put(\dcnta,-0.4){\makebox(0,0)[t]{
		{\scalefont\number\dcnta}}}
      \advance\dcnta by 1
  \repeat
  \dcnta = 1
  \dcntb=\xmax
  \ifnum\dcntb>9\dcntb=9\fi
  \loop\ifnum\dcnta<\dcntb
      \put(\dcnta,0){\line(0,-1){0.1}}
      \put(\dcnta, -0.4){\makebox(0,0)[t]{
		{\scalefont\number\dcnta}}}
      \advance\dcnta by 1
  \repeat
  \put(\dcntb,0){\line(0,-1){0.1}}
  \put(\dcntb, -0.4){\makebox(0,0)[t]{{\scalefont\number\dcntb}}}
}
\def\makeyscale{
   \dcnta = \ymin
   \loop\ifnum\dcnta<0
      \put(0,\dcnta){\line(-1,0){0.1}}
      \put(-0.3, \dcnta){\makebox(0,0)[r]{
		{\scalefont\number\dcnta}}}
      \advance\dcnta by 1
   \repeat
   \dcnta = 1
   \loop\ifnum\dcnta<\ymax
      \put(0,\dcnta){\line(-1,0){0.1}}
      \put(-0.3, \dcnta){\makebox(0,0)[r]{
		{\scalefont\number\dcnta}}}
      \advance\dcnta by 1
   \repeat
	\put(0,\ymax){\line(-1,0){0.1}}
   \put(-0.3, \ymax){\makebox(0,0)[r]{{\scalefont\number\ymax}}}
}

% macro for captions:
\def\gridcaption#1{
	\dcnta=\xmin
	\dcntb=\gridwidth
	\divide\dcntb by 2
	\advance\dcnta by \dcntb
	% \advance\dcnta by 1
	\dcntb=\ymax
	% \advance\dcntb by 2
	\advance\dcntb by 1
	\put(\dcnta,\dcntb){	\begin{picture}(1,1)(0,0)
								\put(0.5,0.5){\makebox(0,0){#1}}
								\end{picture}
							}
}

% macros for drawing skew dotted lines
\newsavebox{\skewdotteddownbox}
\newsavebox{\skewdottedupbox}
\savebox{\skewdotteddownbox}(1,1)[lb]{
	\put(0.2,-0.2){\circle*{0.01}}
   \put(0.4,-0.4){\circle*{0.01}}
 	\put(0.6,-0.6){\circle*{0.01}}
	\put(0.8,-0.8){\circle*{0.01}}
}
\savebox{\skewdottedupbox}(1,1)[lb]{
	\put(0.2,0.2){\circle*{0.01}}
   \put(0.4,0.4){\circle*{0.01}}
 	\put(0.6,0.6){\circle*{0.01}}
	\put(0.8,0.8){\circle*{0.01}}
}
\def\downskewline#1#2#3{
	\multiput(#1,#2)(1,-1){#3}{\usebox{\skewdotteddownbox}}
}
\def\upskewline#1#2#3{
	\multiput(#1,#2)(1,1){#3}{\usebox{\skewdottedupbox}}
}

% Special macros for drawing lattice paths:
\newsavebox{\hplainbox}
\savebox{\hplainbox}{\put(0,0){\line(1,0){1}}}
\newsavebox{\hdottedbox}
\savebox{\hdottedbox}(1,0)[bl]{
	\put(0.2,0){\circle*{0.01}}
	\put(0.4,0){\circle*{0.01}}
	\put(0.6,0){\circle*{0.01}}
	\put(0.8,0){\circle*{0.01}}
}
\newsavebox{\vplainupbox}
\savebox{\vplainupbox}{\put(0,0){\line(0,1){1}}}
\newsavebox{\vdottedupbox}
\savebox{\vdottedupbox}(0,1)[bl]{
	\put(0,0.2){\circle*{0.01}}
	\put(0,0.4){\circle*{0.01}}
	\put(0,0.6){\circle*{0.01}}
	\put(0,0.8){\circle*{0.01}}
}
\newsavebox{\vplaindownbox}
\savebox{\vplaindownbox}{\put(0,0){\line(0,-1){1}}}
\newsavebox{\vdotteddownbox}
\savebox{\vdotteddownbox}(0,1)[bl]{
	\put(0,-0.2){\circle*{0.01}}
	\put(0,-0.4){\circle*{0.01}}
	\put(0,-0.6){\circle*{0.01}}
	\put(0,-0.8){\circle*{0.01}}
}
\newsavebox{\skewupbox}
\savebox{\skewupbox}{\put(0,0){\line(1,1){1}}}
\newsavebox{\skewdownbox}
\savebox{\skewdownbox}{\put(0,0){\line(1,-1){1}}}
\newsavebox{\hstep}		% horizontal step
\newsavebox{\vstep}		% vertical step
\newsavebox{\sstep}		% skew step
\newsavebox{\rvstep}		% reverse vertical step
\newsavebox{\rsstep}		% reverse skew step
\newcount\updownincrement

% Macros for automatic labelling of steps:
\newif\iflabelled
\newif\ifelabel
\newcount\labelinfty
\def\setlabelinfty#1{\global\labelinfty=#1}
\newcount\labelcount
\setlabelinfty{1000}
\def\printlabel{
	\ifnum\labelcount<\labelinfty{\scalefont\number\labelcount}
	\else$\scriptstyle\infty$
	\fi
}

\def\uppath#1#2#3{
	\global\labelcount=1	% If path starts with horizontal step -> 1
	\savebox{\hstep}{\usebox{\hplainbox}}
	\savebox{\vstep}{\usebox{\vplainupbox}}
	\savebox{\sstep}{\usebox{\skewupbox}}
	\savebox{\rvstep}{\usebox{\vplaindownbox}}
	\savebox{\rsstep}{\usebox{\skewdownbox}}
	\updownincrement=1
	\dcnta=#1
	\dcntb=#2
	\thickpoint{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\thickpoint{\dcnta}{\dcntb}
}
\def\updottedpath#1#2#3{
	\global\labelcount=1	% If path starts with horizontal step -> 1
	\savebox{\hstep}{\usebox{\hdottedbox}}
	\savebox{\vstep}{\usebox{\vdottedupbox}}
	\savebox{\sstep}{\usebox{\skewdottedupbox}}
	\savebox{\rvstep}{\usebox{\vdotteddownbox}}
	\savebox{\rsstep}{\usebox{\skewdotteddownbox}}
	\updownincrement=1
	\dcnta=#1
	\dcntb=#2
	\thickcircle{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\thickcircle{\dcnta}{\dcntb}
}
\def\downpath#1#2#3#4{
	\global\labelcount=#4 % If path starts with horizontal step -> labeled #4
	\savebox{\hstep}{\usebox{\hplainbox}}
	\savebox{\vstep}{\usebox{\vplaindownbox}}
	\savebox{\sstep}{\usebox{\skewdownbox}}
	\savebox{\rvstep}{\usebox{\vplainupbox}}
	\savebox{\rsstep}{\usebox{\skewupbox}}
	\updownincrement=-1
	\dcnta=#1
	\dcntb=#2
	\thickpoint{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\thickpoint{\dcnta}{\dcntb}
	\elabelfalse	% for Giambelli special case only!
}
\def\downdottedpath#1#2#3#4{
	\global\labelcount=#4 % If path starts with horizontal step -> labeled #4
	\savebox{\hstep}{\usebox{\hdottedbox}}
	\savebox{\vstep}{\usebox{\vdotteddownbox}}
	\savebox{\sstep}{\usebox{\skewdotteddownbox}}
	\savebox{\rvstep}{\usebox{\vdottedupbox}}
	\savebox{\rsstep}{\usebox{\skewdottedupbox}}
	\updownincrement=-1
	\dcnta=#1
	\dcntb=#2
	\thickcircle{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\thickcircle{\dcnta}{\dcntb}
	\elabelfalse	% for Giambelli special case only!
}
% For segments of paths
\def\segment#1#2#3{
	\labelledfalse
	\savebox{\hstep}{\usebox{\hplainbox}}
	\savebox{\vstep}{\usebox{\vplainupbox}}
	\savebox{\sstep}{\usebox{\skewupbox}}
	\updownincrement=1
	\dcnta=#1
	\dcntb=#2
	\gridpoint{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\gridpoint{\dcnta}{\dcntb}
}
\def\dottedsegment#1#2#3{
	\labelledfalse
	\savebox{\hstep}{\usebox{\hdottedbox}}
	\savebox{\vstep}{\usebox{\vdottedupbox}}
	\savebox{\sstep}{\usebox{\skewdottedupbox}}
	\updownincrement=1
	\dcnta=#1
	\dcntb=#2
	\gridcircle{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\gridcircle{\dcnta}{\dcntb}
}
% For ``unbounded'' paths running to infinity:
\def\uparrowpath#1#2#3{
	\global\labelcount=1	% If path starts with horizontal step -> 1
	\savebox{\hstep}{\usebox{\hplainbox}}
	\savebox{\vstep}{\usebox{\vplainupbox}}
	\savebox{\sstep}{\usebox{\skewupbox}}
	\savebox{\rvstep}{\usebox{\vplaindownbox}}
	\savebox{\rsstep}{\usebox{\skewdownbox}}
	\updownincrement=1
	\dcnta=#1
	\dcntb=#2
	\thickpoint{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\put(\dcnta,\dcntb){\vector(0,1){0.3}}
}
\def\updottedarrowpath#1#2#3{
	\global\labelcount=1	% If path starts with horizontal step -> 1
	\savebox{\hstep}{\usebox{\hdottedbox}}
	\savebox{\vstep}{\usebox{\vdottedupbox}}
	\savebox{\sstep}{\usebox{\skewdottedupbox}}
	\savebox{\rvstep}{\usebox{\vdotteddownbox}}
	\savebox{\rsstep}{\usebox{\skewdotteddownbox}}
	\updownincrement=1
	\dcnta=#1
	\dcntb=#2
	\thickcircle{\dcnta}{\dcntb}
	\dosteplist#3\endlist
	\put(\dcnta,\dcntb){\vector(1,0){0.3}}
}

\newsavebox{\dummybox}
\def\xboxes{\savebox{\dummybox}{\usebox{\vstep}}
				\savebox{\vstep}{\usebox{\rvstep}}
				\savebox{\rvstep}{\usebox{\dummybox}}
				\savebox{\dummybox}{\usebox{\sstep}}
				\savebox{\sstep}{\usebox{\rsstep}}
				\savebox{\rsstep}{\usebox{\dummybox}}
}
				
\def\endlist{\endlist}
\def\dosteplist{\afterassignment\handlenextstep\let\next=}
\def\handlenextstep{
	\ifx\next\endlist
   	\let\next=\relax
	\else
		\ifx\next-
			\put(\dcnta,\dcntb){\usebox{\hstep}}
			\iflabelled\put(\dcnta,\dcntb){\makebox(1,0.6){\printlabel}}\fi
			\ifelabel\advance\labelcount by \updownincrement\fi
			\advance\dcnta by 1
		\else
			\ifx\next|
				\advance\labelcount by \updownincrement
				\put(\dcnta,\dcntb){\usebox{\vstep}}
				\advance\dcntb by\updownincrement
			\else
				\ifx\next/
					\put(\dcnta,\dcntb){\usebox{\sstep}}
					\iflabelled
						\ifnum\updownincrement>0
							\advance\labelcount by 1 % special asymmetric labelling...
							\put(\dcnta,\dcntb){\begin{picture}(1,1)(0,0)
								\put(-0.5,0.5){\printlabel}\end{picture}}
						\else
							\put(\dcnta,\dcntb){\begin{picture}(1,1)(0,0)
								\put(0.3,-0.5){\printlabel}\end{picture}}
						\fi
					\fi
					\advance\dcnta by 1
					\advance\dcntb by\updownincrement
				\else
					\ifx\next* % Up/down change for Giambelli lattice paths!
						\ifelabel\elabelfalse\else\elabeltrue\fi
						\xboxes	% exchange up/down boxes, e-/h-labelling! 
						\updownincrement=-\updownincrement
					\else
						\errmessage{PATHS: Wrong symbol.}
					\fi
				\fi
			\fi
		\fi
		\let\next=\dosteplist
	\fi
	\next
}
 
% draw thick point:
\def\thickpoint#1#2{\put(#1, #2){\circle*{0.4}}}

% draw thick circle
\def\thickcircle#1#2{\put(#1, #2){\circle{0.4}}}

% draw ``medium'' point:
\def\gridpoint#1#2{\put(#1, #2){\circle*{0.2}}}
 
% draw ``medium'' circle:
\def\gridcircle#1#2{\put(#1, #2){\circle{0.2}}}

% label single horizontal step
\def\hlabel#1#2#3{\put(#1,#2){\makebox(1,0.6){\scalefont #3}}}

% label thick point above or below:
\def\labelpointal#1#2#3{							% above, to the left
	\put(#1,#2){
		\begin{picture}(1,1)(0,0)
			\put(-0.85,0.35){$\scriptstyle #3$}
		\end{picture}
	}
}

\def\labelpointar#1#2#3{							% above, to the right
	\put(#1,#2){
		\begin{picture}(1,1)(0,0)
			\put(-0.15,0.35){$\scriptstyle #3$}
		\end{picture}
	}
}

\def\labelpointbl#1#2#3{							% below, to the left
	\put(#1,#2){
		\begin{picture}(1,1)(0,0)
			\put(-0.85,-0.8){$\scriptstyle #3$}
		\end{picture}
	}
}

\def\labelpointbr#1#2#3{							% below, to the right
	\put(#1,#2){
		\begin{picture}(1,1)(0,0)
			\put(-0.15,-0.8){$\scriptstyle #3$}
		\end{picture}
	}
}

\def\endgrid{\end{picture}}

\def\drawrect#1#2#3#4{
	\put(#1,#2){%
		\begin{picture}(#3,#4)(0,0)\thicklines%
			\put(0,0){\line(0,1){#4}}
			\put(0,0){\line(1,0){#3}}
			\put(#3,#4){\line(0,-1){#4}}
			\put(#3,#4){\line(-1,0){#3}}
		\end{picture}
	}
}

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%%%@@@ end input from paths

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%  TABLEAUX.TEX: Begonnen am 20. Juli 1993                                 %
% Dieses kleine Macropaket soll einfache Befehle zur Darstellung von       %
% Tableaux  einfuehren. (Nur fuer AMSLaTeX|)                               %
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\newcount\rowcount
\newcount\columncount

\def\begintableau#1#2{									% rows, columns
\global\rowcount=#1
\begin{picture}(#2,#1)(0,0)
\thinlines
% \put(0,0){\line(0,1){#1}}
% \put(0,#1){\line(1,0){#2}}
}

\def\endtableau{\end{picture}}

\def\row#1{
	\global\advance\rowcount by -1
	\global\columncount=0
	\dorowlist#1\endlist
}

\def\endlist{\endlist}

\def\dorowlist{\afterassignment\handlenextentry\let\next=}

\def\handlenextentry{
	\ifx\next\endlist
		\let\next=\relax
	\else
		\ifx\next=-
			\put(\columncount,\rowcount){\framebox(1,1){\space}}
			\global\advance\columncount by 1
		\else
			\put(\columncount,\rowcount){\framebox(1,1){$\next$}}
			\global\advance\columncount by 1
		\fi
		\let\next=\dorowlist
	\fi
	\next
}
 

   % Makros zur einfachen Eingabe von Tableaux...              %
%%%@@@ end input from tableaux
 
\begin{document}
 
\title[Characters of symplectic and orthogonal groups]
{Lattice path proofs for determinantal formulas for symplectic and orthogonal
characters}
 
\author{Markus Fulmek}
\address{Dept.~of Economics\\University of Vienna\\Hohenstaufengasse 9\\
	A-1010 Vienna, Austria}
\author{Christian Krattenthaler$^\dagger$}
\address{Institut f\"ur Mathematik\\Universit\"at Wien\\Strudlhofgasse~4\\
	A-1090 Vienna, Austria}
\thanks{$^\dagger$ Supported in
part by EC's Human Capital and Mobility Program, grant
CHRX-CT93-0400, the second author was also supported by the
Austrian Science Foundation FWF, grant P10191-MAT}
%\date{\today}
 
\maketitle
\begin{abstract}
We give bijective proofs for Jacobi--Trudi-type and Giambelli-type
identities for symplectic and orthogonal characters. These proofs
base on interpreting King and El-Sharkaway's symplectic tableaux, Proctor's odd
and intermediate symplectic tableaux, Proctor's and King and Welsh's
orthogonal tableaux, and Sundaram's odd orthogonal tableaux
in terms of certain families of nonintersecting
lattice paths. This work is intended to be the counterpart of the
Gessel--Viennot proof of the Jacobi--Trudi identities for Schur
functions for the case of symplectic and orthogonal characters.
\end{abstract}
 
\section{Introduction}\label{Intro}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%    Einleitender Ueberblick: Was soll in diesem Papier gezeigt werden...     %
%                                                                             %
%    Notizen zum Gespraech vom 21. Oktober 1993:                              %
%    In der Einleitung: Verweis auf Gessel-Viennot-Artikel, genauer im Buch   %
%    von Sagan, p154ff: Von diesem approach fuer normale Tableaux ausgehend   %
%    besonders huebsche e-formel mit den symplektischen Tableaux von King     %
%    und El-Sharkaway ... liefert insight ... h-Formel leider weniger schoen: %
%    Uebergang von Christians algebraischen Umformungen der Determinanten     %
%    zum bijektiven Beweis erklaeren.                                         %
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%	Definitionen schon in Einleitung bringen? Momentan geschieht das nicht|    %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\message{introduction: what will happen here...} 
Schur functions, which are irreducible general linear characters, can be
combinatorially defined by means of semistandard Young tableaux (cf.\
section~\ref{Tableaux} for all background information). There are several
determinant formulas for Schur functions. Those which are relevant
for this paper are the Jacobi--Trudi identity and its dual form, the
N\"agelsbach--Kostka formula, and
the Giambelli identity (see (\ref{thm:h-Jacobi-Trudi}),
(\ref{thm:e-Jacobi-Trudi}), (\ref{thm:Giambelli})). In their
well-known (yet unpublished) paper 
%\cite[see also
%\catcode`\@=11
%\csname b@Stembridge:Nonintersecting-paths\endcsname
%\catcode`\@=12]{Gessel-Viennot:Determinants-paths}
\cite{Gessel-Viennot:Determinants-paths} (cf\@.
\cite[sec.~7]{Stembridge:Nonintersecting-paths})
Gessel and Viennot give a beautiful bijective proof for the
Jacobi--Trudi identities for Schur functions. It bases on
interpreting semistandard tableaux as families of nonintersecting
lattice paths. As was shown by Stembridge
\cite[sec.~9]{Stembridge:Nonintersecting-paths}, also the Giambelli
identity allows a bijective proof by using nonintersecting lattice
paths. (The first bijective proof of the Giambelli identity is due to
E\u gecio\u glu and Remmel \cite{Egecioglu-Remmel:A-combinatorial}.)

There are Jacobi--Trudi-type and Giambelli-type determinant formulas
for irreducible symplectic and orthogonal characters (see
(\ref{thm:h-symplectic}), (\ref{thm:e-symplectic}),
(\ref{thm:Giambelli-symplectic})
(\ref{thm:h-orthogonal}), (\ref{thm:e-orthogonal}),
(\ref{thm:Giambelli-orthogonal})), too. Since there are also tableau
descriptions for symplectic and orthogonal characters, it is natural
to ask for bijective proofs of the symplectic and orthogonal
Jacobi--Trudi identities (being partly due to Weyl
\cite{Weyl:Classical-Groups} and partly to Koike and Terada 
\cite{Koike-Terada:BnCnDn}) and Giambelli identities (being due to
Abramsky, Jahn and King \cite{Abramsky-Jahn-King:Frobenius}). 
First attempts in this direction for the Jacobi--Trudi identities
were made by Bressoud and Wei \cite{Bressoud-Wei}, and more
successfully by Okada \cite{Okada:Lattice-Path}. 
However, the determinant formulas that Okada
proved bijectively are different from the Jacobi--Trudi identities
(but are interesting in their own right).
Additional algebraic steps were necessary to prove the Jacobi--Trudi
identities itself.

In this paper we solve the problem completely for the symplectic case
and almost completely for the orthogonal case. 

For the bijective proofs of
the symplectic identities we utilize lattice path interpretations of
the tableaux given by King and El-Sharkaway
\cite{King-El-Sharkaway:Standard}, which are translations of the
Gelfand patterns of Zhelobenko \cite{Zhelobenko:classical}. Particularly nice is the bijective
proof for the {\em dual\/} symplectic Jacobi--Trudi identity (see
(\ref{thm:e-symplectic})), which
combines the Gessel--Viennot method with a 
``modified" reflection
principle. This idea originates from \cite{Krat:Major} and
is also exploited in
\cite{Krat:Oscill,Krat:Noncrossing,Krat-Sagan:Nonintersecting}.
A bijective proof for the ``ordinary" symplectic
Jacobi--Trudi identity (see (\ref{thm:h-symplectic})) requires 
more work. 
We give a proof that uses the concept of
dual lattice paths due to Gessel and Viennot
\cite{Gessel-Viennot:Binomial-determinants} and 
relies on 
our bijection for the symplectic
dual Jacobi--Trudi identity. 
In \cite{Fulmek-Kratt} we provided a different bijective proof 
which is inspired by Okada's combinatorial-algebraic proof
\cite[Lemma~3.1C, Corollary~4.2(2)]{Okada:Lattice-Path}. 
On the other hand, the
bijective proof of the symplectic Giam\-bel\-li identity is almost
trivial. In addition, we provide bijective proofs for all of Proctor's
\cite{Proctor:Odd,Proctor:Intermediate,Proctor:Young-tableaux}
Jacobi--Trudi identities for his odd symplectic and
intermediate symplectic characters (see
(\ref{thm:h-intermediate-symplectic}), 
(\ref{thm:e-intermediate-symplectic}), 
(\ref{thm:e-odd-symplectic})). Besides we add a new Giambelli-type
identity (\ref{thm:Giambelli-intermediate-symplectic})
for his intermediate symplectic characters.

There are several candidates for orthogonal tableaux. We show that
Proctor's \cite{Proctor:Young-tableaux} and King and Welsh's 
\cite{King-Welsh:Orthogonal} orthogonal tableaux are the
``right" tableaux for proving the orthogonal Jacobi--Trudi
identities. As in the symplectic case, a bijective proof of 
the orthogonal dual Jacobi--Trudi identity (see
(\ref{thm:e-orthogonal})) 
is easier found. It also employs the Gessel--Viennot method and some
kind of reflection argument. Our bijective proof of the ``ordinary" orthogonal
Jacobi--Trudi identity (see (\ref{thm:h-orthogonal})) 
uses once more Gessel and Viennot's idea of
dual lattice paths and relies on 
our bijection for the orthogonal
dual Jacobi--Trudi identity.
Unfortunately, we are not able to use
the same tableaux for a bijective proof of the orthogonal
Giambelli identity.
As a substitute, by using Sundaram's tableaux
\cite{Sundaram:Orthogonal} we give at least a bijective proof for the odd
orthogonal Giambelli identity. There are other determinant formulas which
can be proved by means of Sundaram's tableaux. They are
only slight variations of the odd orthogonal Jacobi--Trudi
identities. Since this work also led to explicit bijections between
the various odd orthogonal tableaux mentioned here
\cite{Fulmek-Krat:Bijections}, the only formula which we could not
prove bijectively is the even orthogonal Giambelli identity.

It is remarkable that, while in case of Schur functions the bijective
proofs for the ``ordinary" and the dual Jacobi--Trudi identities are
more or less identical, in the symplectic and orthogonal case the
situation is quite different. Here only the combinatorics of the {\em
dual\/} Jacobi--Trudi identities is transparent while the
combinatorics of the ``ordinary" Jacobi--Trudi identities is more
intricate. 

Our paper is organized in the following way. In the next section we
introduce some standard notation concerning partitions and lattice
paths. All relevant information about tableaux
and Jacobi--Trudi and Giambelli identities is listed in
section~\ref{Tableaux}. This section is intended to serve as a
reference. So it may be skipped first and used if it is
referred to in later sections.
Subsequently, in section~\ref{Lattice} we
briefly review the Gessel--Viennot proof of the Jacobi--Trudi
identity for Schur functions and Stembridge's proof of the Giambelli
identity for Schur functions. This section can be skipped by readers
who are familiar with the Gessel--Viennot business.
Our bijective proofs for the
(even) symplectic identities are the contents of section~\ref{symplectic}. 
Bijective proofs for Proctor's Jacobi--Trudi identities for odd and
intermediate symplectic characters and the new Giambelli identity for
intermediate symplectic characters are the topic of
section~\ref{oddsymplectic}.
Section~\ref{orthogonal} discusses the combinatorics of Proctor's 
and King and Welsh's orthogonal tableaux and how they can be used to
provide bijective 
proofs for the orthogonal Jacobi--Trudi identities. Finally, in
section~\ref{Sundaram} we
exhibit which determinant formulas result from Sundaram's odd
orthogonal tableaux, including the odd orthogonal Giambelli identity.
As a curious by-product of our methods 
we define ``even Sundaram tableaux" which
turn out to enumerate the dimensions of irreducible representations
of the even orthogonal groups.

\section{Notation}\label{Defs}
As usual, 
an $r$-tuple $\lambda = \parof{\lambda_1, \lambda_2,\dots,
\lambda_r}$ with $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_r> 0$
is called a {\em partition of length $r$\/}. We denote the length of
the partition $\lambda $ by $\ell(\lambda)$. The {\em Ferrers
board\/} of $\lambda$
is an array of cells with $\ell(\lambda)$ left-justified rows and $\lambda_i$
cells in row $i$. The {\em conjugate of\/} $\lambda$ is the partition 
$(\lambda^\prime_1, \dots, \lambda^\prime_{\lambda_1})$ where
$\lambda_j'$ is the length of the $j$-th column in the Ferrers board of
$\lambda$.
 
Let $\lambda=\parof{\lambda_1,\dots,\lambda_r}$ be a partition. Call
$s=\numof{\setof{i:\;\lambda_i\geq i}}$ the
{\em rank\/} of $\lambda$ ($s$ is the number of diagonal cells in the
Ferrers board of $\lambda$). Define $\alpha =
\parof{\alpha_1,\dots,\alpha_s}$ and $\beta = \parof{\beta_1,\dots,\beta_s}$
by $\alpha_i\defeq\lambda_i-i$, $\beta_j\defeq\lambda^\prime_j-j$. Clearly,
$\alpha,\beta$ provide another unique encoding of partitions $\lambda$.
It is called the {\em Frobenius notation\/} of $\lambda$, denote it by
$\parof{\alpha|\beta}$. A partition of rank 1 is called a {\em hook\/}.
Let $\Ferrer$ be the Ferrers board of a partition 
$\lambda$. We define the $k$-th
{\em principal hook\/} of $\Ferrer$ to be the set of cells
$\Ferrer_{i,j}$ such that $i=k\leq j$ or $j=k\leq i$. The parameters
$\alpha_i$ and $\beta_i$ can then be interpreted as the number of cells
in the $i$-th principal hook that are above or below the main diagonal,
respectively.
 
By {\em paths\/} we usually mean lattice paths in the plane integer
lattice $\Bbb Z^2$ consisting of unit horizontal and vertical steps
in the positive direction, unless we explicitly allow other
steps. Given points $u$ and $v$, we denote the set of all lattice
paths from $u$ to $v$ by $\fpath\of{u,v}$. If $\bold u=(u_1,\dots,u_m)$
and $\bold v=(v_1,\dots,v_m)$ are vectors of points, we denote the
set of all $m$-tuples $(\path_1,\dots,\path_m)$ of paths, where
$\path_i$ runs from $u_i$ to $v_i$, $i=1,\dots,m$, by
$\fpath\of{\bold u,\bold v}$. 
A set of paths is said to be {\em nonintersecting\/} if no
two paths of this set have a point in common, otherwise it is called
{\em intersecting\/}. If to each horizontal edge $a$ in $\Bbb Z^2$ a weight
$w(a)$ is assigned, the weight $w(\path)$ of a path $\path$ is defined
to be the product of the weights of all its horizontal steps. The
weight $w(\bold\path)$ of an $m$-tuple
$\bold\path=(\path_1,\dots,\path_m)$ of paths 
is defined to be the product $\prod
_{i=1} ^{m}w(\path_i)$ of the weights of all the paths in the
$m$-tuple.
Given any weight function $w$ defined on a set $\cal A$,
by the generating function $\GF{\cal A}$ we mean $\sum
_{x\in\cal A} ^{}w(x)$. 

If $\bold u=(u_1,u_2,\dots,u_m)$ is an $m$-tuple of points and $\si\in\SG{m}$ 
($\SG{m}$ is the permutation group on $\{1,2,\dots,m\}$) we
write $\bold u_\si$ for the permuted $m$-tuple
$(u_{\si(1)},u_{\si(2)},\dots,u_{\si(m)})$. Another multi-notation which will be
frequently used is the following. If some reflection $\reflect$ is
given and $\ep\in\{1,-1\}^m$, by $\bold u^{(\ep)}$ we mean the $m$-tuple
$(u_1^{(\ep)},u_2^{(\ep)},\dots,u_m^{(\ep)})$ where the $i$-th
component $u_i^{(\ep)}$ is ${u}_i$ if
$\epsilon_i=1$ and $\reflect\of{{u}_i}$ if $\ep_i=-1$. 


\section{Tableaux, Jacobi--Trudi identities, Giambelli
identities}\label{Tableaux}
Let $\lambda$ be a partition. A filling
of the cells of the Ferrers board of $\lambda$ with elements of
the set $\{1,2,\dots,n\}$
which is {\em nondecreasing\/} in rows and {\em strictly increasing\/} 
in the columns
is called an {\em $n$-semistandard (Young) tableau of shape $\lambda$\/}.
Frequently we shall say {\em tableau\/} for short.
For a tableau $\tableau$ the filling of the cell $(i,j)$ is denoted by
$\tableau_{ij}$.
  The following example shows such a semistandard tableau of shape
$(4,3,2)$. 
\begin{Figure}\label{FA}
\begin{picture}(15,5)(0,0)
\put(6,0.5){
	\begintableau{3}{4}
	\row{2346}
	\row{446}
	\row{56}
	\endtableau
}
\end{picture}
\end{Figure} 
Let $\bold{x} = \seqof{x_1, x_2, \dots,x_n}$ be
a sequence of $n$ variables. The weight $\bold x^\tableau$ 
of an $n$-semistandard tableau $\tableau$ is defined as
\begin{equation}\label{weight:Schur}
 \bold{x}^{\tableau}\defeq\prod
x_{\tableau_{i,j}} ,
\end{equation}
where the product is over all entries $\tableau_{ij}$ of $\tableau$.
The {\em Schur function\/} $\schf{n}{\la}$ associated to a partition
$\lambda$ is combinatorially defined by (see
\cite[I, (5.12)]{Macdonald:Symmetric-Functions},
\cite[Def.~4.4.1]{Sagan:The-Symmetric}, \cite[Def.~5.1]{Stanley:Plane})
\begin{equation}
 \schf{n}{\lambda} = \sum_{\tableau}\bold{x}^{\tableau}, 
\end{equation}
where the sum is over all $n$-semistandard
tableaux $\tableau$ of shape $\lambda$.
 
With these definitions, the ``classical'' Jacobi-Trudi
identity reads as follows:
Let $\lambda$ be a partition of length $r\le n$, $\bold{x}$ a set of variables.
The Schur function $\schf{n}{\lambda}$ can be expressed as the
following $r\times r$-determinant (see
\cite[I, (3.4)]{Macdonald:Symmetric-Functions}):
\begin{equation}\label{thm:h-Jacobi-Trudi}
\schf{n}{\lambda} =
	\detof{r}{{\hsf_{\lambda_j-j+i}}\of{\bold{x}}}\ ,
\end{equation}
where $\hsf_m\of{\bold x}=\sum _{1\le i_1\le \dots\le i_m\le n}
^{}x_{i_1}\cdots x_{i_m}$ denotes the {\em $m$-th complete homogeneous
symmetric function\/}.
For obvious reasons, we
call (\ref{thm:h-Jacobi-Trudi}) the ``$h$-formula".
 
The ``dual'' Jacobi-Trudi identity, the N\"agelsbach--Kostka formula (see
\cite[I, (3.5)]{Macdonald:Symmetric-Functions}),
looks quite similar (here
the conjugate partition $\lambda^\prime$ is involved):
\begin{equation}\label{thm:e-Jacobi-Trudi}
\schf{n}{\lambda} =
	\detof{\lambda_1}{{\esf_{\lambda^\prime_j-j+i}}\of{\bold{x}}}\
 ,
\end{equation}
where $\esf_m\of{\bold x}=\sum _{1\le i_1< \dots< i_m\le n}
^{}x_{i_1}\cdots x_{i_m}$ denotes the {\em $m$-th 
elementary symmetric function\/}.
Call this the ``$e$-formula".
 

With the Frobenius notation for partitions which was explained in
section~\ref{Defs}
the Giambelli identity reads (see \cite[p.~30, 
Ex.~9]{Macdonald:Symmetric-Functions})
\begin{equation}
\label{thm:Giambelli}
\schf{n}{\parof{\alpha|\beta}} = \detof{s}{\schf{n}{\parof{\alpha_i|\beta_j}}}.
\end{equation}

\bigskip
Now we turn to symplectic Schur functions (symplectic characters). 
Let again $\lambda = \parof{\lambda_1,\dots,\lambda_r}$ be a partition
of length $r\le n$. A 
tableau $\tableau$ of shape $\lambda$ is called a {\em $2n$-symplectic\/}
{\em tableau\/} if its entries are elements of $\{1,2,\dots,2n\}$
and if it obeys the additional constraint
\begin{equation}
\label{eq:symplectic-constraint}
\tableau_{i,j} \ge 2i-1.
\end{equation}
These tableaux were introduced by King and El-Sharkaway
\cite{King-El-Sharkaway:Standard}.
Figure~\ref{FB} shows a symplectic tableau of shape $(4,3,2)$.
\begin{Figure}\label{FB}
\begin{picture}(15,4)(0,0)
\put(6,0.5){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
	\endtableau
}
\end{picture}
\end{Figure}
For the symplectic case let 
$\bold{x} = \seqof{x_1, x_1^{-1},x_2,x_2^{-1},\dots,x_n,x_n^{-1}}$. 
Here the weight of a $2n$-symplectic tableau $\tableau$ is given by
\begin{equation}\label{weight:symplectic}
\bold{x}^\tableau = \prod_{l=1}^{n}x_l^{\numof{\setof{\tableau_{i,j} = 2l-1}} -
	\numof{\setof{\tableau_{i,j} = 2l}}}.
\end{equation}
So an odd entry $2i-1$ of $\tableau$ contributes $x_i$ to the weight
of $\tableau$, while an even entry $2i$ of $\tableau$ contributes 
$x_i^{-1}$ to the weight of $\tableau$.
The {\em symplectic Schur function\/} 
associated to $\lambda$ is combinatorially defined by (see 
\cite[Theorem~2.3]{Sundaram:Tableaux-in})
\begin{equation}
 \sschf{2n}{\lambda} =
	\sum_{\tableau}\bold{x}^{\tableau}, 
\end{equation}
where the sum is over all $2n$-symplectic tableaux $\tableau$ of shape
$\lambda$.
 
The Jacobi--Trudi-type identities for symplectic Schur functions read
as follows. The analogue of the ``$h$-formula"
(\ref{thm:h-Jacobi-Trudi}) is (see \cite[Prop.~24.22]{Fulton-Harris:Representation-Theory})
\begin{equation} \label{thm:h-symplectic}
\sschf{2n}{\lambda} =
\detof{r}{
	\firstcolumn{\hsf_{\lambda_j-j+1}\of{\bold{x}}}
	\hsf_{\lambda_j-j+i}\of{\bold{x}} +
		\hsf_{\lambda_j-j-i+2}\of{\bold{x}}}. 
\end{equation} 
The notation of the determinant means
that the first expression gives the entries of the first row and
the second the entries for the remaining rows, $i\ge2$.
 
The symplectic analogue of the ``$e$-formula"
(\ref{thm:e-Jacobi-Trudi}) is (see \cite[Cor.~24.24]{Fulton-Harris:Representation-Theory})
\begin{equation} \label{thm:e-symplectic}
\sschf{2n}{\lambda} =
	\detof{\lambda_1}{\esf_{\lambda_j^\prime-j+i}\of{\bold{x}} -
		\esf_{\lambda_j^\prime-j-i}\of{\bold{x}}}. 
\end{equation} 
Finally, with the Frobenius notation, which was explained in
section~\ref{Defs}, the symplectic analogue of the Giambelli identity
(\ref{thm:Giambelli}) reads (see \cite[(24.47)]{Fulton-Harris:Representation-Theory})
\begin{equation}
\label{thm:Giambelli-symplectic}
\sschf{2n}{\parof{\alpha|\beta}} = \detof{s}{\sschf{2n}{\parof{\alpha_i|\beta_j}}}.
\end{equation}

\smallskip
Next we consider Proctor's intermediate symplectic Schur functions
\cite{Proctor:Intermediate,Proctor:Odd}.
Let $n,m$ be nonnegative integers, and let $\bold
x=\seqof{x_1,x_1^{-1},x_2,x_2^{-1},\dots,x_n,x_n^{-1}}$ and $\bold
z=\seqof{z_1,z_2,\dots,z_m}$. In \cite{Proctor:Intermediate} Proctor
defined polynomials $\isschf \la \mu $ which interpolate between
Schur functions and (ordinary) symplectic characters. To
be more precise, for $n=0$ they are Schur functions, and for $m=0$
they are (ordinary) symplectic characters. These intermediate
symplectic Schur functions are indexed by {\em circled partitions\/},
which are denoted by a pair $(\la,\mu)$ of partitions. Here, $\la$ is
a partition of length at most $n+m$, in which some parts are circled.
Which parts of $\la$ are {\em not\/} circled is indicated by the
partition $\mu$ of length at most $n$. If there are both uncircled
and circled parts of $\la$ of the same size, then the circled parts
are listed after the uncircled parts of that size. Thus the notation
$(\la,\mu)$ uniquely determines which parts in $\la$ are circled. For
example, the circled partition which is denoted by $((5,4,4,3,1),(5,4,1))$ 
means $(5,4,\overset \circ 4,\overset \circ 3,1)$, 
where $^\circ$ denotes the circled parts. The intermediate symplectic
Schur functions $\isschf \la \mu$ are defined by means of
$(n,m)$-symplectic tableaux of shape $(\la,\mu)$. An
{\em $(n,m)$-symplectic tableau of shape\/} $(\la,\mu)$ is a tableau
$\tableau$ in the usual sense with entries from $\{1,2,\dots,2n+m\}$
which obeys the symplectic constraint (\ref{eq:symplectic-constraint})
for the first $n+1$ rows, i.e\@., for $i\le n+1$, and satisfies another
rather intricate constraint which involves the jeu de taquin. We
shall not give the general definition since we are only interested in
special cases where the second constraint is either superfluous or
very simple.

The first case which we are interested in is the case that $\mu$ consists
of the first $n$ parts of $\la$, i.e\@., that all parts $\la_i$ with
$i\le n$ are uncircled and all parts $\la_i$ with $i>n$ are circled.
Proctor denotes this special subpartition $\mu=(\la_1,\dots,\la_n)$
of $\la$ by $\la^{\text {trnc}}$ \cite[p.~678]{Proctor:Intermediate}.
For the shape $(\la,\la^{\text {trnc}})$ the second constraint is
superfluous so that an $(n,m)$-symplectic tableau of shape $(\la,\la^{\text
{trnc}})$ is a tableau in the usual sense with entries from
$\{1,2,\dots,2n+m\}$ such that (\ref{eq:symplectic-constraint}) holds
for $i\le n+1$. The left tableau in Figure~\ref{FU} is a
$(3,2)$-symplectic tableau of shape $((5,4,4,3,1),(5,4,4))$.
\begin{Figure}\label{FU}
\begin{picture}(21,6)(0,0)
\put(1,0.5){
	\begintableau{5}{5}
		\row{22445}
		\row{3456}
		\row{5777}
		\row{788}
		\row{8}
	\endtableau
}
\put(14,0.5){
	\begintableau{5}{5}
		\row{22449}
		\row{3456}
		\row{6777}
		\row{799}
		\row{9}
	\endtableau
}
\end{picture}
\end{Figure}

The weight of an $(n,m)$-symplectic tableau is defined by
\begin{equation}\label{weight:intermediate-symplectic}
(\bold{x},\bold z)^\tableau = \prod_{l=1}^{n}x_l^{\numof{\setof{\tableau_{i,j} = 2l-1}} -
	\numof{\setof{\tableau_{i,j} = 2l}}}
        \prod_{k=1}^{m}z_k^{\numof{\setof{\tableau_{i,j} = 2n+k}}}.
\end{equation}
The {\em $(n,m)$-symplectic Schur function\/} associated to  
$(\la,{\la^{\text {trnc}}})$ is combinatorially defined by 
\cite[p.~665]{Proctor:Intermediate}
\begin{equation}\isschf \la {\la^{\text {trnc}}}=
        \sum _{\tableau} ^{}(\bold x,\bold z)^\tableau,
\end{equation}
where the sum is over all $(n,m)$-symplectic tableaux of shape
$(\la,\la^{\text {trnc}})$.

If $\ell(\la)=r\le n+1$ there are Jacobi--Trudi-type identities for
$\isschf\la {\la^{\text {trnc}}}$. The analogue of the ``$h$-formula"
(\ref{thm:h-Jacobi-Trudi}) is \cite[p.~317]{Proctor:Odd}
\begin{equation}\label{thm:h-intermediate-symplectic}
\isschf\la {\la^{\text {trnc}}}=
\detof{r}{
	\firstcolumn{\hsf_{\lambda_j-j+1}\of{\bold{x},\bold z}}
	\hsf_{\lambda_j-j+i}\of{\bold{x},\bold z} +
		\hsf_{\lambda_j-j-i+2}\of{\bold{x},\bold z}}.
\end{equation}
Here, $h_m(\bold x,\bold z)$ means the $m$-th complete homogeneous symmetric
function in $x_1,x_1^{-1},\break\dots,x_n, x_n^{-1},z_1,\dots,z_m$.

We state Proctor's analogue \cite[Prop.~8.1]{Proctor:Intermediate} of 
the ``$e$-formula" (\ref{thm:e-Jacobi-Trudi}) in the following 
equivalent form,
\begin{equation} \label{thm:e-intermediate-symplectic}
\isschf{\lambda}{\la^{\text {trnc}}} =
        \detof{\lambda_1}{\esf_{\lambda_j^\prime-j+i}\of{\bold{x},\bold z} -
		\esf_{\lambda_j^\prime-j-i}\of{\bold{x},\bold z}}. 
\end{equation} 

If $\ell(\la)\le n+1$, 
we also find a Giambelli identity for $\isschf \la {\la^{\text
{trnc}}}$, which is new. 
Let the Frobenius notation of $\la$ be $(\alpha|\beta)$.
The intermediate symplectic analogue of the Giambelli identity
(\ref{thm:Giambelli}) reads 
\begin{equation}
\label{thm:Giambelli-intermediate-symplectic}
\isschf{\parof{\alpha|\beta}} {(\alpha|\beta)^{\text {trnc}}} =
\detof{s}{\isschf{\parof{\alpha_i|\beta_j}}
{(\alpha_i|\min\{\beta_j,n-1\})}}.
\end{equation}

The second case which we are interested in is the case that $m=1$ and $\mu$
consists only of one part of $\la$, $\la_d$ say. I.e\@., all
parts of $\la$ except one, namely the first part of $\la$ of size $\la_d$, are
circled.  In this case the above mentioned
second constraint translates into the following: An $(n,1)$-symplectic 
tableau of shape $(\la,\la_d)$ is a tableau in the
usual sense with entries from $\{1,2,\dots,2n+1\}$ such that 
(\ref{eq:symplectic-constraint}) holds
for $i\le n+1$ and such that the last entries in columns
$1,2,\dots,\la_d$ are equal to $2n+1$. The right tableau in Figure~\ref{FU} is a
$(4,1)$-symplectic tableau of shape 
$((5,4,4,3,1),3)$.

The weight of an $(n,1)$-symplectic tableau is defined by
\begin{equation}\label{weight:odd-symplectic}
(\bold{x}, z)^\tableau = z^{\numof{\setof{\tableau_{i,j} = 2n+1}}}
        \prod_{l=1}^{n}x_l^{\numof{\setof{\tableau_{i,j} = 2l-1}} -
	\numof{\setof{\tableau_{i,j} = 2l}}}.
\end{equation}
The {\em $(n,1)$-symplectic\/} {\em Schur function\/} associated to  
$(\la,{\la_d})$ is combinatorially defined by 
\cite[p.~665]{Proctor:Intermediate}
\begin{equation}\osschf \la {\la_d}=
        \sum _{\tableau} ^{}(\bold x, z)^\tableau,
\end{equation}
where the sum is over all $(n,1)$-symplectic tableaux of shape
$(\la,\la_d)$. The $(n,1)$-symplectic Schur functions are (partially
conjectural) characters for Proctor's odd symplectic groups
\cite{Proctor:Odd}.

Let $\ell(\la)=r$. Since $m=1$ we must have $r\le n+1$.
There is also a Jacobi--Trudi-type identity for
$\osschf\la {\la_d}$. It is an analogue of the ``$e$-formula"
(\ref{thm:e-Jacobi-Trudi}).
We state Proctor's formula \cite[Prop.~9.1]{Proctor:Intermediate} 
in the following equivalent form,
\begin{multline} \label{thm:e-odd-symplectic}
\osschf{\lambda}{\la_d} \\=
        \detof{\lambda_1}{z\esf_{\lambda_j^\prime-j+i-1}\of{\bold{x}} -
		z\esf_{\lambda_j^\prime-j-i-1}\of{\bold{x}}
\quad \vdots\quad 
                \esf_{\lambda_j^\prime-j+i}\of{\bold{x}, z} -
		\esf_{\lambda_j^\prime-j-i}\of{\bold{x}, z}}. 
\end{multline} 
The notation of the determinant has to be understood in the sense
that the first expression gives the entries of the columns $j$ for
$1\le j\le \la_d$ and
the second the entries for the remaining columns $j$, $j>\la_d$.
 
\smallskip
The definition of orthogonal tableaux is more complicated than that
of (ordinary) symplectic tableaux.
Actually there are several different candidates
\cite{King-El-Sharkaway:Standard,King-Welsh:Orthogonal,%
Koike-Terada:Young,Okada:A-Robinson,%
Proctor:Young-tableaux,Proctor:A-Schensted,% 
Sundaram:Orthogonal}. We shall use Proctor's 
\cite{Proctor:Young-tableaux,Proctor:A-Schensted}
and King and Welsh's \cite{King-Welsh:Orthogonal}
orthogonal tableaux in the first place.
These can be divided into two groups, let us call them {\em coarse
orthogonal tableaux\/} and {\em fine orthogonal tableaux\/}.
The number of coarse orthogonal tableaux of shape $\lambda$ 
equals the
dimension of the irreducible representation of the orthogonal group
indexed by $\lambda$. However, coarse orthogonal tableaux are not very well
suited for describing the irreducible characters. This task is better
performed by the {\em fine\/} orthogonal tableaux, which even appear
in 4 variations, to be more precise, 2 for the even case and 4 for the odd case.

Let us begin with the definition of coarse orthogonal tableaux. 
Let $\lambda=\parof{\lambda_1,\dots,\lambda_r}$ be a partition of
length $r$. 
A tableau is called {\em coarse orthogonal\/} if it satisfies the 
{\em $c$-th orthogonal condition\/} for all $c\in\Bbb N$. A tableau
$\tableau$ of shape $\lambda$ is said to satisfy the {$c$-th
orthogonal condition\/} if the number of entries $\leq
c$ in
$\tableau$'s first two columns does not exceed $c$, i.e\@.,
\begin{equation}
\label{cond:coarse}
\numof{\setof{\tableau_{i,j} \leq c:\; 1\le j\le 2;\ 1\leq i\leq\lambda^\prime_j}}\leq
c.
\end{equation}
 
The following tableau satisfies the $1$-st, $3$-rd and $5$-th 
orthogonal condition, but not the $2$-nd and $4$-th.
\begin{Figure}\label{FC}
\begin{picture}(15,4)(0,0)
\put(6,0.5){
	\begintableau{3}{5}
		\row{12244}
		\row{245}
		\row{4}
	\endtableau
}
\end{picture}
\end{Figure}


To define fine orthogonal tableaux, we need to introduce two further
conditions, the {\em $c$-th protection condition\/} and
the {\em $c$-th broken strip condition\/}. A tableau is said to satisfy
the {\em $c$-th protection condition\/} if 
whenever $i+j=c$ and $\tableau_{i,1} = c-1$ and
$\tableau_{j,2}=\tableau_{j,3}=\dots=\tableau_{j,h-1} = c-1$ and
$\tableau_{j,h}=c$, we have $\tableau_{j-1, h} = c-1$. 
Stated in a more formally manner, this is
\begin{align}
i+j=c &\land \tableau_{i,1}=c-1 \notag\\
		&\land
\tableau_{j,2}=\dots=\tableau_{j,h-1} = c-1 \hbox{ (for $h\geq2$)} \notag\\
	&\land
\tableau_{j,h} = c \notag\\
	\limp&
\tableau_{j-1,h} = c-1.\label{cond:fine}
\end{align}
We say that the entry $\tableau_{j,h}=c$ is {\em protected above by
$c-1$\/}.

 
The following tableau satisfies the $6$-th protection
condition if and only if $5$ is inserted instead of $*$. (Choose $i=4$
and $j=2$.)
\begin{Figure}\label{FD}
\begin{picture}(15,5)(0,0)
\put(6,0.5){
	\begintableau{4}{5}
		\row{133*6}
		\row{2556}
		\row{378}
		\row{5}
	\endtableau
}
\end{picture}
\end{Figure}

A tableau is said to satisfy the {\em $c$-th broken strip condition\/}
if whenever $i+j=c$, $\tableau_{i,1}=c$ and $\tableau_{i-1,1}\ne c-1$
then we have $\tableau_{j,2}\ne c-1$.

Now, fine orthogonal tableaux are defined by mixing the orthogonal,
protection, and broken strip conditions appropriately. We call
a tableau {\em $N$-orthogonal of the first kind\/}, if its entries are
elements of $\{1,2,\dots,N\}$, and if it satisfies the
$2c$-th orthogonal condition for all $c$, the $N$-th orthogonal
condition (for even $N$ this is already included in the first
requirement, it is a new requirement only for odd $N$), and the
$2c$-th protection condition for all $c$. The tableau in
Figure~\ref{FD} is a fine orthogonal tableau of the first kind if $5$
is inserted instead of $*$.
We call
a tableau {\em $N$-orthogonal of the second kind\/}, if its entries are
elements of $\{1,2,\dots,N\}$, and if it satisfies the
$2c$-th orthogonal condition for all $c$, the $N$-th orthogonal
condition, the
$2c$-th broken strip condition for all $c$, and if
$\tableau_{c,1}=2c-1$
then all entries $2c$ in the $c$-th row have to be
protected above by $2c-1$. (The last requirement is the $2c$-th
protection condition
for the special case $i=j=c$.)
Next we define the weight for these tableaux. If $N$ is even,
$N=2n$, let 
$\bold{x} = \seqof{x_1, x_1^{-1},\dots x_n,x_n^{-1}}$. 
If $N$ is odd, $N=2n+1$, let
$\bold{x} = \seqof{x_1, x_1^{-1},\dots x_n,x_n^{-1},1}$. 
As in the symplectic case,
the weight of an $N$-orthogonal tableau $\tableau$ of the first or
second kind is given by
\begin{equation}\label{weight:orthogonal}
\bold{x}^\tableau = \prod_{l=1}^{n}x_l^{\numof{\setof{\tableau_{i,j} = 2l-1}} -
	\numof{\setof{\tableau_{i,j} = 2l}}}.
\end{equation}
Note that in the odd case $N=2n+1$ the entries $2n+1$ do not contribute
anything to the weight of the tableau.

Now let $N$ be odd, $N=2n+1$. We call
a tableau {\em $(2n+1)$-orthogonal of the third kind\/}, if its entries are
elements of $\{1,2,\dots,2n+1\}$, and if it satisfies the
$(2c+1)$-st orthogonal condition for all $c$, 
and the $(2c+1)$-st protection condition for all $c$. 
We call
a tableau {\em $(2n+1)$-orthogonal of the fourth kind\/}, if its entries are
elements of $\{1,2,\dots,2n+1\}$, and if it satisfies the
$(2c+1)$-st orthogonal condition for all $c$, 
and the $(2c+1)$-st broken strip condition for all $c$.

With $\bold{x} = \seqof{x_1, x_1^{-1},\dots x_n,x_n^{-1},1}$, 
the weight of an $N$-orthogonal tableau $\tableau$ of the third or
fourth kind is given by
\begin{equation}\label{weight:orthogonalodd}
\bold{x}^\tableau = \prod_{l=1}^{n}x_l^{\numof{\setof{\tableau_{i,j} = 2l}} -
	\numof{\setof{\tableau_{i,j} = 2l+1}}}.
\end{equation}
Note that here entries $1$ do not contribute anything to the weight,
entries $2$ contribute $x_1$, entries $3$ contribute $x_1^{-1}$, etc.

All the orthogonal tableaux just described are due to Proctor except
for the fine orthogonal tableaux of the second kind, which are due to
King and Welsh.

Both the coarse orthogonal tableaux with entries being at most $N$
and the various fine $N$-orthogonal tableaux satisfy the $N$-th orthogonal
condition. This implies that the shape $\la$ for
such tableaux has at most $N$ cells in the first two columns. We call
a partition with at most $N$ cells in the first two columns an {\em
$N$-orthogonal partition\/}. In particular, the length of an
$N$-orthogonal partition is at most $N$.


The {\em orthogonal Schur function\/} 
associated to an $N$-orthogonal partition
$\lambda$ is combinatorially defined by (see
\cite[Theorems~6.1--6.3]{Proctor:Young-tableaux} or
\cite[p.~35]{Proctor:A-Schensted})
\begin{equation}
\oschf{N}{\lambda} =
\sum_{\tableau}\bold{x}^\tableau, 
\end{equation}
where the sum is over all $N$-orthogonal tableaux $\tableau$ of the
$\kappa$-th kind which are of shape $\lambda$, where $\kappa$ is any
(fixed) number out of $\{1,2,3,4\}$.
 
As is well-known,
when we replace each $x_i$ by 1 in $\oschf{N}\lambda$ we obtain
the dimension of the corresponding irreducible representation. 
As mentioned above, this dimension is equinumerous with 
the coarse orthogonal tableaux of shape $\lambda$. Therefore, we may
write
\begin{equation}\label{thm:equinumerous}
\oschftriv{N}{\lambda} =
\numof{\setof{\tableau:\;\tableau\hbox{ coarse orthogonal of shape
$\lambda$, }\tableau_{i,j}\leq
N}}.
\end{equation}
  
The Jacobi--Trudi-type identities for orthogonal Schur functions read
as follows. The analogue of the $h$-formula (\ref{thm:h-Jacobi-Trudi}) is
(see \cite[Ex.~24.46]{Fulton-Harris:Representation-Theory})
\begin{equation}
\label{thm:h-orthogonal}
\oschf{N}{\lambda} =
\detof{r}{
	\hsf_{\lambda_j-j+i}\of{\bold{x}} -
		\hsf_{\lambda_j-j-i}\of{\bold{x}}}.
\end{equation}
 
The analogue of the ``$e$-formula"
(\ref{thm:e-Jacobi-Trudi}) is
(see \cite[Cor.~24.35]{Fulton-Harris:Representation-Theory})
\begin{equation}\label{thm:e-orthogonal}
\oschf{N}{\lambda} = 
\detof{\lambda_1}{
	\firstrow{\esf_{\lambda^\prime_j-j+1}\of{\bold{x}}}
	\esf_{\lambda_j^\prime-j+i}\of{\bold{x}} +
		\esf_{\lambda_j^\prime-j-i+2}\of{\bold{x}}}. 
\end{equation}
As in (\ref{thm:h-symplectic}),
the notation of the determinant means
that the first expression gives the entries of the first row and
the second the entries for the remaining rows, $i\ge2$.
Recall that for both (\ref{thm:h-orthogonal}) and (\ref{thm:e-orthogonal})
the sequence $\bold x$ is different for $N$ even or odd.
 

Finally, with the Frobenius notation, which was explained in
section~\ref{Defs}, the orthogonal analogue of the Giambelli identity
(\ref{thm:Giambelli}) reads (see \cite[(24.47)]{Fulton-Harris:Representation-Theory})
\begin{equation}
\label{thm:Giambelli-orthogonal}
\oschf{N}{\parof{\alpha|\beta}} = \detof{s}{\oschf{N}{\parof{\alpha_i|\beta_j}}}.
\end{equation}
\smallskip

For the case $N=2n+1$, Sundaram
\cite{Sundaram:Orthogonal} defined different orthogonal tableaux.
These {\em $(2n+1)$-Sundaram tableaux\/} are
tableaux with entries from the alphabet
$1<2<\dots<2n<\infty$ in the usual sense,
except that column-strictness
does not extend to symbol $\infty$ (i.e., symbol
$\infty$ may occur {\em more than once in a column\/}), 
which have at most one entry $\infty$ in each {\em row\/},
and which obey the
symplectic constraint (\ref{eq:symplectic-constraint}). 
The weight of such a tableau is defined by (\ref{weight:orthogonal}), 
entries $\infty$ do not contribute to the weight. Then, if $\la$ is a
partition with $\ell(\la)\le n+1$, the odd
orthogonal Schur functions $\oschf{2n+1}{\la}$ may be also written as
\cite[Def.~2.3]{Sundaram:Orthogonal})
\begin{equation}
\oschf{2n+1}{\lambda} =
\sum_{\tableau}\bold{x}^\tableau,
\end{equation}
where the sum is over all Sundaram tableaux $\tableau$ of shape $\lambda$.


\medskip
We should remark that different symplectic and orthogonal tableaux,
which we do not consider at all
in this paper, have been introduced by DeConcini, Procesi, Lakshmibai, Musili, Seshadri
\cite{DeConcini:symplectic,DeConcini-Procesi,Lakshmibai,%
Lak-Mus-Sesh,Seshadri}. These tableaux are particularly useful for
formulating a Littlewood--Richardson rule for all classical groups
\cite{Littelmann}. The symplectic tableaux of these authors and the
symplectic tableaux of King and El-Sharkaway, which are considered in
this paper, have recently been related by Sheats \cite{Sheats:Jeu}.



\section{Lattice path proofs for Schur functions}\label{Lattice}
In this section we review the Gessel--Viennot method 
\cite{Gessel-Viennot:Determinants-paths} (see 
\cite[ch.~4]{Sagan:The-Symmetric}) of proving the
Jacobi--Trudi identities (\ref{thm:h-Jacobi-Trudi}) and
(\ref{thm:e-Jacobi-Trudi})
for Schur functions and Stembridge's lattice path proof 
\cite[sec.~9]{Stembridge:Nonintersecting-paths} for the
Giambelli identity for Schur functions. (A different combinatorial
proof for the Giambelli identity has been given by E\u gecio\u glu and 
Remmel \cite{Egecioglu-Remmel:A-combinatorial}).
 
The idea of Gessel and Viennot is to convert tableaux into
nonintersecting lattice paths. Subsequently, from the nonintersecting
lattice paths the determinants (\ref{thm:h-Jacobi-Trudi}) and
(\ref{thm:e-Jacobi-Trudi})
can be easily deduced.

For proving the $h$-formula (\ref{thm:h-Jacobi-Trudi}), 
given an $n$-semistandard tableau $\tableau$ of shape
$\lambda=(\lambda_1,\dots,\lambda_r)$, 
$\lambda$ being a partition of length $r\le n$, 
we map $\tableau$ to an $r$-tuple
$(\path_1,\dots,\path_r)$ of nonintersecting lattice paths by
associating to row $i$ of $\tableau$ the path $\path_i$ from $u_i=(-i,1)$ to
$v_i=(\lambda_i-i,n)$, $i=1,2,\dots,r$, such that the entries in the $i$-th 
row are the heights of
horizontal steps in the path $\path_i$. That this $r$-tuple of paths
is indeed nonintersecting comes from the column-strictness of
$\tableau$.
Figure~\ref{FE} shows the triple of paths associated to a
$6$-semistandard 
tableau $\tableau$ of shape $\la=\parof{4,3,2}$.


%%%@@@ input from figfe
\begin{Figure}\label{FE}
\begin{picture}(20,10)(0,0)
\put(12,5){
\begintableau{3}{4}
\row{2356}
\row{346}
\row{45}
\put(-2,1.5){$T$ = }
\endtableau
}
\put(0,1){
\begingrid{-3}{0}{3}{8}
\makeaxes
\makexscale

\elabelfalse
\labelledtrue

% Die Pfade:
\uppath{-1}{1}{|-|-||-|-}
\uppath{-2}{1}{||-|-||-}
\uppath{-3}{1}{|||-|-|}
\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfe
 
Conversely, given a set of nonintersecting lattice paths
$(\path_1,\dots,\path_r)$ where $\path_i$ runs from $u_i=(-i,1)$ to
$v_i=(\lambda_i-i,n)$, $i=1,2,\dots,r$, 
we can reconstruct the tableau by labelling each
horizontal step of the paths by its height and then read off the
$i$-th row of the tableau from the labels of the $i$-th path.
Following \cite{Sagan:The-Symmetric} we call
this labelling of horizontal steps by their height the 
{\em $h$-labelling\/} (see Figure~\ref{FF}).
 

%%%@@@ input from figff
\begin{Figure}\label{FF}
\begin{picture}(20,8)(0,0)
\put(2,1){
	\begingrid{0}{0}{5}{5}
		\makeaxes
		\makexscale
		\makeyscale
		\gridcaption{$h$-labelling:}
		\def\printhlabels#1{
			\begingroup
			\global\dcnta=0
			\loop\ifnum\dcnta<5
				\hlabel{\dcnta}{#1}{#1}
				\global\advance\dcnta by 1
      		\put(\dcnta,#1){\circle*{0.2}}
			\repeat
			\endgroup
}
		\begingroup
   	\global\dcntb=1
		\loop\ifnum\dcntb<6
			\put(0,\dcntb){\line(1,0){5}}
			\printhlabels{\number\dcntb}
   	   \global\advance\dcntb by 1
   	\repeat
   	\endgroup
	\endgrid
}
 
\put(9.2,1.8){level 1}
\put(9.2,2.8){level 2}
\put(9.2,3.8){level 3}
\put(9.2,4.8){level 4}
\put(9.2,5.8){level 5}
 
\put(12,1){
	\begingrid{-1}{0}{4}{5}
	\makeaxes
	\makexscale
	\makeyscale
	\gridcaption{$e$-labelling:}
		\def\printelabels#1{
			\begingroup
			\dcnta=0
			\dcntc=#1
			\loop\ifnum\dcnta<4
				\advance\dcntc by 1
      		\hlabel{\dcnta}{#1}{\number\dcntc}
				\advance\dcnta by 1
				\put(\dcnta,#1){\circle*{0.2}}
			\repeat
			\endgroup
}
		\begingroup
   	\global\dcntb=0
		\loop\ifnum\dcntb<6
			\put(0,\dcntb){\line(1,0){4}}
			\printelabels{\number\dcntb}
   	   \global\advance\dcntb by 1
   	\repeat
   	\endgroup
		\downskewline{-1}{1}{1}
		\downskewline{-1}{2}{2}
		\downskewline{-1}{3}{3}
		\downskewline{-1}{4}{4}
		\downskewline{-1}{5}{5}
		\downskewline{0}{5}{4}
		\downskewline{1}{5}{3}
		\downskewline{2}{5}{2}
		\downskewline{3}{5}{1}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figff

For proving the $e$-formula (\ref{thm:e-Jacobi-Trudi}), we have to
introduce yet another labelling. Now we label each horizontal step
from $(x-1,y)$ to $(x,y)$ by $x+y$. Call this labelling the
{\em $e$-labelling\/} (see Figure~\ref{FF}).
Given an $n$-semistandard tableau $\tableau$ of shape $\lambda$,
we map $\tableau$ to a $\lambda_1$-tuple
$(\path_1,\dots,\path_{\lambda_1})$ of nonintersecting lattice paths by
associating to column $i$ of $\tableau$ a path $\path_i$ from
$u_i=(-i+1,i-1)$ to
$v_i=(\lambda_i'-i+1,n-\lambda_i'+i-1)$, $i=1,2,\dots,\la_1$,
such that the entries of the $i$-th
column can be read off from the labels of the horizontal steps of
$\path_i$. That the paths are nonintersecting in this case comes from
the fact that entries are nondecreasing along rows of $\tableau$.
Figure~\ref{FG} shows the quadruple of paths associated to a
$6$-semistandard tableau $\tableau$ of shape $\la=\parof{4,3,2}$.


%%%@@@ input from figfg
\begin{Figure}\label{FG}
\begin{picture}(20,12)(0,0)
\put(13,5){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
		\put(-1.9,1.5){$T$ =}
	\endtableau
}
\put(1,1){
	\begingrid{-4}{0}{5}{10}
		\makeaxes
		\makexscale
		\downskewline{-4}{4}{4}
		\downskewline{-4}{5}{5}
		\downskewline{-4}{6}{6}
		\downskewline{-4}{7}{7}
		\downskewline{-4}{8}{8}
		\downskewline{-4}{9}{9}
		\downskewline{-4}{10}{9}
		% Pfade
		\elabeltrue
		\labelledtrue
		\uppath{0}{0}{|-|--|}
		\uppath{-1}{1}{||--|-}
		\uppath{-2}{2}{|||-|-}
		\uppath{-3}{3}{|||||-}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfg
 
Let $l$ be any of these two labellings. We define the {\em weight\/}
$w_l(\horedge)$ of a
horizontal step $\horedge$ to be $w_l(\horedge)\defeq x_{l(\horedge)}$. 
This defines a weight for paths and $m$-tuples of paths (see 
section~\ref{Defs}).
Obviously, both of the above mappings from
tableaux to nonintersecting lattice paths are weight-preserving.
  
The proof for both the $h$- and the $e$-formula can now be given in
a uniform manner. In both cases we are interested in finding the
generating function (using the $h$-labelling in the first case, 
respectively the $e$-labelling in the second case) for $m$-tuples
$(\path_1,\dots,\path_m)$ of 
nonintersecting lattice paths, where $\path_i$ runs from $u_i$ to
$v_i$, $i=1,2,\dots,m$,
($m=r$ or $m=\lambda_1$, respectively, $u_i$ and $v_i$ are
defined in each of the two cases as above.)
We claim that these
generating functions can be written as the determinant
\begin{equation}\label{detGF}
\detof{m}{\GF{\fpath\of{u_i,v_j}}}.
\end{equation}
 
To see this, with the notation of section~\ref{Defs} we
expand the determinant in (\ref{detGF}) into
\begin{align}\notag
\detof{m}{\GF{\fpath\of{u_i,v_j}}} &=
\sum_{\sigma\in\SG{m}}\sgn\of{\sigma}\prod _{i=1} ^{m}
\GF{\fpath\of{u_i,v_{\sigma(i)}}}\\
\notag
&=\sum_{\sigma\in\SG{m},\
\bold\path\in\fpath\of{\bold{u},\bold{v}_{\sigma}}}\sgn\of{\sigma}
w_l(\bold \path).
\end{align}
Thus the determinant in (\ref{detGF}) is expressed as a generating function 
for $m$-tuples\linebreak
$(\path_1,\dots,\path_m)$ of paths where $\path_i$ runs from 
$u_i$ to $v_{\sigma(i)}$, $i=1,2,\dots,m$, for some permutation $\sigma$.
We shall find a weight-preserving but sign-reversing
involution on the set of all those $m$-tuples of paths that are 
{\em intersecting\/}. Given such an involution,
after cancelling all such $m$-tuples of paths by that involution, only the
nonintersecting $m$-tuples will survive, as was claimed. And, because
of the location of the starting and end points of the paths, the
associated permutation for such $m$-tuples must be the identity. As we saw
above, these $m$-tuples of paths exactly correspond to
$n$-semistandard tableaux of shape $\la$.
It is easy to see that
$\GF{\fpath\of{u_i,v_j}}$ evaluates to $\hsf_{\lambda_j-j+i}\of{\bold{x}}$
in the case of the $h$-labelling, and to $\esf_{\lambda_j'-j+i}\of{\bold{x}}$
in the case of the $e$-labelling. This would prove the assertions.
 
Consider an $m$-tuple $(\path_1,\dots,\path_m)\in\fpath\of{\bold
u,\bold v_\sigma}$ of
paths that is intersecting.
In the lexicographic order of the integer lattice
look for the 
largest point of intersection of this $m$-tuple, $p$ say.
Let $\path_i,\path_j$ be the paths having point $p$
in common.
Now interchange terminal portions %
(starting at $p$) of
$\path_i$ and $\path_j$, i.e\@.,
replace $\path_i$ by
$$\path_i^\prime\defeq\brkof{
\hbox{subpath of }\path_i\hbox{ from }u_i\hbox{ to }p
\hbox{ joined with }
\hbox{subpath of }\path_j\hbox{ from }p\hbox{ to }v_{\si(j)}}$$
and $\path_j$ by
$$\path_j^\prime\defeq\brkof{
\hbox{subpath of }\path_j\hbox{ from }u_j\hbox{ to }p
\hbox{ joined with }
\hbox{subpath of }\path_i\hbox{ from }p\hbox{ to }v_{\si(i)}}.$$
Clearly, this is an involution that preserves weight
and reverses sign (since it corresponds to multiplying the initial permutation
$\sigma$ with the transposition $\parof{i,j}$). We call this involution
the {\em Gessel--Viennot involution\/}.
\QED
\bigskip

Now we turn to Stembridge's \cite[sec.~9]{Stembridge:Nonintersecting-paths}
bijective proof of the Giambelli identity (\ref{thm:Giambelli}).
We present Stembridge's construction in a slightly modified fashion.
Again, we consider
paths in the integer lattice consisting of unit horizontal and
vertical steps. While the direction of horizontal steps is always
positive, the direction of a vertical step, however, is
only {\em positive\/} if it is {\em strictly to the right\/} of the
$y$-axis, and is
{\em negative\/} if it does {\em not\/} lie strictly to the right of the
$y$-axis. As we shall see, this change of direction corresponds to change of 
increasing/decreasing behaviour when reading the entries of a tableau's
principal hooks.
 
To the left of the horizontal axis we consider a
``shifted'' $h$-labelling by assigning label $i+1$ to an edge at height $i$,
while we consider the ``usual'' $e$-labelling to the right (recall
Figure~\ref{FF}). As
before, the weight of a horizontal edge with label $l$ is defined to
be $x_l$.
Given an $n$-semistandard tableau of shape $\parof{\alpha|\beta}$,
we associate to its
$i$-th principal hook a path $\path_i$ from $\parof{-\alpha_i,n-1}$ to
$\parof{\beta_i+1,n-\beta_i-1}$ by interpreting the entries of the hook
(read from ``right to bottom'') as labels of the corresponding
steps of $\path_i$.
 
Figure~\ref{FH} shows the pair of nonintersecting lattice paths
associated to a $6$-semistandard
tableau $\tableau$ of shape $\lambda=\parof{4,3,2}$ (rank $2$,
Frobenius notation $\parof{\alpha|\beta} = \parof{3,1|2,1}$).


%%%@@@ input from figfh
\begin{Figure} \label{FH}
\begin{picture}(20,10)(0,0)
\put(12,5){$T$ =}
\put(14,3){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
	\endtableau
}
\put(1,1){
	\begingrid{-4}{0}{4}{6}
		\makexaxis
		\makexscale
		\makeyaxis
		\downskewline{0}{2}{2}
		\downskewline{0}{3}{3}
		\downskewline{0}{4}{4}
		\downskewline{0}{5}{4}
		\downskewline{0}{6}{4}
		% Pfade
		\elabelfalse
		\labelledtrue
		\downpath{-3}{5}{-||-|-|*-|--|}{6}
		\downpath{-1}{5}{-||*-|-}{6}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfh

The construction might look complicated at first sight, but note that the
right half of the picture is simply the same as in the case
of associated ``$e$-paths", while the left half
corresponds precisely to associated ``$h$-paths", shifted one unit down
and  reflected in some vertical line. Hence nonintersecting
$s$-tuples (here, $s$ is the rank of the partition $\la=(\alpha\vert\beta)$) 
of such paths bijectively
correspond to tableaux. This is clear
by the above observation for both half-planes and by the fact that
steps contained in the vertical axis are downwards directed. (The picture's
left and right half correspond to upper and lower part of the partition's
Ferrers board cut in two just above the diagonal. Nonintersecting paths in
both halves imply nondecreasing rows and strictly increasing columns
for both of these
parts; downward directed vertical steps on the $y$-axis imply
nondecreasing entries along the rows when glued together, 
from which column-strictness follows immediately.)
 
In particular, 
this means that for a hook $(a\vert b)$ (a partition with rank $1$)
the corresponding
Schur function $\schf{n}{(a\vert b)}$ is just the generating function of
the family of lattice paths starting in $\parof{-a,n-1}$
and ending in $\parof{b+1,n-b-1}$. Hence application of
the Gessel--Viennot method immediately leads
to the Giambelli identity (\ref{thm:Giambelli}).
\QED
 
 

\section{Symplectic identities: The tableaux of King and
El-Sharkaway}\label{symplectic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%      Der symplektische Fall mit den tableaux von King und El-Sharkaway      %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\message{the symplectic case...} 
In this section we give bijective proofs for (\ref{thm:h-symplectic}),
(\ref{thm:e-symplectic}), and (\ref{thm:Giambelli-symplectic}). 
We start with the proof of the ``$e$-formula"
(\ref{thm:e-symplectic}).
 
\bigskip
{\sc Bijective proof of the symplectic dual Jacobi--Trudi identity
(\ref{thm:e-symplectic})}. 
Let $\tableau$ be a $2n$-symplectic tableau of shape $\lambda$, where $\la$
is a partition of length $r\le n$. In the
same way as in section~\ref{Lattice} (see Figure~\ref{FG}), with $\tableau$ we associate a
$\lambda_1$-tuple $(\path_1,\dots,\path_{\lambda_1})$ of
nonintersecting lattice paths where $\path_i$ runs from $u_i=(-i+1,i-1)$
to $v_i=(\lambda_i'-i+1,2n-\lambda'_i+i-1)$, $i=1,2,\dots,\lambda_1$, 
by reading $\path_i$ off the $i$-th column. Obviously, $\tableau$
obeys the symplectic constraint if and only if the first of the associated
paths (and hence all of the associated paths) does not cross the line
$y=x-1$. Figure~\ref{FI} gives an example for $n=3$ and $\lambda=(4,3,2)$.


%%%@@@ input from figfi
\begin{Figure}\label{FI}
\begin{picture}(20,10)(0,0)
\put(16,3){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
		\put(-2,1.5){$T$ =}
	\endtableau
}
\put(1,1){
	\begingrid{-4}{0}{5}{8}
		\makeaxes
		\makexscale
		% Pfade
		\elabeltrue
		\labelledtrue
		\uppath{0}{0}{|-|--|}
		\uppath{-1}{1}{||--|-}
		\uppath{-2}{2}{|||-|-}
		\uppath{-3}{3}{|||||-}
		% levels auszeichnen
		\downskewline{-4}{4}{4}
		% Gerade y=x-1
      \thinlines
		\put(1,0){\line(1,1){5}}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfi

If we define the weight $w(a)$ of a
horizontal edge $a$ with $e$-label (cf\@. Figure~\ref{FF}) 
$2i-1$ to be $x_i$, and the weight $w(a)$ of
a horizontal edge $a$ with $e$-label $2i$ to be $x_i^{-1}$ then
the correspondence depicted in Figure~\ref{FI} is
weight-preserving with respect to the weight (\ref{weight:symplectic}). 
Therefore the
left side of (\ref{thm:e-symplectic}) can be interpreted as generating
function for all $\lambda_1$-tuples $(\path_1,\dots,\path_{\lambda_1})$ of
nonintersecting lattice paths, where $\path_i$ runs from $(-i+1,i-1)$
to $(\lambda_i'-i+1,2n-\lambda'_i+i-1)$, $i=1,2,\dots,\lambda_1$, 
and does not cross $y=x-1$.

Our bijective proof combines the Gessel--Viennot involution with a
modification of the reflection-principle (cf\@. e.g\@. 
 \cite[{p.~22}]{Comtet:Advanced-Combinatorics}). This idea originates
from \cite{Krat:Major}. 
 
First we find a lattice path interpretation of the determinant at the 
right side of (\ref{thm:e-symplectic}) by expanding it. With 
$\reflect$ denoting the reflection in the line $y=x-2$, we have
\begin{align}\label{eq:e-symplectic-expansion}
 \detof{\lambda_1}{\esf_{\lambda_j^\prime-j+i}\of{\bold{x}} -
		\esf_{\lambda_j^\prime-j-i}\of{\bold{x}}}&=
\detof{\lambda_1}{\GF{\fpath\of{u_i,v_j}}-\GF{\fpath\of{\reflect(u_i),v_j}}}\\
&\notag  \hskip-5cm=	{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1,-1\}^{\lambda_1}}} 
	\sgn\of{\sigma}\prod _{i=1} ^{\la_1}\ep_i
\prod_{i,\epsilon_i=1}\GF{\fpath\of{u_i,v_{\si(i)}}}\cdot
\prod_{i,\epsilon_i=-1}\GF{\fpath\of{\reflect(u_i),v_{\si(i)}}}\\
\notag
&\hskip-5cm=	\underset{\bold \path\in
        \fpath\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                        \epsilon\in\{1,-1\}^{\lambda_1}}}
	\sgn\of{\sigma}
	\sgn\of{\epsilon}
	w(\bold \path),
\end{align}
where $\sgn\of\epsilon$ stands for $\prod _{i=1} ^{\lambda_1}\epsilon_i$.
The notation $\bold u^{(\epsilon)}$ is explained in section~\ref{Defs}.
The weight $w(\bold \path)$ is induced
(as described in section~\ref{Defs}) by the edge weight that was introduced 
just before.

By (\ref{eq:e-symplectic-expansion}) we have written the determinant in
(\ref{thm:e-symplectic}) as a generating function for $\lambda_1$-tuples
$(\path_1,\dots,\path_{\lambda_1})$ of lattice paths where $\path_i$ runs
from either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, $i=1,2,\dots,\lambda_1$.
In order to establish (\ref{thm:e-symplectic}),
we shall give an involution that cancels all those $\lambda_1$-tuples of
paths that are either intersecting or contain a path that crosses the
line $y=x-1$.

%Christian
%%%@@@ input from figfj
\begin{Figure}\label{FJ}
\begin{picture}(20,10)(0,0)
\put(1,1){
	\begingrid{-1}{-3}{8}{5}
		\makeaxes
		\makexscale
      %Letzter Schnittpunkt
		\thickcircle{6}{4}
		\put(6.5,3.7){$p$}
		% gerade Punkte und ihre Spiegelbilder
		\gridpoint{0}{0}		\gridcircle{2}{-2}
		\gridpoint{0}{2}		\gridcircle{4}{-2}
		\gridpoint{2}{2}		\gridcircle{4}{0}
		\gridpoint{3}{3}		\gridcircle{5}{1}
		\gridpoint{4}{4}		\gridcircle{6}{2}
		\gridpoint{6}{4}
		\downskewline{0}{0}{2}
		\downskewline{0}{2}{4}
		\downskewline{2}{2}{2}
		\downskewline{3}{3}{2}
		\downskewline{4}{4}{2}
		% Gerade y=x-2
      \thinlines
		\put(-1,-3){\line(1,1){8}}
      % Ungespiegelter Pfad
		\labelledfalse
		\uppath{0}{0}{||--|--|--||}
		% Gespiegelter Pfad
		\updottedpath{2}{-2}{--|||--|||}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfj

%Christian
First, we define the announced modification of the
reflection-principle. The reflection-principle itself could not be
used for our purposes 
since reflection of paths is not weight-preserving with respect
to the weight of paths that we are considering.
A path $\path$ crossing the line $y=x-1$ must meet the line $y=x-2$. Let
$p$ be the
last meeting point. To $\path$'s initial portion up to $p$
we may apply the following modified reflection in $y=x-2$.
(See Figure~\ref{FJ}.)
All points $(x,y)\in \path$ with $x+y\equiv 0 \pmod 2$ (call them {\em even
points\/}) are reflected in the usual way, i.e\@., $(x,y)\mapsto(y+2,x-2)$
(see the points $(0,0)$, $(0,2)$, $(2,2)$, $(3,3)$, $(4,4)$, $(6,4)$
in Figure~\ref{FJ}),
and so are all {\em odd\/} points ($x+y\equiv1\pmod2$) whose adjacent
steps are both vertical or both horizontal (see the points $(0,1)$,
$(1,2)$, $(5,4)$ in Figure~\ref{FJ}).
The remaining case is a ``kink'' in an odd point 
$q$, i.e\@., a horizontal step
of the path meets a vertical one in 
$q$. Here the whole kink is shifted
until its even points have reached their new (reflected) positions.
That is, if 
$q=(x,y)$ is reached by a vertical step and left by a horizontal
step then $(x,y)$ is mapped to $(y+1,x-1)$ (see the point $(2,3)$ in
Figure~\ref{FJ}), and if 
$q=(x,y)$ is reached by a horizontal step and left by a vertical
step then $(x,y)$ is mapped to $(y+3,x-3)$ (see the point $(4,3)$ in
Figure~\ref{FJ}). 
Figure~\ref{FJ} gives an example of this modified reflection.



Now we are able to describe the desired weight-preserving and
sign-reversing involution.
For a $\lambda_1$-tuple $\bold{P}$ containing a path meeting the forbidden
line $y=x-2$, choose $i$ minimal such that $\path_i$ 
meets $y=x-2$, and 
replace $\path_i$'s portion up to the 
last meeting point with the line $y=x-2$
by its {\em modified\/} reflection. Clearly, this mapping is weight-preserving 
and sign-reversing. It reverses sign since it changes the sign
$\sgn\of\epsilon$ of
${\epsilon}$ while leaving $\sigma$ invariant.
On $\lambda_1$-tuples of paths that do not contain any path crossing
$y=x-1$ but are intersecting we apply the Gessel-Viennot involution.
Clearly, we thus obtain again a $\lambda_1$-tuple of paths 
that does not contain any path crossing
$y=x-1$ but is intersecting. It is therefore straight-forward 
to see that this mapping is
a weight-preserving and sign-reversing involution. Thus only those
$\lambda_1$-tuples remain that are nonintersecting and where none of their
paths cross the line $y=x-1$. Because the paths do not cross
$y=x-1$, the starting points must lie above $y=x-1$, hence
$\ep=(1,1,\dots,1)$. Again, since they are nonintersecting,
because of the location of the starting and ending points the associated
permutation must be the identity. But as was exhibited above,
 these $\lambda_1$-tuples
correspond to symplectic tableaux.
\QED

\noindent 
{\sc Remark}.
In \cite[Lemma~3.2C]{Okada:Lattice-Path} basically the same
lattice path interpretation is found but without observing that the
restriction on the paths can be formulated by a non-crossing
condition. Therefore, the reflection argument does not appear, and
the formula (\ref{thm:e-symplectic}) is not proved purely bijectively.
 
\bigskip
{\sc Bijective proof of the symplectic Jacobi--Trudi identity
(\ref{thm:h-symplectic})}. 
Proving the $h$-formula (\ref{thm:h-symplectic}) bijectively is more
difficult. 
In \cite{Fulmek-Kratt} we gave an algorithmic proof 
inspired by Okada's combinatorial-algebraic proof
\cite[Lemma~3.1C, Corollary~4.2(2)]{Okada:Lattice-Path}. In the proof 
that we are going to describe here we
use the concept of dual lattice paths, which is due to
Gessel and Viennot
\cite[sec.~4]{Gessel-Viennot:Binomial-determinants}. The idea is 
to pass from ``$h$-paths" to ``$e$-paths" by ``dualization", 
and then rely on a
variation of the bijection for the symplectic dual Jacobi--Trudi
identity, which we have given just above.
 
Let $\tableau$ be a $2n$-symplectic tableau of shape $\lambda$, where $\la$
is a partition of length $r\le n$. 
In the same way as in section~\ref{Lattice}
(see Figure~\ref{FE}), by reading paths from the rows, we map
$\tableau$ to an $r$-tuple
$(\path_1,\dots,\path_{r})$ of nonintersecting lattice paths,
where $\path_i$ runs from $u_i=(-i,1)$ to
$v_i=(\la_i-i,2n)$, $i=1,2,\dots,r$. Obviously, the symplectic constraint 
translates into the
condition that the first horizontal step of path $\path_i$
must be at least at height $2i-1$.
We define the weight $w(a)$ of a
horizontal edge $a$ with $h$-label (cf\@. Figure~\ref{FF}) 
$2i-1$ to be $x_i$, and the weight $w(a)$ of
a horizontal edge $a$ with $h$-label $2i$ to be $x_i^{-1}$ so that
this correspondence is
weight-preserving with respect to the weight (\ref{weight:symplectic}). 
 
Next we expand the determinant in (\ref{thm:h-symplectic}) and find
its lattice path interpretation. In the following 
$\reflect$ denotes the reflection in the line $x=-1$. 
Using again notation from section~\ref{Defs}, we have
\begin{align}\label{eq:h-symplectic-expansion}
& \detof{r}{\firstcolumn{\hsf_{\lambda_j-j+1}\of{\bold{x}}}
        \hsf_{\lambda_j-j+i}\of{\bold{x}} +
		\hsf_{\lambda_j-j-i+2}\of{\bold{x}}} \\
&\notag\hskip.5cm =\detof{r}{\firstcolumn{\GF{\fpath\of{u_1,v_j}}}
        \GF{\fpath\of{u_i,v_j}} +
		\GF{\fpath\of{\reflect(u_i),v_j}}} \\
&\hskip.5cm\notag  =	
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1\}\times\{1,-1\}^{r-1}}} 
	\sgn\of{\sigma}
\prod_{i,\epsilon_i=1}\GF{\fpath\of{u_i,v_{\si(i)}}}\cdot
\prod_{i,\epsilon_i=-1}\GF{\fpath\of{\reflect(u_i),v_{\si(i)}}}\\
\notag
& \hskip.5cm =	\underset{\bold\path\in
                \fpath\of{{\bold{u}}^{(\epsilon)},{\bold{v}_\si}}}
		{\sum_{\sigma\in \SG{r},\ 
                \epsilon\in\{1\}\times\{1,-1\}^{r-1}}} 
	\sgn\of{\sigma} w(\bold\path),
\end{align}
where the weight $w(\bold\path)$ 
is induced (as explained in section~\ref{Defs}) 
by the edge weight which was introduced
just before. Thus the determinant in (\ref{thm:h-symplectic}) is expressed 
as a generating function for $r$-tuples
$(\path_1,\dots,\path_r)$ of paths where $\path_i$ runs from
either $u_i$ or $\reflect(u_i)$ to $v_{\si(i)}$,
$i=1,2,\dots,r$, and where $\path_1$ {\em has} to start in $u_1$.
We shall give a
weight-preserving and sign-reversing involution that among these
$r$-tuples of paths cancels all those
that are either intersecting, 
or contain a path $\path_i$ the first horizontal
step of which is strictly below the line $y=2i-1$,
or where the associated $\epsilon$ differs from
$\parof{1,1,\dots,1}$. 
Suppose that this had been done. The remaining 
$r$-tuples are nonintersecting and the associated $\ep$ equals
$(1,1,\dots,1)$, so again the associated permutation must be the
identity. Finally, the first horizontal
step of $\path_i$ is at least at height $2i-1$, $i=1,2,\dots,r$. As we saw
above, these $r$-tuples of paths exactly correspond to
$2n$-symplectic tableaux of shape $\la$. This would prove
(\ref{thm:h-symplectic}).

 
Let $\bold\path=(\path_1,\dots,\path_r)$ be one of the $r$-tuples
of paths in $\fpath\of{\bold u^{(\ep)},\bold v_\si}$ which have to be
cancelled. 
Again note that $\path_1$ {\em has} to start in $u_1$.

If $\bold\path$ is intersecting we apply the Gessel--Viennot
involution. If $\bold\path$ is nonintersecting, then in the first
step we map $\bold\path=(\path_1,\dots,\path_r)$ to a $\la_1$-tuple
$\bold Q=(Q_1,\dots,Q_{\la_1})$ of nonintersecting lattice paths,
which we call the {\em dual\/} of $\bold \path$.

The correspondence between a set of paths and its dual is best
described with a picture at hand.
To start with, add a vertical step at the beginning of each of the paths of
$\bold\path$ so that, in abuse of notation, we obtain an $r$-tuple
$\bold\path$ of nonintersecting lattice paths where $\path_i$ starts
at either $u_i=(-i,0)$ or $\reflect(u_i)$ 
with a vertical step and
runs to $v_{\si(i)}$. The (bold) paths in
Figure~\ref{Fdual}.a show a $6$-tuple $\bold\path$ of
nonintersecting lattice paths with
$\la=(4,4,4,4,2,1)$, $\si=(2,3,1,4,5,6)$, and $\ep=(1,1,-1,1,1,1)$,
$n$ is chosen to be $6$.
%%%@@@ input from figfdual
\begin{figure} % Ueberfluessig, wenn Figure ohnehin = figure!
\begin{Figure}\label{Fdual}
	\begin{picture}(26,43)(0,0)
%Pfeile zwischen den Bildern:
				\put(13.5,36){\vector(1,0){1}}
				\put(22,24.5){\vector(0,-1){1}}
				\put(14,14){\vector(-1,0){1}}
%Bilderbeschriftung
        \put(4,27){\small a. The ,\kern-0.8pt,$h$-paths`` $\bold\path$}
        \put(16,27){\small b. Picture reflected in $x=0$}
        \put(15,3){\small c. Marking of paths interchanged}
        \put(-2,0){\small d. After deformation of the plane, the
                                ``$e$-paths" $\bold Q$,}
        \put(-2,-1){\small \hphantom{d. }the dual of $\bold\path$}
                %e-paths
		\put(0,1){
			\begingrid{-3}{-7}{6}{15}
            \makeaxes
				% The reflection line:
            \thinlines
				\put(-4,-6){\line(1,1){12}}
            \thicklines
				\put(-3,-6){\!\!\!\small$y=x-2$}
				% The paths:
				\uppath{0}{0}{---|||-|||||}
				\labelpointbl{0}{0}{\overline u_1}
				\labelpointar{4}{8}{\overline v_2}
				\uppath{-2}{2}{--|--|||||||}
				\labelpointbl{-2}{2}{\overline u_3}
				\labelpointar{2}{10}{\overline v_3}
				\uppath{3}{-3}{||-||-|||-||}
				\labelpointbl{3}{-3}{\reflect{\overline u_2}}
				\labelpointar{6}{6}{\overline v_1}
				\uppath{-3}{3}{|||-||-|||--}
				 \labelpointbl{-3}{3}{{\overline u_4}}
				 \labelpointar{1}{11}{\overline v_4}
				% The dual paths
				\downdottedpath{-3}{15}{||||||||/|||/|}{0}
				\labelpointbl{-1}{1}{\reflect{{u}_3}}
				\labelpointar{-3}{15}{{v}_1}
			   \downdottedpath{-2}{14}{|||||/|||||/|/|}{0}
				\labelpointbl{1}{-1}{{{u}_1}}
				\labelpointar{-2}{14}{{v}_2}
			   \downdottedpath{-1}{13}{|/|||||||/||/||}{0}
				\labelpointbl{2}{-2}{{{u}_2}}
				\labelpointar{-1}{13}{{v}_3}
			   \downdottedpath{0}{12}{/|||||||/||//|||}{0}
				\labelpointbl{4}{-4}{{{u}_4}}
				\labelpointar{0}{12}{{v}_4}
			   \downdottedpath{3}{9}{|||||/|/||||||}{0}
				\labelpointbl{5}{-5}{{{u}_5}}
				\labelpointar{3}{9}{{v}_5}
				\downdottedpath{5}{7}{||/||||||||||}{0}
				\labelpointbl{6}{-6}{{{u}_6}}
				\labelpointar{5}{7}{{v}_6}
			\endgrid
		}
      %paths interchanged
		\put(15,6){
			\begingrid{-3}{-2}{6}{12}
            \makeaxes
				% The reflection line:
                                \thinlines
				\put(1,-2){\line(0,1){15}}
                                \thicklines
				\put(0.3,-2.5){\small$x=1$}
				% The paths:
				\uppath{0}{0}{///|||/|||||}
				\labelpointbl{0}{0}{\ \overline u_1}
				\labelpointal{4}{12}{\ \overline v_2}
				\uppath{-2}{0}{//|//|||||||}
				\labelpointbl{-2}{0}{\ \overline u_3}
				\labelpointal{2}{12}{\ \overline v_3}
				\uppath{3}{0}{||/||/|||/||}
				\labelpointbl{3}{0}{\reflect{\overline u_2}}
				\labelpointal{6}{12}{\ \overline v_1}
				\uppath{-3}{0}{|||/||/|||//}
				\labelpointbl{-3}{0}{{\ \overline u_4}}
				\labelpointal{1}{12}{\ \overline v_4}
				% The dual paths
				\downdottedpath{-3}{12}{||||||||-|||-|}{0}
				\labelpointbl{-1}{0}{\reflect{{u}_3}}
				\labelpointal{-3}{12}{\ {v}_1}
			   \downdottedpath{-2}{12}{|||||-|||||-|-|}{0}
				\labelpointbl{1}{0}{\ {{u}_1}}
				\labelpointal{-2}{12}{\ {v}_2}
			   \downdottedpath{-1}{12}{|-|||||||-||-||}{0}
				\labelpointbl{2}{0}{{\ {u}_2}}
				\labelpointal{-1}{12}{\ {v}_3}
			   \downdottedpath{0}{12}{-|||||||-||--|||}{0}
				\labelpointbl{4}{0}{{\ {u}_4}}
				\labelpointal{0}{12}{\ {v}_4}
			   \downdottedpath{3}{12}{|||||-|-||||||}{0}
				\labelpointbl{5}{0}{{\ {u}_5}}
				\labelpointal{3}{12}{\ {v}_5}
				\downdottedpath{5}{12}{||-||||||||||}{0}
				\labelpointbl{6}{0}{{\ {u}_6}}
				\labelpointal{5}{12}{\ {v}_6}
			\endgrid
		}
      %h-paths reflected in x=0
		\put(15,29){
			\begingrid{-3}{-2}{6}{12}
            \makeaxes
				% The reflection line:
                                \thinlines
				\put(1,-2){\line(0,1){15}}
                                \thicklines
				\put(0.3,-2.5){\small$x=1$}
				% The paths:
				\updottedpath{0}{0}{///|||/|||||}
				\labelpointbl{0}{0}{\ \overline u_1}
				\labelpointal{4}{12}{\ \overline v_2}
				\updottedpath{-2}{0}{//|//|||||||}
				\labelpointbl{-2}{0}{\ \overline u_3}
				\labelpointal{2}{12}{\ \overline v_3}
				\updottedpath{3}{0}{||/||/|||/||}
				\labelpointbl{3}{0}{\reflect{\overline u_2}}
				\labelpointal{6}{12}{\ \overline v_1}
				\updottedpath{-3}{0}{|||/||/|||//}
				\labelpointbl{-3}{0}{{\ \overline u_4}}
				\labelpointal{1}{12}{\ \overline v_4}
				% The dual paths
				\downpath{-3}{12}{||||||||-|||-|}{0}
				\labelpointbl{-1}{0}{\reflect{{u}_3}}
				\labelpointal{-3}{12}{\ {v}_1}
			   \downpath{-2}{12}{|||||-|||||-|-|}{0}
				\labelpointbl{1}{0}{\ {{u}_1}}
				\labelpointal{-2}{12}{\ {v}_2}
			   \downpath{-1}{12}{|-|||||||-||-||}{0}
				\labelpointbl{2}{0}{{\ {u}_2}}
				\labelpointal{-1}{12}{\ {v}_3}
			   \downpath{0}{12}{-|||||||-||--|||}{0}
				\labelpointbl{4}{0}{{\ {u}_4}}
				\labelpointal{0}{12}{\ {v}_4}
			   \downpath{3}{12}{|||||-|-||||||}{0}
				\labelpointbl{5}{0}{{\ {u}_5}}
				\labelpointal{3}{12}{\ {v}_5}
				\downpath{5}{12}{||-||||||||||}{0}
				\labelpointbl{6}{0}{{\ {u}_6}}
				\labelpointal{5}{12}{\ {v}_6}
			\endgrid
		}
      %h-paths
		\put(2,29){
			\begingrid{-6}{-2}{3}{12}
            \makeaxes
				% The reflection line:
                                \thinlines
				\put(-1,-2){\line(0,1){15}}
                                \thicklines
				\put(-2.3,-2.5){\small$x=-1$}
				% The paths:
				\downdottedpath{-4}{12}{|||||/|||///}{0}
				\labelpointbl{0}{0}{\ \overline u_1}
				\labelpointal{-4}{12}{\ \overline v_2}
				\labelpointbl{0}{-0.7}{\ w_2}
				\downdottedpath{-2}{12}{|||||||//|//}{0}
				\labelpointbl{2}{0}{\ \overline u_3}
				\labelpointal{-2}{12}{\ \overline v_3}
				\labelpointbl{2}{-0.7}{\ w_3}
				\downdottedpath{-6}{12}{||/|||/||/||}{0}
				\labelpointbl{-3}{0}{\reflect{\overline u_2}}
				\labelpointal{-6}{12}{\ \overline v_1}
				\labelpointbl{-3}{-0.7}{\ w_1}
				\downdottedpath{-1}{12}{//|||/||/|||}{0}
				\labelpointbl{3}{0}{{\ \overline u_4}}
				\labelpointal{-1}{12}{\ \overline v_4}
				\labelpointbl{3}{-0.7}{{\ w_4}}
				% The dual paths
				\uppath{1}{0}{|-|||-||||||||}
				\labelpointbl{1}{0}{\reflect{{u}_3}}
				\labelpointal{3}{12}{\ {v}_1}
			   \uppath{-1}{0}{|-|-|||||-|||||}
				\labelpointbl{-1}{0}{\ {{u}_1}}
				\labelpointal{2}{12}{\ {v}_2}
			   \uppath{-2}{0}{||-||-|||||||-|}
				\labelpointbl{-2}{0}{{\ {u}_2}}
				\labelpointal{1}{12}{\ {v}_3}
			   \uppath{-4}{0}{|||--||-|||||||-}
				\labelpointbl{-4}{0}{{\ {u}_4}}
				\labelpointal{0}{12}{\ {v}_4}
			   \uppath{-5}{0}{||||||-|-|||||}
				\labelpointbl{-5}{0}{{\ {u}_5}}
				\labelpointal{-3}{12}{\ {v}_5}
				\uppath{-6}{0}{||||||||||-||}
				\labelpointbl{-6}{0}{{\ {u}_6}}
				\labelpointal{-5}{12}{\ {v}_6}
			\endgrid
		}
	\end{picture}
\end{Figure}
\end{figure} % Ueberfluessig, wenn Figure ohnehin = figure!

%%%@@@ end input from figfdual

The starting points of $\bold\path$ occupy $r$ lattice points on the
line $y=0$, the end points occupy $r$ lattice points on the line
$y=2n$. Now, in the region $-r\le x\le \la_1-1$ let
$w_1,w_2,\dots,w_{\la_1}$ (from left to right) be the lattice points
on $y=0$ not occupied by $\bold\path$, and $\overline
v_1,\overline v_2,\dots,\overline v_{\la_1}$ be the lattice points on
$y=2n$ not occupied by $\bold\path$. 
Then we connect $w_i$ to $\overline v_i$ by a lattice path,
$i=1,2,\dots,\la_1$, by starting at $w_i$ and going up vertically
unless we would meet a lattice path of $\bold\path$. In the latter
case we proceed by an upward diagonal step in direction $(-1,1)$. The
dotted paths in Figure~\ref{Fdual}.a show the result in our example.
Besides, in Figure~\ref{Fdual}.a the starting points
of the new paths are also labelled in the following way: The points
$(j-1,0)$, $j=1,2,\dots,\la_1$, which are not occupied by $\bold\path$
are labelled by $\overline u_j$, and the points $(-j-1,0)$,
$j=1,2,\dots,r-1$, which are not occupied by $\bold\path$ are
labelled by $\reflect(\overline u_j)$.

In the next step we reflect the whole picture in the line $x=0$, see
Figure~\ref{Fdual}.b. Then we exchange the roles of $\bold \path$,
the old paths, and the new paths. In
Figure~\ref{Fdual}.c this is indicated by marking the new paths with bold
lines and the old paths with dotted lines. In the final step we deform
the plane by the linear transformation $(x,y)\to(x,y-x)$, see
Figure~\ref{Fdual}.d. 

The $\la_1$-tuple $\bold Q$ of paths (bold in Figure~\ref{Fdual}.d)
that we finally obtained by this procedure is defined to be 
the {\em dual\/} of $\bold\path$. The bold paths in 
Figure~\ref{Fdual}.d should remind us
of the objects that were involved in the bijective proof of the
symplectic {\em dual\/} Jacobi--Trudi identity. Indeed, let
$\overline u_i=(-i+1,i-1)$ and $\overline
v_i=(\la_i'-i+1,2n-\la_i'+i-1)$, and let $\overline \reflect$ denote
the reflection in the line $y=x-2$. Then the $\la_1$-tuple of
paths $\bold Q=(Q_1,\dots,Q_{\la_1})$ is nonintersecting, and,
for some permutation
$\overline \si\in\SG{\la_1}$ and some sign vector $\overline
\ep\in\{1,-1\}^{\la_1}$, the path $Q_i$ runs from
$\overline \reflect^{(1-\overline \ep_i)/2}(\overline u_i)$ (i.e\@., either
$\overline u_i$ or $\overline\reflect(\overline u_i)$ depending on whether
$\overline\ep_i=1$ or $-1$) to $\overline
v_{\overline\si(i)}$. In the example of Figure~\ref{Fdual} we have 
$\la'=(6,5,4,4)$,
$\overline\si=(2,1,3,4)$ and $\overline\ep=(1,-1,1,1)$.
Moreover, there holds
\begin{equation}\label{signum}
\sgn\of \si=\sgn\of{\overline \si}\cdot \sgn\of{\overline\ep}.
\end{equation}
This is easily checked in our example, but
before we verify this fact in general,
we finish the description of the involution.

The basic idea is to apply the mapping involving the
``modified reflection" used in the proof of
the symplectic dual Jacobi--Trudi identity
(i.e., choose $i$ minimal such that $Q_i$ meets $y=x-2$, and 
replace $Q_i$'s portion up to the last meeting point with the line $y=x-2$
by its modified reflection; cf\@. Fig.~\ref{FJ}) 
and after that map the resulting $\la_1$-tuple of paths to its
preimage under dualization. However, the application of modified
reflection 
may introduce new points of intersections of paths, which we cannot
permit, since dualization does not work with intersecting paths.

There is an easy way to come around this problem. Let $\bold A$ denote
the mapping involving the
``modified reflection", and let $\bold B$ denote
the Gessel--Viennot involution. Now, starting with $\bold Q$, apply
$\bold A$, then $\bold B$, then $\bold A$ again, etc., as long as
possible. Formally this is $\bold{B}^{[0\text { or }1]}\circ
\parof{\bold{A}\circ\bold{B}}^m\circ\bold{A}$, for some $m$.
The algorithm terminates because the set of $\la_1$-tuples
under consideration is finite and there cannot be any cycle. 
For, both $\bold A$ and $\bold B$ are actually involutions on the set
of $\la_1$-tuples of paths containing at least one path that 
meets the line $y=x-2$, $\bold B$ being not defined for
nonintersecting $\la_1$-tuples. 
Now think of this set of $\la_1$-tuples as the vertices of a graph where two
vertices $O_1$ and $O_2$ are connected by an edge if $O_1$ and $O_2$
are mapped to each other by either $\bold A$ or $\bold B$. This graph
has maximal degree 2, hence consists of paths and cycles. The
``starting objects" for our algorithm, which are {\em nonintersecting\/}
$\la_1$-tuples, all have degree 1 in this graph
(only operation $\bold A$ is applicable), therefore we cannot run
into a cycle. Since $\bold A$ can always be applied,
the algorithm has to stop with an application of $\bold A$.
But that means that we finally obtained some {\em nonintersecting\/}
$\la_1$-tuple, $\bold Q'$ say. 

Next note that, if $\si'$ is the permutation
corresponding to $\bold Q'$ and if $\ep'$ is the sign vector
corresponding to $\bold Q'$, then we have
\begin{equation}\label{signreverse}
\sgn\of{\overline
\si}\cdot\sgn\of{\overline\ep}=-\sgn\of{\si'}\cdot\sgn\of{\ep'}.
\end{equation}
This is because we started from $\bold Q$ by an application of $\bold A$
and arrived at $\bold Q'$ by an application of $\bold A$, therefore, to go
from $\bold Q$ to $\bold Q'$, we applied an odd number of operations $\bold
A$ and $\bold B$, $\bold A$ changing the sign of the corresponding
sign vector, $\bold B$ changing the sign of the corresponding permutation.

In the last step we determine the preimage of $\bold Q'$ under
dualization, $\hat{\bold\path}$ say. In fact, it is easy to check that
each step of the dualization process can be reversed so that
$\hat{\bold\path}$ is well-defined.

Obviously, the map from $\bold\path$ to $\hat{\bold\path}$ is
weight-preserving. To check that it is also sign-reversing, let $\hat\si$
be the permutation corresponding to
$\hat{\bold\path}$. Because of (\ref{signum}), which
also applies to $\hat\si, \si',\ep'$, and (\ref{signreverse}), we
have
$$\sgn\of\si=\sgn\of{\overline\si}\cdot\sgn\of{\overline\ep} 
=-\sgn\of{\si'}\cdot\sgn\of{\ep'}=-\sgn\of{\hat\si}.$$
Thus the mapping from $\bold\path$ to $\hat{\bold\path}$ is
sign-reversing.

Finally, we have to prove (\ref{signum}). Recall that $\bold\path$
and $\bold Q$ are related by dualization as exhibited in
Figure~\ref{Fdual}. Let $s$ and $\overline s$ be the number of
reflected paths of $\bold\path$ and $\bold Q$, respectively. Since
$\bold\path$ is nonintersecting, the permutation $\si$ corresponding
to $\bold\path$ must satisfy the inequalities
$$\si^{-1}(1)>\si^{-1}(2)>\dots>\si^{-1}(s);\
\si^{-1}(s+1)<\si^{-1}(s+2)<\dots<\si^{-1}(r).$$
Likewise, the permutation $\overline\si$ corresponding to $\bold Q$
must satisfy
$$\overline \si^{-1}(1)>\overline \si^{-1}(2)>\dots>\overline
\si^{-1}(
%Christian
\bar s);\
\overline \si^{-1}(
%Christian
\bar s+1)<\overline \si^{-1}(
%Christian
\bar s+2)<\dots<\overline
\si^{-1}(\la_1).$$
Therefore the number of inversions of $\si^{-1}$ is
\begin{equation}\label{inv-sigma}
\inv(\si^{-1})=\sum _{k=1} ^{s}(\si^{-1}(k)-1),
\end{equation}
while the number of inversions of $\overline \si^{-1}$ is
\begin{equation}\label{inv-sigmabar}
\inv(\overline \si^{-1})=\sum _{k=1} ^{\overline s}(\overline \si^{-1}(k)-1).
\end{equation}

How are $s$ and $\overline s$, $\si^{-1}(k)$ and $\overline
\si^{-1}(k)$ related? By construction of dualization, the reflected
starting points of $\bold\path$ are exactly the reflections of the
reflected starting points of $\bold Q$, see Figure~\ref{Fdual}. In
fact, if a path of $\bold\path$ starts at some point then there
cannot be a path of $\bold\path$ starting at its reflection, hence it
must be occupied by a starting point of a path of $\bold Q$, and vice
versa. Therefore the number of reflected starting points in $\bold
\path$ and $\bold Q$ is the same, which simply says that $s=\overline s$.
Besides, because of the particular numbering of the starting points
of $\bold \path$ and $\bold Q$ (which is caused by the fact that
$\bold\path$'s first starting point, $u_1$, {\em has\/} to be
on the reflection line, and, hence,
$\bold Q$'s starting points avoid the reflection line) we have
\begin{equation}\label{sigma-relation}
\si^{-1}(k)-1=\overline \si^{-1}(k)\quad \quad \text {for
$k=1,2,\dots,s$.}
\end{equation}
A combination of (\ref{inv-sigma}), (\ref{inv-sigmabar}),
(\ref{sigma-relation}), and $s=\overline s$, gives
\begin{align} 
\notag
\sgn\of\si&=(-1)^{\inv(\si^{-1})}\\
\notag
&=(-1)^{\inv(\overline \si^{-1})+s}\\
\notag
&=\sgn\of{\overline \si}\cdot \sgn\of{\overline\ep},
\end{align}
which confirms (\ref{signum}). This finishes the proof.
\QED


\bigskip
% \subsection{The symplectic Giambelli-identity}
{\sc Bijective proof of the symplectic Giambelli identity
(\ref{thm:Giambelli-symplectic})}.
We use the same encoding of symplectic tableaux as was used in the
proof of the ``ordinary" Giambelli identity (\ref{thm:Giambelli}) for
tableaux (recall Figure~\ref{FH}). That is, we map a symplectic
tableau of shape $(\alpha,\beta)$, where $(\alpha,\beta)$ is the
Frobenius notation of a partition of rank $s$, to an $s$-tuple
$(\path_1,\dots,\path_s)$ of lattice paths where $\path_i$ runs from 
$\parof{-\alpha_i,2n-1}$ to
$\parof{\beta_i+1,2n-\beta_i-1}$, $i=1,2,\dots,s$.
Figure~\ref{FR} shows an example
(with $n=3$) of this
correspondence between nonintersecting lattice paths and symplectic
tableaux.


%%%@@@ input from figfr
\begin{Figure} \label{FR}
\begin{picture}(20,10)(0,0)
\put(13,6.5){$T$ =}
\put(15,5){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
	\endtableau
}
\put(1,1){
	\begingrid{-4}{-1}{4}{6}
		\makexaxis
		\makexscale
		\makeyaxis
		\downskewline{0}{2}{2}
		\downskewline{0}{4}{4}
		\downskewline{0}{6}{4}
		\labelledtrue
		\elabelfalse
		% Pfade
		\downpath{-3}{5}{-||-|-|*-|--|}{6}
		\downpath{-1}{5}{-||*-|-}{6}
		% Gerade y=x-1
     \thinlines
		\put(1,0){\line(1,1){5}}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfr

As is clear from the picture, the symplectic constraint 
translates into the condition that the first path must not cross the
line $y=x-1$. If, as before, we define the weight of a
horizontal edge with label $2i-1$ to be $x_i$, and the weight of
a horizontal edge with label $2i$ to be $x_i^{-1}$, then
this correspondence is
weight-preserving with respect to the weight (\ref{weight:symplectic}).

Obviously, the set of all $s$-tuples of lattice paths
subject to this condition 
is invariant under Gessel--Viennot involution. Therefore
the same arguments as in Stembridge's proof of the ``ordinary"
Giambelli identity, which was reviewed in section~\ref{Lattice}, establish
(\ref{thm:Giambelli-symplectic}).
\QED



\section{Symplectic identities: The odd and intermediate symplectic tableaux of
Proctor}\label{oddsymplectic}
In this short section we give bijective proofs for
(\ref{thm:h-intermediate-symplectic}),
(\ref{thm:e-intermediate-symplectic}), 
(\ref{thm:Giambelli-intermediate-symplectic}), and
(\ref{thm:e-odd-symplectic}). We are going to rely on the bijections
of the previous section, so there is almost no additional work to be
done. 

\bigskip
{\sc Bijective proof of the $(n,m)$-symplectic dual Jacobi--Trudi identity
(\ref{thm:e-intermediate-symplectic})}.
In the same way as in the proof of the (ordinary) symplectic dual
Jacobi--Trudi identity (\ref{thm:e-symplectic}) given in the previous
section (see Figure~\ref{FI}), we map $(n,m)$-symplectic tableaux of
shape $(\la,\la^{\text {trnc}})$ to $\la_1$-tuples
$(\path_1,\dots,\path_{\la_1})$ of nonintersecting lattice paths,
where $\path_i$ runs from $(-i+1,i-1)$ to
$(\la_i'-i+1,2n+m-\la_i'+i-1)$, $i=1,2,\dots,\lambda_1$. 
In general, the symplectic constraint
would translate into the
condition that  the path $\path_1$ (and hence all other paths) must
not cross the line segment $\{(x,y):y=x-1\text { and }x\le n+1\}$.
However, since for (\ref{thm:e-intermediate-symplectic}) we have
to assume $\ell(\la)\le n+1$, the number of horizontal steps of each
path cannot exceed $n+1$, and therefore all the paths do not cross
the {\em complete\/} line $y=x-1$. This shows that for $\ell(\la)\le
n+1$ the $(n,m)$-symplectic tableaux of
shape $(\la,\la^{\text {trnc}})$ correspond to $\la_1$-tuples
$(\path_1,\dots,\path_{\la_1})$ of nonintersecting lattice paths,
where $\path_i$ runs from $u_i=(-i+1,i-1)$ to
$v_i=(\la_i'-i+1,2n+m-\la_i'+i-1)$, $i=1,2,\dots,\lambda_1$, and does not cross $y=x-1$.
If we define the weight $w(a)$ of a
horizontal edge $a$ with $e$-label (cf\@. Figure~\ref{FF}) 
$2i-1$, $i\le n$, to be $x_i$, the weight $w(a)$ of
a horizontal edge $a$ with $e$-label $2i$, $i\le n$, to be $x_i^{-1}$,
and the weight of a horizontal edge with $e$-label $i$, $i> 2n$, to be
$z_{i-2n}$, then the above correspondence is
weight-preserving with respect to the weight
(\ref{weight:intermediate-symplectic}). 

Let $\reflect$ be the reflection in the line $y=x-2$.
Again, the determinant in (\ref{thm:e-intermediate-symplectic})
may be written as a generating function for $\lambda_1$-tuples
$(\path_1,\dots,\path_{\lambda_1})$ of lattice paths where $\path_i$ runs
from either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, $i=1,2,\dots,\lambda_1$,
$$ \detof{\lambda_1}{\esf_{\lambda_j^\prime-j+i}\of{\bold{x},\bold z} -
		\esf_{\lambda_j^\prime-j-i}\of{\bold{x},\bold z}} =
	\underset{\bold\path\in
        \fpath\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1,-1\}^{\lambda_1}}}
	\sgn\of{\sigma}
	\sgn\of{\epsilon} 
        w(\bold\path).
$$

The involution that cancels $\la_1$-tuples $\bold \path$ where a path of
$\bold \path$ crosses $y=x-1$ is defined in the same way as in the proof
of (\ref{thm:e-symplectic}) in the previous section. This is a
weight-preserving mapping since the right-most possible crossing
point of a path in $\bold \path$ with $y=x-1$ is $(n,n-1)$, and hence
the steps that
are involved in the modified reflection, depicted in Figure~\ref{FJ},
all have $e$-label at most $2n$. Of course, for the intersecting
$\la_1$-tuples we apply the Gessel--Viennot involution.
\QED

\bigskip
{\sc Bijective proof of the $(n,m)$-symplectic Jacobi--Trudi identity
(\ref{thm:h-intermediate-symplectic})}. 
In the same way as in the proof of the (ordinary) symplectic 
Jacobi--Trudi identity (\ref{thm:h-symplectic}) given in the previous
section (see Figure~\ref{FE}), we map $(n,m)$-symplectic tableaux of
shape $(\la,\la^{\text {trnc}})$ to $r$-tuples
$(\path_1,\dots,\path_{r})$ of nonintersecting lattice paths,
where $\path_i$ runs from $(-i,1)$ to
$(\la_i-i,2n+m)$, $i=1,2,\dots,r$. In general, the symplectic constraint 
would translate into the
condition that the first horizontal step of path $\path_i$, $i\le
n+1$, must be at least at height $2i-1$.
However, since also for (\ref{thm:h-intermediate-symplectic}) we have
to assume $r=\ell(\la)\le n+1$, this condition holds for {\em all\/}
paths $\path_i$.
Thus for $\ell(\la)\le
n+1$ the $(n,m)$-symplectic tableaux of
shape $(\la,\la^{\text {trnc}})$ correspond to $r$-tuples
$(\path_1,\dots,\path_{r})$ of nonintersecting lattice paths,
where $\path_i$ runs from $u_i=(-i,1)$ to
$v_i=(\la_i-i,2n+m)$, $i=1,2,\dots,r$, and the first horizontal step of $\path_i$ is
at least at height $2i-1$.
If we define the weight of a
horizontal edge with $h$-label (cf\@. Figure~\ref{FF}) 
$2i-1$, $i\le n$, to be $x_i$, the weight of
a horizontal edge with $h$-label $2i$, $i\le n$, to be $x_i^{-1}$,
and the weight of a horizontal edge with $h$-label $i$, $i> 2n$, to be
$z_{i-2n}$, then the above correspondence is
weight-preserving with respect to the weight
(\ref{weight:intermediate-symplectic}). 

For the proof of (\ref{thm:h-intermediate-symplectic}) we let
$\reflect$ be the reflection in the line $x=-1$, as in the proof of
(\ref{thm:h-symplectic}) in the previous section.
Then the determinant in (\ref{thm:h-intermediate-symplectic})
may be written as a generating function for $r$-tuples
$(\path_1,\dots,\path_{r})$ of lattice paths where $\path_i$ runs
from either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, $i=1,2,\dots,r$,
namely
\begin{multline}\notag
 \detof{r}{\firstcolumn{\hsf_{\lambda_j-j+1}\of{\bold{x},\bold z}}
        \hsf_{\lambda_j-j+i}\of{\bold{x},\bold z} +
		\hsf_{\lambda_j-j-i+2}\of{\bold{x},\bold z}} \\
\notag=
	\underset{\bold\path\in
        \fpath\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{r},\ 
                \epsilon\in\{1\}\times\{1,-1\}^{r-1}}} 
	\sgn\of{\sigma}
        w(\bold\path).
\end{multline}


For the rest of the proof we proceed as in the
proof of (\ref{thm:h-symplectic}) given in the previous
section. 
\QED

\bigskip

{\sc Bijective proof of the $(n,m)$-symplectic Giambelli identity
(\ref{thm:Giambelli-intermediate-symplectic})}.
In the same way as in section~\ref{symplectic}, $(n,m)$-symplectic
tableaux of shape $((\alpha,\beta),\break (\alpha|\beta)^{\text {trnc}})$, 
where $(\alpha,\beta)$ is the
Frobenius notation of a partition of rank $s$, correspond to
$s$-tuples
$(\path_1,\dots,\path_s)$ of lattice paths where $\path_i$ runs from 
$\parof{-\alpha_i,2n+m-1}$ to
$(\beta_i+1,\break 2n+m-\beta_i-1)$, $i=1,2,\dots,s$, 
and does not cross the line $y=x-1$ 
(see Figure~\ref{FR}). The weight of a horizontal edge
with label $2i-1$, $i\le n$, is defined to be $x_i$, the weight of
a horizontal edge with label $2i$, $i\le n$, is defined to be $x_i^{-1}$,
and the weight of a horizontal edge with label $i$, $i> 2n$, is
defined to be $z_{i-2n}$. Then this correspondence is
weight-preserving with respect to the weight
(\ref{weight:intermediate-symplectic}).

The rest of the proof is 
identical with the proof of the (ordinary) symplectic
Giambelli identity (\ref{thm:Giambelli-symplectic}) given in 
section~\ref{symplectic}. 
\QED

\bigskip
{\sc Bijective proof of the odd symplectic dual Jacobi--Trudi identity
(\ref{thm:e-odd-symplectic})}. 
Again, as in the proof of the (ordinary) symplectic dual
Jacobi--Trudi identity (\ref{thm:e-symplectic}) given in the previous
section (see Figure~\ref{FI}), we map $(n,1)$-symplectic tableaux of
shape $(\la,\la_d)$ to $\la_1$-tuples
$(\path_1,\dots,\path_{\la_1})$ of nonintersecting lattice paths,
where $\path_i$ runs from $(-i+1,i-1)$ to
$(\la_i'-i+1,2n-\la_i'+i)$, $i=1,2,\dots,\lambda_1$. 
Also here, because of $\ell(\la)\le
n+1$, the symplectic constraint
translates into the condition that all the paths do not cross
the line $y=x-1$. Besides, there is the constraint that 
the last entries in columns
$1,2,\dots,\la_d$ of $(n,1)$-symplectic tableaux are equal to $2n+1$.
It is easy to see that altogether this means that 
$(n,1)$-symplectic tableaux of
shape $(\la,\la_d)$ correspond to $\la_1$-tuples
$(\path_1,\dots,\path_{\la_1})$ of nonintersecting lattice paths,
where $\path_i$ runs from $u_i=(-i+1,i-1)$ to
$v_i=(\la_i'-i+1,2n-\la_i'+i)$, $i=1,2,\dots,\lambda_1$, and does not cross $y=x-1$, and
in addition the paths $\path_1,\dots,\path_{\la_d}$ end with a horizontal
step (which has $e$-label $2n+1$).
If we define the weight of a
horizontal edge with $e$-label (cf\@. Figure~\ref{FF}) 
$2i-1$, $i\le n$, to be $x_i$, the weight of
a horizontal edge with $e$-label $2i$, $i\le n$, to be $x_i^{-1}$,
and the weight of a horizontal edge with $e$-label $2n+1$
to be $z$, then the above correspondence is
weight-preserving with respect to the weight
(\ref{weight:odd-symplectic}). 

 Again it is easy to see that 
the determinant in (\ref{thm:e-odd-symplectic})
may be written as a generating function for certain $\lambda_1$-tuples 
$(\path_1,\dots,\path_{\lambda_1})$ of lattice paths where $\path_i$ runs
from either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, $i=1,2,\dots,\lambda_1$,
\begin{multline} \notag
\detof{\lambda_1}{z\esf_{\lambda_j^\prime-j+i-1}\of{\bold{x}} -
		z\esf_{\lambda_j^\prime-j-i-1}\of{\bold{x}}
\quad \vdots\quad 
                \esf_{\lambda_j^\prime-j+i}\of{\bold{x}, z} -
		\esf_{\lambda_j^\prime-j-i}\of{\bold{x}, z}}
\\=	\underset{\bold\path\in
                \fpath^{(1)}\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1,-1\}^{\lambda_1}}}
	\sgn\of{\sigma}
	\sgn\of{\epsilon}
	\GF{\fpath^{(1)}\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}.
\end{multline}         
Here $\fpath^{(1)}$ denotes the additional constraint that paths
$\path_{\si^{-1}(1)},\dots,\path_{\si^{-1}(\la_d)}$ end with a horizontal step.
The rest of the proof is identical with the proof for
(\ref{thm:e-intermediate-symplectic}). One only has to note that 
the modified reflection as well as the Gessel--Viennot involution
preserve the additional constraint regarding the last steps of
$\path_{\si^{-1}(1)},\dots,\path_{\si^{-1}(\la_d)}$.
\QED



\section{Orthogonal identities: Proctor's tableaux}\label{orthogonal}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%              Der orthogonale Fall mit den Tableaux von Proctor              %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$N$-orthogonal tableaux must satisfy rather complicated constraints, compared
to the simple symplectic tableaux. So we start with the easier case of coarse
orthogonal tableaux and give a proof of a ``weak form'' of the orthogonal
$e$-formula, where all the variables involved are set to $1$. Recall that
(\ref{thm:e-orthogonal}), when evaluated at $x_1 =\dots =x_n=1$, just reads
\begin{multline}\label{thm:weak-e-orthogonal}
\oschftriv{N}{\lambda}  \\=
\detof{\lambda_1}{
	\firstrow{\esf_{\lambda^\prime_j-j+1}\of{1,1,\dots,1}}
	\esf_{\lambda_j^\prime-j+i}\of{1,1,\dots,1} +
		\esf_{\lambda_j^\prime-j-i+2}\of{1,1,\dots,1}}.
\end{multline}
By (\ref{thm:equinumerous}), the left side of the equation equals the
{\em number\/} of coarse orthogonal tableau.
 
\bigskip
{\sc Bijective proof of the dimension formula
(\ref{thm:weak-e-orthogonal})}.
First we describe the lattice path interpretation of coarse
orthogonal tableaux. A coarse orthogonal tableau $\tableau$ 
with entries $\le N$ and of
shape $\lambda=(\la_1,\dots,\la_r)$, where $\la$ is an $N$-orthogonal
partition,
is mapped to a $\la_1$-tuple
$(\path_1,\dots,\path_{\la_1})$ of lattice paths, where $\path_i$
runs from $(-i+1,i-1)$ to $(\lambda_i'-i+1,N-\lambda'_i+i-1)$, 
$i=1,2,\dots,\lambda_1$, by
using the correspondence of section~\ref{Lattice} that based on the
$e$-labelling, i.e\@., by
reading the path $\path_i$ from the $i$-th column of $\tableau$.
Figure~\ref{FS} depicts an example of this
correspondence where $\la=(4,3,2,1)$ and the entries of the tableau
are bounded by $N=9$.


%%%@@@ input from figfs
\begin{Figure}\label{FS}
\begin{picture}(20,12)(0,0)
\put(13,7){$T$ =}
\put(15,5){
	\begintableau{4}{4}
	\row{1445}
	\row{266}
	\row{47}
   \row{6}
	\endtableau
}
\put(1,1){
\begingrid{-3}{0}{5}{11}
\makexaxis
\makexscale
 
% Pfade
\elabeltrue
\labelledtrue
\uppath{0}{0}{--|-|-|||}\hlabel{3}{1}{$\path_1$}
\uppath{-1}{1}{|||-|--||}\hlabel{2}{5}{$\path_2$}
\uppath{-2}{2}{|||-|-|||}\hlabel{0}{7}{$\path_3$}
\uppath{-3}{3}{||||-||||}\hlabel{-2}{7}{$\path_4$}
% gespiegelter Pfad
\labelledfalse
\updottedpath{0}{0}{||-|-|---}
% Gerade y=x
\thinlines
\put(-1,-1){\line(1,1){7}}
\endgrid
}
\end{picture}
\end{Figure} 

%%%@@@ end input from figfs

It is obvious that this $\la_1$-tuple must be nonintersecting because
$\tableau$ is a tableau. Let $\reflect$ denote the reflection in
the line $y=x$. This notation holds for the remainder of this
section. We claim that the coarse
orthogonal condition (\ref{cond:coarse}) translates into the condition that
{\em the second path $\path_2$ and the reflection
$\reflect\of{\path_1}$ of the first path in the line $y=x$ must not
intersect\/}. This is seen as follows.

Let $c$ be fixed.
Denote the number of elements in the $i$-th column of $\tableau$ that are 
smaller or equal $c$ 
by $\mu_i$, $i=1,2$. The coarse orthogonal condition then says
$$\mu_1+\mu_2\leq c.$$
If $(\path_1,\path_2,
\dots,\path_{\la_1})$ corresponds to $\tableau$ then the meeting
point of ${\path_1}$ with the antidiagonal line $x+y=c$
is $p_1=(\mu_1,c-\mu_1)$ and the meeting
point of ${\path_2}$ with $x+y=c$ is $p_2=(\mu_2-1,c-\mu_2+1)$.
The reflection $\reflect\of{p_1}$ 
of $p_1$ in the line $y=x$ is the point $(c-\mu_1,\mu_1)$. 
Obviously we have that $\mu_1+\mu_2\le c$
if and only if $\reflect\of{{p}_1}$ lies strictly south-east
of $p_2$.
In other words: $\tableau$ obeys the $c$-th
orthogonal condition for {\em all\/} $c$ if and only if for {\em
all\/} $c$ the reflected
meeting point of $\path_1$ and $x+y=c$ lies strictly south-east
of the meeting point of 
 $\path_2$ and $x+y=c$. This is equivalent to our claim.
 
If we can show that the number of all $\la_1$-tuples  
$(\path_1,\dots,\path_{\la_1})$ of nonintersecting 
lattice paths, where $\path_i$
runs from $(-i+1,i-1)$ to $(\lambda_i'-i+1,N-\lambda'_i+i-1)$, 
$i=1,2,\dots,\lambda_1$, and
where the reflection of $\path_1$ does not intersect $\path_2$ equals
the determinant in (\ref{thm:weak-e-orthogonal}), then 
(\ref{thm:weak-e-orthogonal}) is established. In fact, we are going
to prove a more general path counting result from which
(\ref{thm:weak-e-orthogonal}) immediately follows. 

\medskip
\noindent {\bf Theorem 1}. {\sl
Let $\reflect$ denote the reflection in the line $y=x$.
Let $u_1,\dots,u_m$ be pairwise
distinct points where
$u_{i+1}$ lies to the north-west
of $u_{i}$, $i=1,2,\dots,m-1$, and in addition $u_2$ lies to the
north-west of $\reflect\of{u_1}$. Let $v_1,\dots,v_m$ be another
collection of pairwise distinct points with the same property.
Besides, $u_i$ is assumed to lie
to the south-west of $v_i$, $i=1,2,\dots,m$.
Then the number  of nonintersecting $m$-tuples
$(\path_1,\dots,\path_m)$
of lattice paths, where $\path_1$ runs from either $u_1$ or
$\reflect\of{u_1}$ 
to $v_1$,
and for $i\ge2$ the paths $\path_i$ run from $u_i$ to 
$v_i$, 
and where {\rm in addition\/} there is no intersection of
$\path_2$ with $\reflect\of{\path_1}$, is given by
\begin{equation}\label{Anzahlformel}
\big\vert
\,\absof{\fpath\of{u_i,v_j}} +
\absof{\fpath\of{\reflect\of{u_i},v_j}}\,
\big\vert_{m\times m}.
\end{equation}
(As usual, $\absof{\cal A}$ denotes the number of elements of the set
$\cal A$.)
If $u_1$ lies {\em on\/} the line $y=x$ then $\reflect\of{u_1}$, 
the reflection of $u_1$ in $y=x$, is considered to be {\em
different\/}
from $u_1$. Thus in this case every $m$-tuple is counted twice.
}

\medskip
\noindent {\sc Remark.} For the $\la_1$-tuples that result from coarse
orthogonal tableaux the point $u_1$ is $(0,0)$ which lies on $y=x$.
Hence in this case the last remark applies and therefore the actual
number that we are interested in is the determinant
(\ref{Anzahlformel}) divided by two. This two can be cancelled in the
first row which consists of entries of the form
$2\absof{\fpath\of{u_1,v_j}}$. Thus (\ref{thm:weak-e-orthogonal}) is
established once the theorem is proved.

\medskip
\proof
With $\reflect$ being the reflection in the line $y=x$,
we expand the determinant,
\begin{align}\notag
&\big\vert{\,
\absof{\fpath\of{u_i,v_j}} + \absof{\fpath\of{\reflect\of{u_i},v_j}}
\,}\big\vert_{m\times m}\\
\notag
&\hskip2cm={
{\sum _{\si\in\SG m,\ \ep\in\{1,-1\}^m}} ^{}}\sgn\of\si\prod _{i,\ep_i=1} ^{}
\absof{\fpath\of{u_i,v_{\si(i)}}}\prod _{i,\ep_i=-1} ^{}
\absof{\fpath\of{\reflect\of{u_i},v_{\si(i)}}}\\
\notag &\hskip2cm=
   \underset{\bold\path\in\fpath\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
        {\sum_{\si\in\SG{m},\ \ep\in\{1,-1\}^m}} 
          ^{}\sgn\of\si.
\end{align}
(The meaning of $\bold u^{(\ep)}$ is that of section~\ref{Defs}.)
We have expressed the determinant as a weighted sum over
$m$-tuples $\bold\path=(\path_1,\dots,\path_{m})$ of lattice
paths, where $\path_i$ runs from either $u_i$ or $\reflect(u_i)$ 
to $v_{\si(i)}$, $i=1,2,\dots,m$.
We shall give a sign-reversing involution that cancels all $m$-tuples
$(\path_1,\dots,\path_m)$ that are either intersecting or contain two
paths $\path_i,\path_j$, $i\ne j$, such that $\path_i$
intersects $\reflect\of{\path_j}$. 
Suppose that this had been done. The remaining
$m$-tuples are nonintersecting and for all pairs $\path_i,\path_j$
the path $\path_i$ and the reflected path $\reflect\of{\path_j}$ do
not intersect. In particular, the paths $\path_i$, $i\ge 2$, neither
meet $\path_1$ nor $\reflect\of{\path_1}$. A moment's thought shows
that, because of the order of the end points, this implies that the
end point of $\path_1$ must be $v_1$. Since for $l\ge 2$ the end
points $v_l$ lie to the north-west of $v_1$ and $\reflect\of{v_1}$, 
the complete paths
$\path_i$, $i\ge 2$, must lie to the north-west of $\path_1$ and
$\reflect\of{\path_1}$, not intersecting both of them. Since also
$\path_2,\dots,\path_m$ are nonintersecting, because of the order of
the starting and end points, $\path_i$ must run from $u_i$ to $v_i$
for $i\ge 2$.
In other words, the remaining $m$-tuples are those for which $\si=\text
{id}$ and $\ep=(1,\dots,1)$ or $(-1,1,\dots,1)$, 
which are nonintersecting, and where
$\reflect\of{\path_1}$ 
does not meet $\path_i$ for $i\ge 2$. This would prove the
assertion.
 

%%%@@@ input from figfk
\begin{Figure}\label{FK}
\begin{picture}(20, 9)(0,0)
\put(1,1){
	\begingrid{-1}{-1}{7}{7}
		\thinlines
		\put(-1,-1){\line(1,1){8}}
		%labels
      \put(-2.2,4.6){$u_i$}
      \put(2.2,0.6){$u_j$}
      \hlabel{4}{1}{$\path_j$}
      \hlabel{0}{2}{$\reflect\of{u_j}$}
      \put(2.2,6.3){$\path_i$}
      \hlabel{3}{4}{$\hskip-4pt\reflect\of{\path_j}$}
      \hlabel{1}{5}{$\hskip3pt p$}
		\labelledfalse
		\uppath{-1}{5}{---|---}
		\updottedpath{1}{3}{|-|--||}
		\uppath{3}{1}{-|-||--}
		\put(2,5){\circle{0.4}}
		\put(5,2){\circle{0.4}}
	\endgrid
}
 
\put(11,4){$\longleftrightarrow$}

\put(13,1){
	\begingrid{-1}{-1}{7}{7}
      \thinlines
		\put(-1,-1){\line(1,1){8}}
      %labels
      \put(-2.2,4.6){$u_i$}
      \hlabel{3.2}{-1.5}{$\reflect\of{u_i}$}
      \put(5.3,0.4){$\reflect\of{\path_i'}$}
      \hlabel{0}{2}{$\reflect\of{u_j}$}
      \put(2,6.5){$\path_j'$}
      \hlabel{3}{4}{$\hskip-4pt{\path_i'}$}
      \hlabel{1}{5}{$\hskip3pt p$}
		\updottedpath{-1}{5}{-----||}
		\uppath{1}{3}{|-||---}
		\uppath{5}{-1}{|||||--}
		\put(2,5){\circle{0.4}}
		\put(5,2){\circle{0.4}}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfk

Now we define the involution. 
Given an $m$-tuple of paths,
look for the highest level $x+y=c$ that contains a point
$p$ where two paths $\path_i$ and $\path_j$, $i\ne j$, 
of the $m$-tuple intersect or 
where some $\path_i$ and $\reflect\of{\path_j}$, $i\ne j$, intersect.
If there is a point of (ordinary) intersection between two paths on
this level, then choose $(i,j)$, $i\ne j$, to be minimal
in lexicographic order such that $\path_i$ and 
${\path_j}$ intersect on this level and
 apply the Gessel-Viennot involution.
If not, then choose $(i,j)$, $i\ne j$, to be minimal
in lexicographic order such that $\path_i$ and 
$\reflect\of{\path_j}$ intersect on this level.
Now, as in the Gessel--Viennot involution, interchange terminal portions
of $\reflect\of{{\path}_j}$ and $\path_i$ beginning from $p$,
obtaining $\path_i^\prime$ and ${\path_j}^\prime$. Then
reflect back $\path_i^\prime$, thus obtaining $\reflect\of{\path_i'}$
and $\path_j'$. Figure~\ref{FK} illustrates this
operation.
Thus, from the original ${m}$-tuple of paths we obtain a new ${m}$-tuple
by replacing $\path_i$ by $\reflect\of{\path_i'}$ and $\path_j$ by
$\path_j'$.
What actually happens is that, beginning from the meeting point $p$,
respectively its reflection (both points are circled in
Figure~\ref{FK}), terminal portions of paths are
interchanged, while, up to $p$, respectively its reflection, initial
portions are reflected. For later use, let us call this operation {\em
re\-flec\-tion-Ges\-sel-Vien\-not mapping\/}.

The above described mapping is sign-reversing since
the associated permutations differ by the transposition $(i,j)$. 
Finally, the mapping is an involution since the
Gessel--Viennot involution is and since the reflection-Gessel-Viennot
mapping does not change anything on the level $x+y=c$ or on higher
levels and so does not introduce points of (ordinary) 
intersection between paths there. Hence, renewed application of the
mapping will give back the original $m$-tuple.
\QED 

\bigskip

%%%@@@ input from orth
{\sc Bijective proof of the orthogonal dual Jacobi--Trudi identity
(\ref{thm:e-orthogonal})}.
Things are a bit more complicated in the (non-coarse) orthogonal case.
First, because now we have to take care of weights,
we have to use the ``modified'' reflection
that we already used in the symplectic case.
Second, only (approximately) half of the orthogonal conditions hold,
while the odd cases are replaced by protection or/and broken strip
conditions. Actually, all 4 different fine orthogonal tableaux that
were defined in section~\ref{Tableaux} could be used for a bijective
proof of the orthogonal dual Jacobi--Trudi identity. Since the proofs
are very similar, we give all the details only for the orthogonal
tableaux of the first kind and sketches of proofs for the tableaux of
second, third and fourth kind.

\medskip
{\em By fine orthogonal tableaux of the first kind\/}.
First we translate the $N$-orthogonal tableaux of the first kind into the
lattice path language. 
Given an $N$-orthogonal tableau $\tableau$ of the first kind which is of
shape $\lambda=(\la_1,\dots,\la_r)$, where $\la$ is an $N$-orthogonal
partition,
in the same way as for coarse orthogonal tableaux we map it to a $\la_1$-tuple
$(\path_1,\dots,\path_{\la_1})$ of lattice paths, where $\path_i$
runs from $(-i+1,i-1)$ to $(\lambda_i'-i+1,N-\lambda'_i+i-1)$, 
$i=1,2,\dots,\lambda_1$, by
reading the path $\path_i$ from the $i$-th column of $\tableau$.
Figure~\ref{FT} depicts an example of this
correspondence where $\la=(4,4,3,1)$ and $N=8$.

%%%@@@ input from figft
\begin{Figure}\label{FT}
\begin{picture}(20,12)(0,0)
\put(13,4){$T$ =}
\put(15,2){
	\begintableau{4}{4}
		\row{133*}
		\row{2556}
		\row{378}
		\row{5}
	\endtableau
}
\put(15,0){$*=5\text { or }3$}
\put(1,1){
	\begingrid{-3}{0}{4}{9}
		\makexaxis
		\makexscale
		% Gerade y=x
      \thinlines
		\put(-1,-1){\line(1,1){6}}
      % Pfade
		\elabeltrue
		\labelledtrue
		\uppath{0}{0}{---|-|||}\hlabel{3}{0}{$P_1$}
		\uppath{-1}{1}{||-|-|-|}\hlabel{-1}{2}{$P_2$}
		\uppath{-2}{2}{||-|-||-}\hlabel{0}{6}{$P_3$}
		\uppath{-3}{3}{||||--||}\hlabel{-4}{6}{$P_4$}
		\updottedpath{-3}{3}{||-||-||}
		\labelledfalse
		\updottedpath{0}{0}{||-|-|--}\hlabel{2}{3}{$\mreflect\of{P_1}$}
		% levels
		\downskewline{-3}{7}{7}
		\downskewline{-2}{8}{7}
	\endgrid
}
\end{picture}
\end{Figure} 

%%%@@@ end input from figft

The last entry in the first row of this tableau must be 5 in order
that the tableau is orthogonal. (Otherwise the 6-th
protection condition (\ref{cond:fine}) would be violated.) Right
now the reader should ignore all dotted lines in Figure~\ref{FT} and
assume that $*=5$.


It is obvious that the $\la_1$-tuple of paths that results from an
$N$-orthogonal tableau $\tableau$ of the first kind
must be nonintersecting because
$\tableau$ is a tableau. Let $\mreflect$ denote the following
modified reflection in the line $y=x$. If $N$ is even, $\mreflect$ is
the modified reflection
that was explained in section~\ref{symplectic} (see Figure~\ref{FJ})
but in the line $y=x$ instead of $y=x-2$. If $N$ is odd, then on the
region below the line $x+y=N-1$
$\mreflect$ acts as the modified reflection in $y=x$, while it acts
as the usual reflection $\reflect$ between $x+y=N-1$ and $x+y=N$.
This definition holds for the remainder of this
section. 

It was shown at the beginning of this section
that the $2c$-th
orthogonal condition (\ref{cond:coarse}) translates into the condition that
the intersection point $p_1$ of the line $x+y=2c$ with the reflection
$\reflect\of{\path_1}$ of the first path in the line $y=x$ lies
strictly to the south-east of
the intersection point $p_2$ of $\path_2$ and the line $x+y=2c$.
Since in even points, i.e\@., in points $(x,y)$ with $x+y\equiv 0\pmod
2$,
the modified reflection agrees with the ordinary reflection, the
preceding statement is also true if we replace `reflection' by `modified
reflection'. (The same is true for points on level $x+y=N$ if $N$ is odd.)

It remains to see how the $2c$-th protection condition
translates into the lattice path language. Here one is advised to
come back to Figure~\ref{FT} with dotted lines included. If $*$ does
not equal 5 but something less, 3 say, then in the strip 
between the lines $x+y=4$ and $x+y=6$ we have
the situation of Figure~\ref{FM}.

%%%@@@ input from figfm
\begin{Figure}\label{FM}
\begin{picture}(10,8)(0,0)
\put(1,1){
	\begingrid{0}{0}{6}{6}
		\downskewline{0}{5}{5}
		\downskewline{1}{6}{5}
		\segment{1}{4}{|-}
		\segment{2}{3}{-|}
		\segment{3}{2}{-|}
		\dottedsegment{4}{1}{-|}
      \hlabel{5}{0}{$\mreflect\of{\path_1}$}
	\endgrid
}
\end{picture}
\end{Figure}

%%%@@@ end input from figfm
\noindent
That is, in the strip between these two lines the (modified) reflected
first path exhibits an east-north turn,
there are 
(possibly) some more paths, also exhibiting an east-north turn, 
differing from its predecessor by a shift
in direction $(-1,1)$, with the
exception of the last which exhibits a north-east turn whose even
points also differ from the even points of the preceding east-north
turn by a shift in direction $(-1,1)$. 
(Note that violations of the $2c$-th protection condition
might very well look different from the situation described above, but
then would already form a violation of the $2c$-th
{\em orthogonal\/} condition.) So we have
shown that $N$-orthogonal tableaux of the first kind and 
of shape $\la$ correspond to
$\la_1$-tuples $(\path_1,\dots,\path_{\la_1})$ of nonintersecting
lattice paths, where $\path_i$ runs from $(-i+1,i-1)$ to 
$(\lambda_i'-i+1,N-\lambda'_i+i-1)$, $i=1,2,\dots,\lambda_1$, 
where for all $c$ the meeting of
$\mreflect\of{\path_1}$ and $x+y=2c$ lies strictly to the south-east
of the meeting of $\path_2$ with $x+y=2c$, and where for all $c$
in the region
between the lines $x+y=2c-2$ and $x+y=2c$ we never have the situation
that is exemplified in Figure~\ref{FM}.

The weight that we consider here
is basically the same as in the symplectic ``$e$-case'', namely, 
the weight of a horizontal edge with $e$-label (cf\@. Figure~\ref{FF}) 
$2i$ is defined to be $x_i^{-1}$, and the weight of
a horizontal edge with $e$-label $2i-1$ is defined to be $x_i$
except that the weight of a horizontal edge with $e$-label $N$ is
defined to be 1 if $N$ is odd. Obviously, for $N$ being even or odd,
this makes 
the correspondence depicted in Figure~\ref{FT} 
weight-preserving with respect to the weight (\ref{weight:orthogonal}). 

Let $u_i=(-i+1,i-1)$ and $v_i=(\lambda_i'-i+1,N-\lambda'_i+i-1)$,
$i=1,\dots,\la_1$ and let $\reflect$ be the reflection in
$y=x$.
We expand the determinant in (\ref{thm:e-orthogonal}),
\begin{align}\label{eq:e-orthogonal-expansion}
&\detof{\lambda_1}{
	\firstrow{\esf_{\lambda^\prime_j-j+1}\of{\bold{x}}}
	\esf_{\lambda_j^\prime-j+i}\of{\bold{x}} +
		\esf_{\lambda_j^\prime-j-i+2}\of{\bold{x}}}  \\
\notag &\hskip1cm=\detof{\lambda_1}{
	\firstrow{\GF{\fpath\of{u_1,v_j}}}
	\GF{\fpath\of{u_i,v_j}} +
		\GF{\fpath\of{\reflect\of{u_i},v_j}}}  \\
\notag &\hskip1cm=	\underset{
                \epsilon\in\{1\}\times\{1,-1\}^{\lambda_1-1}}
		{\sum_{\sigma\in \SG{\lambda_1}}} 
	\sgn\of{\sigma}
\prod_{i,\epsilon_i=1}\GF{\fpath\of{u_i,v_{\si(i)}}}\cdot
\prod_{i,\epsilon_i=-1}\GF{\fpath\of{\reflect\of{u_i},v_{\si(i)}}}\\
\notag &\hskip1cm=	\underset{\bold\path\in
        \fpath\of{\bold u^{(\ep)},\bold v_\si}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1\}\times\{1,-1\}^{\lambda_1-1}}} 
	\sgn\of{\sigma} w(\bold\path),
\end{align}
which is thereby written as a generating function for $\lambda_1$-tuples
$(\path_1,\dots,\path_{\la_1})$ of paths where $\path_i$ runs from
either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, $i=1,2,\dots,\lambda_1$.
Again, the notation $\bold u^{(\ep)}$ is from section~\ref{Defs}.
Note that here, in contrast to Theorem~1, we do not
consider reflections of $u_1$. 

We have to construct a weight-preserving and sign-reversing involution
such that after the cancellation
only those $\la_1$-tuples $\bold\path=(\path_1,\dots,\path_{\la_1})$ 
survive which correspond to $N$-orthogonal tableaux of the first kind.
In order to make this more precise, we introduce a few terms. 
We say that $\bold\path$
exhibits a {\em reflected intersection\/} in point $p$ if there are two paths
$\path_i$ and $\path_j$, $i\ne j$, such that $\mreflect\of{\path_i}$ and
$\path_j$ intersect in $p$.
We say that $\bold\path$
exhibits an {\em odd reflected crossing\/} in point $p$ if there are two paths
$\path_i$ and $\path_j$ such that $\mreflect\of{\path_i}$ and
$\path_j$ cross in an odd point $p$ (i.e\@., in a point $p=(x,y)$ with
$x+y\equiv 1\pmod 2$), in
the way that is exemplified in the left picture in
Figure~\ref{fig:odd-violations1}.

%%%@@@ input from figov1
\begin{Figure}\label{fig:odd-violations1}
\begin{center}{\small odd reflected crossings:}\end{center}
\begin{picture}(24,8)(0,0)
\put(1,1){
	\begingrid{0}{1}{8}{5}
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		\thicklines
%		\put(3,0.1){$\scriptstyle y=x$}
		\put(1.6,4.5){$\path_j$}
		\put(-1.4,3){$\mreflect(\path_i)$}
      
		\labelledfalse
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		\segment{2}{2}{||}
		\drawrect{0.5}{1.5}{3}{3}
 		\thickpoint{2}{3}
		\put(2.3,2.4){$p$}
		
	\endgrid
}
\put(13,0){
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		\thicklines
		\put(7,4.1){$\scriptstyle y=x$}
		\put(1.6,4.5){$\path_j$}
		\put(-1.4,3){$\mreflect(\path_i)$}
		\put(4.6,1.5){$\path_i$}
		\put(6.4,0){$\mreflect(\path_j)$}
      
		\labelledfalse
		\dottedsegment{1}{3}{--}
		\segment{2}{2}{||}
		\dottedsegment{4}{0}{--}
		\segment{5}{-1}{||}
%		\drawrect{0.5}{1.5}{3}{3}
		
	\endgrid
}


\end{picture}
\end{Figure}

%%%@@@ end input from figov1
\noindent
For later reference we remark that if there is an odd reflected
crossing of $\mreflect(\path_i)$ and $\path_j$ {\em above\/} $y=x$
then there is an odd reflected
crossing of $\mreflect(\path_j)$ and $\path_i$ {\em below\/} $y=x$,
and vice versa, see the right picture in
Figure~\ref{fig:odd-violations1}. We call an odd reflected crossing
{\em special\/} if the situation is as exemplified in
Figure~\ref{fig:special-odd-reflected-violation}. That is, the odd
reflected crossing is located {\em above\/} $y=x$, and in the strip
between lines $x+y=2c-2$ and $x+y=2c$ containing this odd reflected
crossing there are (possibly) some more paths, exhibiting an
east-north turn, each differing from its predecessor by a shift
in direction $(-1,1)$, the turning point of the first differing from
the odd reflected crossing also by a shift in direction $(-1,1)$, and
then follows a point (marked by a circle in
Figure~\ref{fig:special-odd-reflected-violation}) that is not occupied by
any path and differs from the turning point of the last east-north
turn again by a shift in direction $(-1,1)$. It will be important
later to observe that an odd reflected crossing in the strip between lines
$x+y=2c-2$ and $x+y=2c$ can only be {\em non-special\/} if there are
(ordinary) intersections or even reflected intersections in this
strip. This is straight-forward for odd reflected crossings 
{\em above\/} $y=x$. If we encounter an odd reflected crossing {\em
below\/} $y=x$,
we just have to remember that this is equivalent to an odd reflected
crossing {\em above\/} $y=x$, so we are in the same case again.
\begin{Figure}\label{fig:special-odd-reflected-violation}
\begin{center}\small special odd reflected crossing:\end{center}
\begin{picture}(28,11)(0,0)
\put(8,1){
	\begingrid{-1}{1}{9}{9}
		\labelledfalse
		\downskewline{-1}{8}{7}
		\downskewline{0}{9}{8}
		\thinlines
		\put(1,0){\line(1,1){8}}
		\thicklines
		\put(2,0.1){$\scriptstyle y=x$}
		\put(1,8.2){$\path_{i_h}$}
		\put(3,6.2){$\path_{i_2}$}
		\put(4.3,5){$\mreflect\of{\path_{i_1}}$}

		\thickcircle{0}{8}
		
		\segment{0}{7}{-|}
		\segment{1}{6}{-|}
		\segment{3}{4}{||}
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		\thickpoint{3}{5}
		\put(3.3,4.8){$p$}
		
		
	\endgrid
}



\end{picture}
\end{Figure}



We say that $\bold\path$
exhibits an {\em odd reflected violation\/} in point $p$ if there are paths
$\path_{i_1},\path_{i_2},\dots, \path_{i_h}$, $h\ge 2$,
such that in some region between lines $x+y=2c-2$ and $x+y=2c$ and
{\em above\/} $y=x$ (this means that the even points of the reflected
path $\mreflect\of{\path_{i_1}}$ have to be above or on $y=x$)
the paths $\mreflect\of{\path_{i_1}},\path_{i_2},\dots,\path_{i_h}$ 
exhibit the situation that is exemplified in 
Figure~\ref{fig:odd-violations2} (compare Figure~\ref{FM}). 
The ``reference point" $p$, which is for later
use, is marked by a bold dot.

%%%@@@ input from figov2
\begin{Figure}\label{fig:odd-violations2}
\begin{center}\small odd reflected violation:\end{center}
\begin{picture}(28,11)(0,0)
\put(8,1){
	\begingrid{-1}{1}{9}{9}
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		\put(-1,0){\line(1,1){9}}
		\thicklines
		\put(0,0.1){$\scriptstyle y=x$}
		\put(1,8.2){$\path_{i_h}$}
		\put(3,6.2){$\path_{i_2}$}
		\put(4.3,5.2){$\mreflect\of{\path_{i_1}}$}
		
		\segment{0}{7}{|-}
		\segment{1}{6}{-|}
		\segment{2}{5}{-|}
		\dottedsegment{3}{4}{-|}
		\drawrect{-0.5}{3.5}{5}{5}
		\thickpoint{3}{5}
		\put(3.3,4.8){$p$}
		
		
	\endgrid
}



\end{picture}
\end{Figure}

%%%@@@ end input from figov2

We shall give a weight-preserving and sign-reversing involution that
cancels all $\la_1$-tuples $(\path_1,\dots,\path_{\la_1})$ 
that are either intersecting, or exhibit a
reflected intersection {\em in an even point\/} (i.e\@., in a point $(x,y)$ with
$x+y\equiv 0\pmod 2$),
 or a special odd reflected crossing, or an odd
reflected violation. 
(We shall call any of these situations sometimes simply a
``violation'' for short.) 
Suppose that this had
been done. The remaining 
$\la_1$-tuples are nonintersecting. The modified
reflection $\mreflect\of{\path_1}$ does not touch the path
$\path_2$ in even points. In particular, $\mreflect\of{\path_1}$ and
$\path_2$ cannot {\em cross\/} in even points.
Moreover, $\mreflect\of{\path_1}$ and $\path_2$ cannot cross in an odd
point. For, either this is already an odd reflected crossing (cf\@.
Figure~\ref{fig:odd-violations1}), or it is a crossing as exemplified
in Figure~\ref{fig:not-odd-reflected-crossing}. In the latter case there
has to be an odd reflected crossing 
%Christian
or an even reflected intersection of $\mreflect\of{\path_1}$ and
$\path_2$ ``later", because, by 
%Christian
$N$-orthogonality of $\la$, the end point of
$\path_2$, which is $u_2$, lies in the north-west of the end point of 
$\mreflect\of{\path_1}$, which is $\reflect\of{u_1}$. But, an odd
reflected crossing is either special or, if non-special, implies an
(ordinary) intersection or an even reflected intersection, as we observed
before. 
Hence, $\mreflect\of{\path_1}$ and $\path_2$ cannot 
cross at all and do not even touch in even points and on the line
$x+y=N$. Besides, 
$\mreflect\of{\path_1},\path_2,\dots,\path_h$ in any region between
lines $x+y=2c-2$ and $x+y=2c$ cannot exhibit the situation that is
exemplified in Figure~\ref{fig:odd-violations2}. 

\begin{Figure}\label{fig:not-odd-reflected-crossing}
\begin{center}{\small not an odd reflected crossing:}\end{center}
\begin{picture}(23,8)(0,0)
\put(7,1){
	\begingrid{0}{1}{8}{5}
      \downskewline{3}{5}{4}
      \downskewline{2}{4}{4}
		\thinlines
%		\put(2,0){\line(1,1){5}}
		\thicklines
%		\put(3,0.1){$\scriptstyle y=x$}
		\put(3,4.5){$\mreflect(\path_1)$}
		\put(2.1,3){$\path_2$}
      
		\labelledfalse
		\segment{3}{3}{--}
		\dottedsegment{4}{2}{||}
		\drawrect{2.5}{1.5}{3}{3}
		
	\endgrid
}



\end{picture}
\end{Figure}

Summarizing, by essentially the same arguments as in the proof of 
Theorem~1, what this tells 
is that only $\la_1$-tuples remain for which 
$\si=\text{id}$ and $\ep=(1,\dots,1)$, which are nonintersecting,
where $\mreflect\of{\path_1}$ is always strictly to the south-east of $\path_2$
on even levels (i.e\@., on lines of the form $x+y=2c$), 
and where for $\mreflect\of{\path_1},\path_2,\dots,\path_h$ in any
region between $x+y=2c-2$ and $x+y=2c$ never 
the situation that is
exemplified in Figure~\ref{FM}.
occurs.
These are exactly those $\la_1$-tuples
which correspond to $N$-orthogonal
tableaux of the first kind. This would prove (\ref{thm:e-orthogonal}).
\medskip

Now we define the involution.
Given a $\la_1$-tuple
$(\path_1,\path_2,\dots,\path_{\la_1})$ that has to be cancelled, 
we look for the highest level $x+y=c$
containing an (ordinary) intersection between two paths or
an even reflected intersection or a special odd reflected crossing
or an odd reflected 
violation in some point, respectively reference point, $p$. 
If there is a point of (ordinary) intersection between two paths on
this level, then choose $(i,j)$, $i\ne j$, to be minimal
in lexicographic order such that $\path_i$ and 
${\path_j}$ intersect on this level and
 apply the Gessel-Viennot involution.
If not, then we basically apply the  
reflection-Gessel-Viennot mapping from the proof of Theorem~1
except that ordinary reflection is replaced by {\em modified\/}
reflection (because we have to preserve the weight). However, the
details are a little bit delicate.

To be precise, let $(i_1,i_2)$ be minimal in lexicographic order
such that $\path_{i_1}$ and $\path_{i_2}$ are the paths that are
``involved" in such an even reflected intersection, special odd reflected
crossing, or odd reflected violation on that level. 
(``Involved" has an obvious
meaning for even reflected intersections.
For special odd reflected crossings and 
odd reflected violations $\path_{i_1}$ and $\path_{i_2}$ are as in
the definition of special odd reflected crossings, see
Figure~\ref{fig:special-odd-reflected-violation}, respectively 
odd reflected violations, see Figure~\ref{fig:odd-violations2}.)
If we are considering an even reflected intersection then we carry out
the reflection-Gessel-Viennot mapping straight-forwardly, with
ordinary reflection replaced 
by modified reflection. If we are considering a special
odd reflected crossing or an odd reflected violation we apply a
further modification. 

Let us start with an odd reflected violation with reference point $p$.
See the left-most picture in
Figure~\ref{fig:crossings-to-protection-violations}. First flip 
the east-north turn of $\mreflect\of{\path_{i_1}}$ into a north-east
turn, and the north-east turn of $\path_{i_h}$ into a east-north turn 
(see the middle picture of
Figure~\ref{fig:crossings-to-protection-violations}), thus obtaining the paths 
$\path_{i_1}'$ and $\path_{i_h}'$. We have introduced a 
meeting point between ${\path_{i_1}'}$ and $\path_{i_2}$ (which could
also be $\path'_{i_h}$ if $h=2$). 
Now interchange terminal portions of ${\path_{i_1}'}$ and
$\path_{i_2}$, thus obtaining ${\path_{i_1}''}$ and $\path_{i_2}'$
(see the right-most picture in
Figure~\ref{fig:crossings-to-protection-violations}). 
Finally apply the modified reflection to $\path_{i_2}'$, thus
obtaining $\mreflect(\path_{i_2}')$. 
From the original $\la_1$-tuple by this
procedure we obtain another $\la_1$-tuple where $\path_{i_1}$ is replaced
by $\path_{i_1}''$, 
$\path_{i_2}$ by $\mreflect(\path_{i_2}')$, and $\path_{i_h}$
by $\path_{i_h}'$.


%%%@@@ input from figctpv
\begin{Figure}
\label{fig:crossings-to-protection-violations}
\begin{picture}(28,9)(0,0)

\put(18,0){
	\begingrid{-1}{3}{5}{10}
	\gridcaption{\small special odd reflected crossing:}
	\downskewline{-1}{9}{5}
	\downskewline{0}{10}{5}
	\thinlines
	\put(1,3){\line(1,1){5}}
	\thicklines
	\put(2,3.1){$\scriptstyle y=x$}
	\labelpointbl{5}{6}{\textstyle{\path'_{i_2}}}
	\labelpointar{3}{7}{\textstyle\path''_{i_1}}
	\labelpointar{1}{9}{\textstyle\path'_{i_h}}
	\thickpoint{3}{6}
	\labelpointbl{3}{6}{\textstyle p}
	\labelpointal{0}{9}{\textstyle q}
	\put(0,9){\circle{0.4}}
	
	\segment{0}{8}{-|}
	\segment{1}{7}{-|}
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	\endgrid
	}

\put(7.5,4){$\leftrightarrow$}

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	\begingrid{-1}{3}{5}{10}
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	\put(1,3){\line(1,1){5}}
	\thicklines
	\put(2,3.1){$\scriptstyle y=x$}
	\labelpointbl{4}{6}{\textstyle{\path'_{i_1}}}
	\labelpointar{3}{7}{\textstyle\path_{i_2}}
	\labelpointar{1}{9}{\textstyle\path'_{i_h}}
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\put(16.5,4){$\leftrightarrow$}

\put(0,0){
	\begingrid{-1}{3}{5}{10}
	\gridcaption{\small odd reflected violation:}
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	\thicklines
	\put(2,3.1){$\scriptstyle y=x$}
	\labelpointbl{5}{5}{\textstyle\mreflect\of{\path_{i_1}}}
	\labelpointar{3}{7}{\textstyle\path_{i_2}}
	\labelpointar{1}{9}{\textstyle\path_{i_h}}
	\thickpoint{3}{6}
	\labelpointbl{3}{6}{\textstyle p}
	
	\segment{0}{8}{|-}
	\segment{1}{7}{-|}
	\segment{2}{6}{-|}
	\dottedsegment{3}{5}{-|}
	
	\endgrid
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\end{picture}
\end{Figure}



%%%@@@ end input from figctpv

It should be noted that we end up with a $\la_1$-tuple that contains a
special odd
reflected {\em crossing\/} in $p$ (compare Figure~\ref{fig:odd-violations1}). 
The point $q$ is circled to indicate that it is not occupied by any
path.

On the other hand, if we start with a special 
odd reflected crossing in $p$, we
simply go in the reverse direction, i.e\@., we go from right to left in
Figure~\ref{fig:crossings-to-protection-violations}. All we have to do
is to determine the point $q$, which is simply the ``first" point on
the same level as $p$ that is not occupied by any path.

For later use, let us call the mapping defined for even reflected 
intersections, special odd reflected crossings, and odd reflected violations
{\em modified-reflection-Gessel-Viennot mapping\/}. It is obvious
that this mapping is weight-preserving and sign-reversing, the latter
because in any case the associated permutations differ by the
transposition $(i_1,i_2)$. It is also an involution since 
we do not introduce new ``violations"
of paths on or above the level that we are considering, as is not
difficult to see.

Thus the construction of our
weight-preserving and sign-reversing involution is complete, which finishes
the proof. \QED
\medskip

{\em By fine orthogonal tableaux of the second kind\/}.
Note that the orthogonal tableaux of the first kind differ
from the ones of the second kind only in that ``most'' of
the protection conditions are replaced by broken strip conditions.
So it is easy to set up a weight-preserving  bijection between these
two types of orthogonal tableaux. Given an orthogonal tableau of the
first kind, we look for all violations of some broken strip condition.
It is not hard to see that in lattice path language they look precisely
as depicted in
Fig.~\ref{fig:broken-strip-violation}. Note that the even points on the
reflected kink must lie {\em strictly\/} above the line $y=x$ to be in
accordance with the definition of the broken strip condition. 

%%%@@@ input from figbsv
\begin{Figure}
\label{fig:broken-strip-violation}
\begin{picture}(29,5)(0,0)

\put(12,0){
	\begingrid{0}{0}{4}{4}
	\downskewline{0}{3}{3}
	\downskewline{1}{4}{3}
	\segment{1}{2}{-|}
	\dottedsegment{2}{1}{|-}
	\endgrid
	}

\end{picture}
\end{Figure}

%%%@@@ end input from figbsv
The situation of Fig.~\ref{fig:broken-strip-violation} appears in the
%Christian
intermediate 
picture in Fig.~\ref{fig:crossings-to-protection-violations} (if we
forget about $y=x$). So we can
transform it into an odd reflected violation {\em strictly above\/}
the line $y=x$ by  the 
operation that leads from the intermediate picture to the left picture in 
Fig.~\ref{fig:crossings-to-protection-violations}, and vice
versa. 
(The case
where these even points on the reflected kink
actually {\em would lie\/} on line $y=x$ precisely
corresponds to the case of the single protection condition that is {\em not\/}
replaced by a broken strip condition.)
This gives a weight-preserving bijection between the set of all
orthogonal tableaux of the first kind and orthogonal tableaux of the second 
kind. This finishes the proof. \QED

The same weight-preserving bijection (formulated in the ``language of
tableaux'', without employing a lattice path interpretation) is given in
\cite[App.~A, p.~276f]{King-Welsh:Orthogonal}. Note that the above operation
mapping odd reflected violations to broken strip violations is not feasible
in situations where the even points of the reflected kinks lie on line
$y=x$, since then it would introduce an intersection of the second path with
the first path itself (which coincides with its own reflection in the
relevant strip in this
case).

\medskip
{\em By fine orthogonal tableaux of the third kind\/}.
The definition of orthogonal tableaux of the third kind is very
similar to the definition of orthogonal tableaux of the first kind.
Therefore we can more or less copy the proof by orthogonal tableaux of
the first 
kind. At each
occurrence,
 $2c$ has to be replaced by $2c+1$, and the
meaning of ``modified reflection" has to be adjusted. Namely, now all
odd points (i.e\@., points $(x,y)$ with $x+y\equiv1\pmod 2$) are
reflected in the usual way and so are all even points 
(i.e\@., points $(x,y)$ with $x+y\equiv0\pmod 2$) whose adjacent steps
are both vertical or both horizontal, while a ``kink" in an even
point is shifted accordingly (compare with Figure~\ref{FJ}). Besides,
steps between the lines $x+y=0$ and $x+y=1$ are reflected in the
usual way.
\QED

\medskip
{\em By fine orthogonal tableaux of the fourth kind\/}.
The definition of orthogonal tableaux of the fourth kind is very
similar to the definition of orthogonal tableaux of the second kind.
Therefore we can map them by a weight-preserving bijection to
orthogonal tableaux
of the third kind in much the same way as we mapped orthogonal
tableaux of the second 
kind to orthogonal tableaux of the first kind.
\QED

Naturally, the question arises why we do not need any protection
conditions here (as was the case for tableaux of the second kind). The
reason for this is that there are no odd points on the line $y=x$; and
odd levels play the role that even levels played for tableaux of the
second kind. Hence {\em all\/} odd reflected violations can be converted
into broken strip violations.

%%%@@@ end input from orth

\bigskip
{\sc Bijective proof of the orthogonal Jacobi--Trudi identity
(\ref{thm:h-orthogonal})}.
Let $\tableau$ be an $N$-orthogonal tableau (of first, second, third,
or fourth kind, respectively) of shape $\la$, where $\la$ is an
$N$-orthogonal partition, i.e\@., $\la_1'+\la_2'\le N$. As in
section~\ref{Lattice} (see Figure~\ref{FE}), by reading paths from
the rows and using the $h$-labelling (see Figure~\ref{FF}) we map
$\tableau$ to an $r$-tuple $(\path_1,\dots,\path_r)$ of
nonintersecting lattice paths, where $\path_i$ runs from $u_i=(-i,1)$
to $v_i=(\la_i-i,N)$, $i=1,2,\dots,r$. The orthogonal, protection, and broken strip
conditions translate into rather complicated constraints on the
heights of horizontal steps of these paths. We do not need to make
this precise since, as in the proof of the symplectic Jacobi--Trudi
identity, we take a detour and pass from ``$h$-paths" to
``$e$-paths" by applying the concept of {\em dual lattice paths\/} due
to Gessel and Viennot
\cite[sec.~4]{Gessel-Viennot:Binomial-determinants}. This allows us to rely
on the bijections for the orthogonal dual Jacobi--Trudi identity,
which were presented just before.

To complete the lattice path interpretation of the left-hand side of
(\ref{thm:h-orthogonal}), we address the weights of orthogonal tableaux and
lattice paths. Since the weight definition (\ref{weight:orthogonal}) for
fine orthogonal tableaux of the first and second kind differs from the
weight definition (\ref{weight:orthogonalodd}) for fine orthogonal
tableaux of the third and fourth kind, we have to split the definition
of edge weights. If we started with an orthogonal tableau of the
first or second kind, we define the weight of a horizontal edge with
$h$-label $2i$ to be $x_i^{-1}$, and the weight of horizontal edge with
$h$-label $2i-1$ to be $x_i$, except that 
the weight of a horizontal edge with $e$-label $N$ is
defined to be 1 if $N$ is odd. This makes the above correspondence
weight-preserving with respect to (\ref{weight:orthogonal}). 
If we started with an orthogonal tableau of the
third or fourth kind, we define the weight of a horizontal edge with
$h$-label $1$ to be $1$, the weight of a horizontal edge with
$h$-label $2i$ to be $x_i$, and the weight of horizontal edge with
$h$-label $2i+1$ to be $x_i^{-1}$.
This makes the above correspondence
weight-preserving with respect to (\ref{weight:orthogonalodd}).

Now we expand the determinant in (\ref{thm:h-orthogonal}) and find
its lattice path interpretation. In the following computation
$\reflect$ is the reflection in the line $x=0$.
\begin{align}\label{eq:h-orthogonal-expansion}
\notag
 &\detof{\lambda_1}{\hsf_{\lambda_j-j+i}\of{\bold{x}} -
		\hsf_{\lambda_j-j-i}\of{\bold{x}}}=
\detof{\lambda_1}{\GF{\fpath\of{u_i,v_j}}-\GF{\fpath\of{\reflect(u_i),v_j}}}\\
&\notag  
\hskip0.1cm=	{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1,-1\}^{\lambda_1}}} 
	\sgn\of{\sigma}
	\sgn\of{\epsilon}
\prod_{i,\epsilon_i=1}\GF{\fpath\of{u_i,v_{\si(i)}}}\cdot
\prod_{i,\epsilon_i=-1}\GF{\fpath\of{\reflect(u_i),v_{\si(i)}}}\\
\notag
&\hskip0.1cm=	\underset{\bold \path\in
        \fpath\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                        \epsilon\in\{1,-1\}^{\lambda_1}}}
	\sgn\of{\sigma}
	\sgn\of{\epsilon}
	w(\bold \path),
\end{align}
where again $\sgn\of\epsilon$ stands for $\prod _{i=1} ^{\lambda_1}\epsilon_i$
and the notation $\bold u^{(\epsilon)}$ is from section~\ref{Defs}.
The weight $w(\bold \path)$ is induced (as explained in section~\ref{Defs}) 
by the edge weight that was introduced 
just before. Thus the determinant
in (\ref{thm:h-orthogonal}) is written as a generating function for
$r$-tuples $\bold\path=(\path_1,\dots,\path_r)$ of lattice paths where $\path_i$
starts at either $u_i$ or $\reflect(u_i)$ and runs to
$v_{\sigma(i)}$, $i=1,2,\dots,r$, for some permutation $\si$. We have
to give a weight-preserving and sign-reversing involution that
cancels all those $r$-tuples of paths that either contain a path in a
reflected starting point, or are intersecting, or (if $\si=\text
{id}$, $\ep=(1,1,\dots,1)$) where the tableau corresponding to
$\bold\path$ violates one of the relevant orthogonal, protection, or
broken strip conditions.

Let $\bold\path=(\path_1,\dots,\path_r)$ be one of these $r$-tuples
of paths in $\fpath\of{\bold u^{(\ep)},\bold v_\si}$ which have to be
cancelled. 
If $\bold\path$ is intersecting we apply the Gessel--Viennot
involution. If $\bold\path$ is nonintersecting,
then in the first step we determine the {\em dual} of $\bold\path$,
compare with 
Figure~\ref{Fdual}. It can be seen that here it is a
$\la_1$-tuple $\bold Q=(Q_1,\dots,Q_{\la_1})$ of
paths, where $Q_i$ runs from either $\overline u_i=(-i+1,i-1)$ or
$\overline\reflect(\overline u_i)$ to $\overline v_{\si(i)}$,
$\overline v_j$ being given by $\overline
v_j=(\la_j'-j+1,N-\la_j'+j-1)$, for some
permutation $\si\in\SG{\la_1}$. In this case $\overline\reflect$ denotes the
reflection in the line $y=x$. In particular, it should be observed
that the first path, $Q_1$, {\em has\/} to start in $\overline
u_1=(0,0)$.
(This is an implication of the fact that no path of $\bold\path$
starts in $(0,0)$.) So the duals are exactly the
nonintersecting objects 
that are involved in the bijective proof of the orthogonal {\em dual}
Jacobi--Trudi identity.

Now we want to apply one of the bijections (depending on which type of
tableaux we want to base our proof) that proved the orthogonal dual
Jacobi--Trudi identity. However, the same problem as in the proof of
the symplectic Jacobi--Trudi identity appears as this may introduce
new points of intersections of paths. Again, this cannot be
permitted, since dualization does not work with intersecting paths.
To come around this problem we use the same trick as in the proof of
the symplectic Jacobi--Trudi identity.
Let $\bold A$ denote
the appropriate mapping involving the
``modified-reflection-Gessel-Viennot mapping" (cf\@.
Fig.~\ref{fig:crossings-to-protection-violations}) used in the proof of
the orthogonal dual Jacobi--Trudi identity, and let $\bold B$ denote
the Gessel--Viennot involution. Again, starting with $\bold Q$, apply
$\bold A$, then $\bold B$, then $\bold A$ again, etc., as long as
possible. By the same arguments as in the proof of the
symplectic Jacobi--Trudi identity 
it is seen that the algorithm terminates.
Since $\bold A$ (the mapping involving the
``modified-reflection-Gessel-Viennot mapping" in $y=x$) can always be applied to these
objects, the algorithm has to stop with an application of $\bold A$.
So we finally obtained some {\em nonintersecting\/}
$\la_1$-tuple, $\bold Q'$ say. In the
last step we determine the preimage of $\bold Q'$ under the
dualization procedure, ${\bold\path'}$ say. That the map from
$\bold\path$ to ${\bold\path'}$ is
weight-preserving is obvious. That it is also sign-reversing is
checked in 
much the same way as it is done in the proof of the symplectic
Jacobi--Trudi identity.
\QED





\section{Orthogonal identities: Sundaram's tableaux}\label{Sundaram}
Sundaram's tableaux are not too far from symplectic tableaux.
Therefore it is possible to modify the bijections of
section~\ref{symplectic} in order to be applicable for Sundaram's
tableaux. Let $\la$ be a partition with $\ell(\la)=r\le n+1$.
Throughout this section $\bold x$ denotes the sequence
$\seqof{x_1,x_1^{-1},\dots,x_n,x_n^{-1},1}$.
The identities which result from lattice path interpretations of
Sundaram's tableaux are: A variant (see 
\cite[Prop.~24.44]{Fulton-Harris:Representation-Theory}) of the odd
orthogonal $h$-formula ((\ref{thm:h-orthogonal}) for $N=2n+1$)
\begin{equation}
\label{thm:h-odd-orthogonal}
\oschf{2n+1}{\lambda} =
\detof{r}{
        \firstrow{\hsf'_{\lambda_j-j+1}\of{\bold{x}}}
	\hsf'_{\lambda_j-j+i}\of{\bold{x}} +
		\hsf'_{\lambda_j-j-i+2}\of{\bold{x}}},
\end{equation}
where $\hsf'_m\of{\bold x}=\hsf_m\of{\bold x}-\hsf_{m-2}\of{\bold x}$,
a variant of the odd orthogonal $e$-formula ((\ref{thm:e-orthogonal})
for $N=2n+1$)
\begin{multline}\label{thm:e-odd-orthogonal}
\oschf{2n+1}{\lambda}\\
 =\detof{\lambda_1}{\sum_{k\geq 0}\esf_{\lambda_j'-j+i-k}\of
        {x_1,x_1^{-1},\dots,x_n,x_n^{-1}} -
	\sum_{k\geq 0}\esf_{\lambda_j'-j-i-k}\of
        {x_1,x_1^{-1},\dots,x_n,x_n^{-1}} 
},
\end{multline}
and the odd orthogonal Giambelli identity
((\ref{thm:Giambelli-orthogonal}) for $N=2n+1$)
\begin{equation}
\label{thm:Giambelli-odd-orthogonal}
\oschf{2n+1}{\parof{\alpha|\beta}} = \detof{s}{\oschf{2n+1}{\parof{\alpha_i|\beta_j}}}.
\end{equation}

The equivalence of (\ref{thm:h-orthogonal}) for $N=2n+1$ and
(\ref{thm:h-odd-orthogonal}), and that of (\ref{thm:e-orthogonal}) for $N=2n+1$ and
(\ref{thm:e-odd-orthogonal}) is easily verified by 
simple row operations.

%\smallskip
%We should remark that the same identities would result from lattice
%path interpretations of Koike and Terada's odd orthogonal tableaux
%\cite[p.~123]{Koike-Terada:Young}. This is no surprise since these
%tableaux are not very different from Sundaram's tableaux. Instead of
%allowing one special entry at the end of each row as Sundaram does, they
%allow one special entry at the beginning of each row. In fact, it is
%very easy to convert one type of these odd tableaux into the other by
%the jeu de taquin (see \cite[\Zitat]{Proctor:Young-tableaux}). Also
%the lattice path interpretations of Koike and Terada's odd orthogonal
%tableaux would not differ too much from the above lattice path
%interpretations of Sundaram's tableaux. Again, ``special entries"
%would be encoded by diagonal steps. The algorithms would again be the
%same.



\bigskip
{\sc Bijective proof for the odd orthogonal dual Jacobi--Trudi-type
identity (\ref{thm:e-odd-orthogonal})}.
Here we shall consider lattice paths consisting of 
arbitrary vertical steps, of horizontal steps lying strictly 
below the line $x+y=2n+1$, and of downward
diagonal steps in direction $(1,-1)$ on the line $x+y=2n+1$.
We label horizontal
edges by the usual $e$-labelling (see Figure~\ref{FF}) and diagonal
edges by $\infty$. The weight of horizontal edges is defined as in
the symplectic case, i.e\@., the weight of a
horizontal edge with label $2i-1$ is $x_i$, and the weight of
a horizontal edge with label $2i$ is $x_i^{-1}$. Diagonal edges do
not contribute to the weight.

Let $\tableau$ be a $(2n+1)$-Sundaram tableau of shape $\lambda$, where $\la$
is a partition of length $r\le n+1$. By reading paths from the columns,
$\tableau$ corresponds to $\la_1$-tuples $(\path_1,\dots,\path_{\la_1})$
of nonintersecting lattice paths, where $\path_i$ runs from $(-i+1,i-1)$ to 
$(\la_i'-i+1,2n-\la_i'+i)$, $i=1,2,\dots,\lambda_1$,
and does not cross the line $y=x-1$, and where $\path_i$ consists of
arbitrary vertical steps, of horizontal steps strictly below
$x+y=2n+1$, and possibly downward diagonal steps on the line
$x+y=2n+1$. As in section~\ref{symplectic}
the non-crossing condition results from the symplectic constraint
(\ref{eq:symplectic-constraint}). Note that with the above
weight this mapping is weight-preserving with respect to 
(\ref{weight:orthogonal}). Basically the same correspondence already
appears in \cite[Lemma~3.2B]{Okada:Lattice-Path}.
An example with $n=3$ and
$\la=(4,4,1,1)$ is given in Figure~\ref{FX}.

%%%@@@ input from figfx
\begin{Figure}\label{FX}
\begin{picture}(20,13)(0,0)
\put(14,5){$T$ =}
\put(16,3){
	\begintableau{4}{4}
		\row{134\infty}
		\row{446\infty}
		\row{\infty}
		\row{\infty}
	\endtableau
}
\put(1,1){
	\begingrid{-3}{-1}{4}{10}
		\makexaxis
		\makexscale
		\makeyaxis

       \downskewline{-3}{4}{4}
       \downskewline{-3}{5}{5}
       \downskewline{-3}{6}{6}
       \downskewline{-3}{7}{7}
       \downskewline{-3}{8}{7}
       \downskewline{-3}{9}{7}

       \thinlines
       \put(0,-1){\line(1,1){5}}
       %labels
		 \labelledtrue
		 \setlabelinfty{7}
		 \elabeltrue
		 \uppath{0}{0}{-||-|||*//}
		 \elabeltrue	% this is needed again, since * changes labeling to h!
		 \uppath{-1}{1}{||--|||}
		 \elabeltrue
		 \uppath{-2}{2}{|||-|-|}
		 \elabeltrue
		 \uppath{-3}{3}{|||||||*//}
	\endgrid
}
 
\end{picture}
\end{Figure}

%%%@@@ end input from figfx
With $\reflect$ being the reflection in the line $y=x-2$,
we write the determinant in (\ref{thm:e-odd-orthogonal})
as a generating function for $\la_1$-tuples 
$(\path_1,\dots,\path_{\lambda_1})$ of lattice paths where $\path_i$ runs
from either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, 
$i=1,2,\dots,\lambda_1$, namely
\begin{multline}
\notag
\detof{\lambda_1}{\sum_{k\geq 0}\esf_{\lambda_j'-j+i-k}\of
        {x_1,x_1^{-1},\dots,x_n,x_n^{-1}} -
	\sum_{k\geq 0}\esf_{\lambda_j'-j-i-k}\of
        {x_1,x_1^{-1},\dots,x_n,x_n^{-1}} 
}\\
\hskip-2cm=	\underset{\bold\path\in
        \fpath^{(2)}\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1,-1\}^{\lambda_1}}}
	\sgn\of{\sigma}
	\sgn\of{\epsilon}
        w(\bold\path).
\end{multline}
Here
$\fpath^{(2)}$ indicates that now we are considering paths
with horizontal steps only strictly below $x+y=2n+1$ and downward
diagonal steps on $x+y=2n+1$.
$\bold u^{(\ep)}$ has again the meaning of section~\ref{Defs}.

The involution that cancels $\la_1$-tuples $\bold \path$ where a path of
$\bold \path$ crosses $y=x-1$ is defined in the same way as in the proof
of (\ref{thm:e-symplectic}) in section~\ref{symplectic}. Again, this is a
weight-preserving mapping since the right-most possible crossing
point of a path in $\bold \path$ with $y=x-1$ is $(n,n-1)$, and hence
the steps that
are involved in the modified reflection, depicted in Figure~\ref{FJ},
all have $e$-label at most $2n$. Of course, for the intersecting
$\la_1$-tuples we apply the Gessel--Viennot involution.
\QED



\bigskip
{\sc Bijective proof for the odd orthogonal Jacobi--Trudi-type
identity (\ref{thm:h-odd-orthogonal})}.
Now we consider lattice paths consisting of 
arbitrary vertical steps, of horizontal steps up to height at most
$2n$, and of possibly {\em one\/} upward
diagonal 
step
in direction $(1,1)$ from some point $(x,2n)$ to
$(x+1,2n+1)$.
We label horizontal
edges by the usual $h$-labelling (see Figure~\ref{FF}) and diagonal
edges by $\infty$. The weight of horizontal edges is defined as
before, i.e\@., the weight of a
horizontal edge with label $2i-1$ is $x_i$, and the weight of
a horizontal edge with label $2i$ is $x_i^{-1}$. As before, diagonal
edges do not contribute to the weight.


This time we apply ``row-wise" reading to Sundaram's tableaux. Thus a
$(2n+1)$-Sundaram tableau of shape $\lambda$, where $\la$
is a partition of length $r\le n$, corresponds to 
$r$-tuples $(\path_1,\dots,\path_{r})$
of nonintersecting lattice paths, where $\path_i$ runs from $(-i,1)$ to 
$(\la_i-i,2n+1)$, $i=1,2,\dots,r$, and where $\path_i$ consists of
arbitrary vertical steps, of horizontal steps at height between $2i-1$
and $2n$, and possibly one upward diagonal step
from height $2n$ to
height $2n+1$. As in section~\ref{symplectic}
the lower bound for the heights of paths 
results from the symplectic constraint
(\ref{eq:symplectic-constraint}). Note that with the above described
labelling this mapping is weight-preserving with respect to 
(\ref{weight:orthogonal}).
An example with $n=3$ and
$\la=(4,4,1,1)$ is given in Figure~\ref{FY}.
\vskip-1cm
%%%@@@ input from figfy
\begin{Figure}\label{FY}
\begin{picture}(20,11)(0,0)
\put(14,5){$T$ =}
\put(16,3){
	\begintableau{4}{4}
		\row{134\infty}
		\row{446\infty}
		\row{\infty}
		\row{\infty}
	\endtableau
}
\put(1,1){
	\begingrid{-5}{0}{3}{7}
		\makexaxis
		\makexscale
		\makeyaxis

      %labels
      \labelledtrue
		\elabelfalse
		\setlabelinfty{7}
		\uppath{-4}{1}{|||||/}
		\uppath{-3}{1}{|||||/}
		\uppath{-2}{1}{|||--||-/}
		\uppath{-1}{1}{-||-|-||/}
	\endgrid
}
 
\end{picture}
\end{Figure}

%%%@@@ end input from figfy

Now let $\reflect$ be the reflection in the line $x=-1$.
A moment's thought shows that
the determinant in (\ref{thm:h-odd-orthogonal})
may be written as a generating function for 
$(\path_1,\dots,\path_{r})$ of lattice paths where $\path_i$ runs
from either $u_i$ or $\reflect(u_i)$ to $v_{\sigma(i)}$, $i=1,2,\dots,r$,
namely
\begin{equation}
\notag
\detof{r}{
        \firstrow{\hsf'_{\lambda_j-j+1}\of{\bold{x}}}
	\hsf'_{\lambda_j-j+i}\of{\bold{x}} +
		\hsf'_{\lambda_j-j-i+2}\of{\bold{x}}}
=	\underset{\bold\path\in
        \fpath^{(3)}\of{{\bold{u}}^{(\epsilon)},{\bold{v}}_\sigma}}
		{\sum_{\sigma\in \SG{\lambda_1},\ 
                \epsilon\in\{1\}\times\{1,-1\}^{r-1}}} 
	\sgn\of{\sigma} w(\bold\path).
\end{equation}
The notation $\fpath^{(3)}$ indicates that here we are considering 
paths with arbitrary vertical steps, with horizontal steps at height 
at most $2n$, and possibly one upward diagonal steps from height $2n$ to
height $2n+1$.

The definition of the involution that cancels those $r$-tuples 
$(\path_1,\dots,\path_r)$ where either the associated $\ep$ is
different from $(1,1,\dots,1)$ or where there is some $\path_i$ whose
first horizontal step is strictly below the line $y=2i-1$ is
basically
identical with the one in the proof of (\ref{thm:h-symplectic}) in
section~\ref{symplectic}. 
We just have to say what to do with the
diagonal steps, representing $\infty$, during the dualization
procedure. Namely, to the dualization rules in the proof of
(\ref{thm:h-symplectic}) (``\dots connect $w_i$ to $\overline v_i$ 
by a lattice path,
$i=1,2,\dots,\la_1$, by starting at $w_i$ and going up vertically
unless we would meet a lattice path of $\bold\path$. In the latter
case proceed by an upward diagonal step in direction $(-1,1)$\dots'')
we have to add that when we meet the end point of a diagonal step we
proceed by a left-handed horizontal step.
\QED





\bigskip
{\sc Bijective proof for the odd orthogonal 
Giambelli identity (\ref{thm:Giambelli-odd-orthogonal})}.
The proof of the odd orthogonal Giambelli identity
by means of
Sundaram's tableaux is similar to the one given for the symplectic
Giambelli identity (\ref{thm:Giambelli-symplectic}) (see
Figure~\ref{FR}). The lattice path
interpretation of Sundaram's tableaux is the same as in the
symplectic case, besides the entries $\infty$ 
are interpreted as
downward diagonal steps.
Thus, if $(\alpha|\beta)$ is a partition (in Frobenius notation) of
rank $s$, $(2n+1)$-Sundaram tableaux of shape $\parof{\alpha|\beta}$ 
correspond to $s$-tuples $(\path_1,\dots,\path_s)$ of nonintersecting
lattice paths where $\path_i$ runs from $\parof{-\alpha_i,2n}$ to
$\parof{\beta_i+1,2n-\beta_i}$, $i=1,2,\dots,s$, and do not cross the line $y=x-1$ 
in the way as is depicted in
Figure~\ref{FZ} below. The labelling is defined as in the proofs of
the ordinary and symplectic Giambelli 
identities; in addition, 
diagonal steps are labelled by $\infty$. 
Again, the weight of a
horizontal edge with label $2i-1$ is defined to be $x_i$, and the weight of
a horizontal edge with label $2i$ is defined to be $x_i^{-1}$.
Again, diagonal edges do not contribute to the weight.
Figure~\ref{FZ} gives an example for $n=4$ and
$(\alpha|\beta)=(3,2\vert 3,0)$. As is clear from the picture and the form
of the
determinant in (\ref{thm:Giambelli-odd-orthogonal}),
application of the Gessel--Viennot method immediately implies
(\ref{thm:Giambelli-odd-orthogonal}).

%%%@@@ input from figfz
\begin{Figure}\label{FZ}
\begin{picture}(20,11)(0,0)
\put(14,5){$T$ =}
\put(16,3){
	\begintableau{4}{4}
		\row{236\infty}
		\row{447\infty}
		\row{\infty}
		\row{\infty}
	\endtableau
}
\put(1,1){
	\begingrid{-3}{0}{4}{9}
		\makexaxis
		\makexscale

		\downskewline{0}{2}{2}
		\downskewline{0}{3}{3}
		\downskewline{0}{4}{4}
		\downskewline{0}{5}{4}
		\downskewline{0}{6}{4}
		\downskewline{0}{7}{4}
		\downskewline{0}{8}{4}

      \thinlines
		\put(1,0){\line(1,1){4}}
      \labelledtrue
		\elabelfalse
		\setlabelinfty{8}
		\downpath{-3}{8}{/||-|||-|*--||||||*//}{8}
		\downpath{-2}{8}{/|-|||*-|||||}{8}
	\endgrid
}
 
\end{picture}
\end{Figure}

%%%@@@ end input from figfz
\QED

\bigskip
{\sc ``Even" Sundaram tableaux}.
Let us define {\em $2n$-Sundaram tableaux\/} to be
tableaux with entries from the alphabet
$1<2<\dots<2n<\infty$ in the usual sense,
except that column-strictness
does not extend to symbol $\infty$ (i.e., symbol
$\infty$ may occur {\em more than once in a column\/}),
which have at most one entry $\infty$ in each {\em row\/},
such that the entries in row $i$ are at least $2i$. 
Note that the only difference to the definition of $(2n+1)$-Sundaram
tableaux is the requirement that the entries in row $i$ have to be at
least $2i$ (instead of only $2i-1$). In particular, the entry $1$
actually cannot appear in a $2n$-Sundaram tableau.
It turns out that $2n$-Sundaram tableaux of a given
shape enumerate the dimension of the corresponding irreducible
representation of the {\em even\/} orthogonal group. However, we were
not able to find appropriate weights for these tableaux such that
they would generate the full characters.

\medskip
\noindent {\bf Theorem 2}. {\sl
Let $\la=(\la_1,\dots,\la_r)$ be partition of length $r\le n$. Then
there holds
\begin{equation}\label{thm:even-Sundaram}
\oschftriv{2n}{\lambda}  =
\numof{\setof{\tableau:\;\tableau\hbox{ $2n$-Sundaram of shape
$\lambda$}}}.
\end{equation}
}

\medskip
\proof
If we use the same arguments as in the proof of the odd orthogonal
dual Jacobi--Trudi-type
identity (\ref{thm:e-odd-orthogonal}), with the modified reflection
in $y=x-2$ replaced by the ``ordinary" reflection in the line
$y=x-1$, we obtain
\begin{multline} 
\notag
\numof{\setof{\tableau:\;\tableau\hbox{ $2n$-Sundaram of shape
$\lambda$}}}\\
=\detof{\lambda_1}{\sum_{k\geq 0}\esf_{\lambda_j'-j+i-k}\of
        {1,1,\dots,1} -
	\sum_{k\geq 0}\esf_{\lambda_j'-j-i-k+1}\of
        {1,1,\dots,1} 
},
\end{multline}
where $(1,1,\dots,1)$ stands for a sequence of exactly $2n$ 1's.
Simple row manipulations turn this determinant into
\begin{equation}
\notag
\detof{\lambda_1}{
	\firstrow{\esf_{\lambda^\prime_j-j+1}\of{\seqof{1,1,\dots,1}}}
	\esf_{\lambda_j^\prime-j+i}\of{\seqof{1,1,\dots,1}} +
	\esf_{\lambda_j^\prime-j-i+2}\of{\seqof{1,1,\dots,1}}},
\end{equation}
which by (\ref{thm:e-orthogonal}) is identified as
$\oschftriv{2n}{\lambda}$.
\QED

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\end{document}
 
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