%%%% format a4 \catcode'32=9 \magnification=1200 \voffset=1cm \hoffset=0cm %\hoffset=1cm \font\tenpc=cmcsc10 %\font\eightpc=cmcsc8 % Charge des fontes de 8 et 6 points : \font\eightrm=cmr8 \font\eighti=cmmi8 \font\eightsy=cmsy8 \font\eightbf=cmbx8 \font\eighttt=cmtt8 \font\eightit=cmti8 \font\eightsl=cmsl8 \font\sixrm=cmr6 \font\sixi=cmmi6 \font\sixsy=cmsy6 \font\sixbf=cmbx6 \skewchar\eighti='177 \skewchar\sixi='177 \skewchar\eightsy='60 \skewchar\sixsy='60 % Chargement des fontes AMS \font\tengoth=eufm10 \font\tenbboard=msbm10 \font\eightgoth=eufm8 \font\eightbboard=msbm8 \font\sevengoth=eufm7 \font\sevenbboard=msbm7 \font\sixgoth=eufm6 \font\fivegoth=eufm5 \newfam\gothfam \newfam\bboardfam \catcode`\@=11 \def\raggedbottom{\topskip 10pt plus 36pt \r@ggedbottomtrue} \def\pc#1#2|{{\bigf@ntpc #1\penalty \@MM\hskip\z@skip\smallf@ntpc #2}} \def\tenpoint{% \textfont0=\tenrm \scriptfont0=\sevenrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\tenrm}% \textfont1=\teni \scriptfont1=\seveni \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\teni}% \textfont2=\tensy \scriptfont2=\sevensy \scriptscriptfont2=\fivesy \textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\tengoth}% \textfont\bboardfam=\tenbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bboard{\fam\bboardfam}% \textfont\itfam=\tenit \def\it{\fam\itfam\tenit}% \textfont\slfam=\tensl \def\sl{\fam\slfam\tensl}% \textfont\bffam=\tenbf \scriptfont\bffam=\sevenbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\tenbf}% \textfont\ttfam=\tentt \def\tt{\fam\ttfam\tentt}% \abovedisplayskip=12pt plus 3pt minus 9pt \abovedisplayshortskip=0pt plus 3pt \belowdisplayskip=12pt plus 3pt minus 9pt \belowdisplayshortskip=7pt plus 3pt minus 4pt \smallskipamount=3pt plus 1pt minus 1pt \medskipamount=6pt plus 2pt minus 2pt \bigskipamount=12pt plus 4pt minus 4pt \normalbaselineskip=12pt \setbox\strutbox=\hbox{\vrule height8.5pt depth3.5pt width0pt}% \let\bigf@ntpc=\tenrm \let\smallf@ntpc=\sevenrm \let\petcap=\tenpc \normalbaselines\rm} \def\eightpoint{% \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0=\fiverm \def\rm{\fam\z@\eightrm}% \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1=\fivei \def\oldstyle{\fam\@ne\eighti}% \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2=\fivesy \textfont\gothfam=\eightgoth \scriptfont\gothfam=\sixgoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\eightgoth}% \textfont\bboardfam=\eightbboard \scriptfont\bboardfam=\sevenbboard \scriptscriptfont\bboardfam=\sevenbboard \def\bboard{\fam\bboardfam}% \textfont\itfam=\eightit \def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl \def\sl{\fam\slfam\eightsl}% \textfont\bffam=\eightbf \scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \textfont\ttfam=\eighttt \def\tt{\fam\ttfam\eighttt}% \abovedisplayskip=9pt plus 2pt minus 6pt \abovedisplayshortskip=0pt plus 2pt \belowdisplayskip=9pt plus 2pt minus 6pt \belowdisplayshortskip=5pt plus 2pt minus 3pt \smallskipamount=2pt plus 1pt minus 1pt \medskipamount=4pt plus 2pt minus 1pt \bigskipamount=9pt plus 3pt minus 3pt \normalbaselineskip=9pt \setbox\strutbox=\hbox{\vrule height7pt depth2pt width0pt}% \let\bigf@ntpc=\eightrm \let\smallf@ntpc=\sixrm \normalbaselines\rm} \let\bb=\bboard \tenpoint %ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ dactylographie franaise ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ \catcode`\;=\active \def;{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font\kern -1.2 \fontdimen3 \font\fi\string;} \catcode`\:=\active \def:{\relax\ifhmode\ifdim\lastskip>\z@\unskip\fi\penalty\@M\ \fi\string:} \catcode`\!=\active \def!{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string!} \catcode`\?=\active \def?{\relax\ifhmode\ifdim\lastskip>\z@ \unskip\fi\kern\fontdimen2 \font \kern -1.1 \fontdimen3 \font\fi\string?} \def\^#1{\if#1i{\accent"5E\i}\else{\accent"5E #1}\fi} \def\"#1{\if#1i{\accent"7F\i}\else{\accent"7F #1}\fi} \frenchspacing %ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ format de sortie ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ \newif\ifpagetitre \newtoks\auteurcourant \auteurcourant={\hfil} \newtoks\titrecourant \titrecourant={\hfil} \def\appeln@te{} \def\vfootnote#1{\def\@parameter{#1}\insert\footins\bgroup\eightpoint \interlinepenalty\interfootnotelinepenalty \splittopskip\ht\strutbox % top baseline for broken footnotes \splitmaxdepth\dp\strutbox \floatingpenalty\@MM \leftskip\z@skip \rightskip\z@skip \ifx\appeln@te\@parameter\indent \else{\noindent #1\ }\fi \footstrut\futurelet\next\fo@t} \pretolerance=500 \tolerance=1000 \brokenpenalty=5000 \newdimen\hmargehaute \hmargehaute=0cm \newdimen\lpage \lpage=13.3cm \newdimen\hpage \hpage=20cm \newdimen\lmargeext \lmargeext=1cm \hsize=11.25cm \vsize=18cm \parskip 0pt \parindent=12pt \def\margehaute{\vbox to \hmargehaute{\vss}}% \def\margebasse{\vss} \output{\shipout\vbox to \hpage{\margehaute\nointerlineskip \corpsdepage\margebasse} \advancepageno \global\pagetitrefalse \ifnum\outputpenalty>-20000 \else\dosupereject\fi} \def\corpsdepage{\hbox to \lpage{\hss\pagetexte\hskip\lmargeext}} \def\pagetexte{\vbox{\makeheadline\pagebody\makefootline}} \headline={\ifpagetitre\titleheadline \else \ifodd\pageno\rightheadline \else\leftheadline\fi\fi} \def\leftheadline{\eightpoint\hfil\the\auteurcourant\hfil} \def\rightheadline{\eightpoint\hfil\the\titrecourant\hfil} \def\titleheadline{\hfill} \pagetitretrue \def\footnoterule{\kern-6\p@ \hrule width 2truein \kern 5.6\p@} % the \hrule is .4pt high \let\rmpc=\sevenrm \def\pd#1#2 {\pc#1#2| } \def\pointir{\discretionary{.}{}{.\kern.35em---\kern.7em}\nobreak \hskip 0em plus .3em minus .4em } \def\resume#1{\vbox{\eightpoint \pc R\'ESUM\'E|\pointir #1}} \def\abstract#1{\vbox{\eightpoint \pc ABSTRACT|\pointir #1}} \def\titre#1|{\message{#1} \par\vskip 30pt plus 24pt minus 3pt\penalty -1000 \vskip 0pt plus -24pt minus 3pt\penalty -1000 \centerline{\bf #1} \vskip 5pt \penalty 10000 } \def\section#1|{\par\vskip .3cm {\bf #1}\pointir} \def\ssection#1|{\par\vskip .2cm {\it #1}\pointir} \long\def\th#1|#2\finth{\par\medskip {\petcap #1\pointir}{\it #2}\par\smallskip} \long\def\tha#1|#2\fintha{\par\medskip {\petcap #1.}\par\nobreak{\it #2}\par\smallskip} \def\cf{{\it cf}} \def\rem#1|{\par\medskip {{\it #1}\pointir}} \def\rema#1|{\par\medskip {{\it #1.}\par\nobreak }} % \def\ieme{\raise 1ex\hbox{\pc{}i\`eme|}} \def\omini{\raise 1ex\hbox{\pc{}o|}} \def\emini{\raise 1ex\hbox{\pc{}e|}} \def\ermini{\raise 1ex\hbox{\pc{}er|}} \def\remini{\raise 1ex\hbox{\pc{}re|}} %reference pour un article : \def\article#1|#2|#3|#4|#5|#6|#7| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3, {\sl #4}, t.\nobreak\ {\bf #5}, {\oldstyle #6}, p.\nobreak\ #7.\par}} %reference pour un livre : \def\livre#1|#2|#3|#4| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir {\sl #3}\pointir #4.\par}} %reference complementaire : \def\divers#1|#2|#3| {{\leftskip=7mm\noindent \hangindent=2mm\hangafter=1 \llap{[#1]\hskip.35em}{#2}\pointir #3.\par}} % \mathchardef\conj="0365 \def\dem{\par{\it D\'emonstration}\pointir} \def\qed{\quad\raise -2pt\hbox{\vrule\vbox to 10pt{\hrule width 4pt \vfill\hrule}\vrule}} \def\virg{\raise 2pt\hbox{,}} % virgule aprs une fraction \def\cqfd{\penalty 500 \hbox{\qed}\par\smallskip} %les differents retraits, voir aussi \item \def\decale#1|{\par\noindent\hskip 28pt\llap{#1}\kern 5pt} \def\decaledecale#1|{\par\noindent\hskip 34pt\llap{#1}\kern 5pt} % pour les titres en deux lignes et les sections sans point-tiret : \def\titrea#1|#2|{\message{#1 #2} \par\vskip.5cm plus .1cm minus .1cm\penalty -1000 \centerline{\bf #1} \centerline{\bf #2} \vskip 5pt \penalty 10000 } \def\sectiona#1|{\par\vskip .3cm {\bf #1.} \par\nobreak\vskip 3pt } \def\ssectiona#1|{\par\vskip .2cm {\it #1.} \par\nobreak\vskip 2pt } \def\rest#1{\ifinner\setbox1=\hbox{$\textstyle{#1}$} \else\setbox1=\hbox{$\displaystyle{#1}$}\fi \dimen1=\ht1 \advance\dimen1 by\dp1 \divide\dimen1 by 2 \box1\lower 2pt\hbox{$\left|\vbox to\dimen1{}\right.$}} \def\displaylinesno#1{\displ@y\halign{ \hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}& \llap{$##$}\crcr#1\crcr}} \def\ldisplaylinesno#1{\displ@y\halign{ \hbox to\displaywidth{$\@lign\hfil\displaystyle##\hfil$}& \kern-\displaywidth\rlap{$##$}\tabskip\displaywidth\crcr#1\crcr}} \def\Eqalign#1{\null\,\vcenter{\openup\jot\m@th\ialign{ \strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil &&\quad\strut\hfil$\displaystyle{##}$&$\displaystyle{{}##}$\hfil \crcr#1\crcr}}\,} \def\lfq{\leavevmode\raise.3ex\hbox{$\scriptscriptstyle \langle\!\langle$}\thinspace} \def\rfq{\leavevmode\thinspace\raise.3ex\hbox{$\scriptscriptstyle \rangle\!\rangle$}} \catcode`\@=12 %format to draw oblique lines \def\grille{\noalign{\nointerlineskip\Grille\nointerlineskip}} %ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ \def\fleche(#1,#2)\dir(#3,#4)\long#5{% \noalign{\nointerlineskip\leftput(#1,#2){\vector(#3,#4){#5}}\nointerlineskip}} %ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ \def\diagram#1{\def\normalbaselines{\baselineskip=0pt\lineskip=5pt} \matrix{#1}} \def\hfl#1#2#3{\smash{\mathop{\hbox to#3{\rightarrowfill}}\limits ^{\scriptstyle#1}_{\scriptstyle#2}}} \def\gfl#1#2#3{\smash{\mathop{\hbox to#3{\leftarrowfill}}\limits ^{\scriptstyle#1}_{\scriptstyle#2}}} \def\vfl#1#2#3{\llap{$\scriptstyle #1$} \left\downarrow\vbox to#3{}\right.\rlap{$\scriptstyle #2$}} %ΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡΡ \message{`lline' & `vector' macros from LaTeX} \catcode`@=11 \def\{{\relax\ifmmode\lbrace\else$\lbrace$\fi} \def\}{\relax\ifmmode\rbrace\else$\rbrace$\fi} \def\newcount{\alloc@0\count\countdef\insc@unt} \def\newdimen{\alloc@1\dimen\dimendef\insc@unt} \def\newwrite{\alloc@7\write\chardef\sixt@@n} \newwrite\@unused \newcount\@tempcnta \newcount\@tempcntb \newdimen\@tempdima \newdimen\@tempdimb \newbox\@tempboxa \def\@spaces{\space\space\space\space} \def\@whilenoop#1{} \def\@whiledim#1\do #2{\ifdim #1\relax#2\@iwhiledim{#1\relax#2}\fi} \def\@iwhiledim#1{\ifdim #1\let\@nextwhile=\@iwhiledim \else\let\@nextwhile=\@whilenoop\fi\@nextwhile{#1}} \def\@badlinearg{\@latexerr{Bad \string\line\space or \string\vector \space argument}} \def\@latexerr#1#2{\begingroup \edef\@tempc{#2}\expandafter\errhelp\expandafter{\@tempc}% %% error help message pieces. \def\@eha{Your command was ignored. ^^JType \space I \space to replace it with another command,^^Jor \space \space to continue without it.} \def\@ehb{You've lost some text. \space \@ehc} \def\@ehc{Try typing \space \space to proceed.^^JIf that doesn't work, type \space X \space to quit.} \def\@ehd{You're in trouble here. \space\@ehc} \typeout{LaTeX error. \space See LaTeX manual for explanation.^^J \space\@spaces\@spaces\@spaces Type \space H \space for immediate help.}\errmessage{#1}\endgroup} \def\typeout#1{{\let\protect\string\immediate\write\@unused{#1}}} % line & circle fonts \font\tenln = line10 \font\tenlnw = linew10 %\font\tencirc = circle10 %\font\tencircw = circlew10 \newdimen\@wholewidth \newdimen\@halfwidth \newdimen\unitlength \unitlength =1pt %\newbox\@picbox %\newdimen\@picht \def\thinlines{\let\@linefnt\tenln \let\@circlefnt\tencirc \@wholewidth\fontdimen8\tenln \@halfwidth .5\@wholewidth} \def\thicklines{\let\@linefnt\tenlnw \let\@circlefnt\tencircw \@wholewidth\fontdimen8\tenlnw \@halfwidth .5\@wholewidth} \def\linethickness#1{\@wholewidth #1\relax \@halfwidth .5\@wholewidth} \newif\if@negarg \def\lline(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax \@linelen=#3\unitlength \ifnum\@xarg =0 \@vline \else \ifnum\@yarg =0 \@hline \else \@sline\fi \fi} \def\@sline{\ifnum\@xarg< 0 \@negargtrue \@xarg -\@xarg \@yyarg -\@yarg \else \@negargfalse \@yyarg \@yarg \fi \ifnum \@yyarg >0 \@tempcnta\@yyarg \else \@tempcnta -\@yyarg \fi \ifnum\@tempcnta>6 \@badlinearg\@tempcnta0 \fi \setbox\@linechar\hbox{\@linefnt\@getlinechar(\@xarg,\@yyarg)}% \ifnum \@yarg >0 \let\@upordown\raise \@clnht\z@ \else\let\@upordown\lower \@clnht \ht\@linechar\fi \@clnwd=\wd\@linechar \if@negarg \hskip -\wd\@linechar \def\@tempa{\hskip -2\wd\@linechar}\else \let\@tempa\relax \fi \@whiledim \@clnwd <\@linelen \do {\@upordown\@clnht\copy\@linechar \@tempa \advance\@clnht \ht\@linechar \advance\@clnwd \wd\@linechar}% \advance\@clnht -\ht\@linechar \advance\@clnwd -\wd\@linechar \@tempdima\@linelen\advance\@tempdima -\@clnwd \@tempdimb\@tempdima\advance\@tempdimb -\wd\@linechar \if@negarg \hskip -\@tempdimb \else \hskip \@tempdimb \fi \multiply\@tempdima \@m \@tempcnta \@tempdima \@tempdima \wd\@linechar \divide\@tempcnta \@tempdima \@tempdima \ht\@linechar \multiply\@tempdima \@tempcnta \divide\@tempdima \@m \advance\@clnht \@tempdima \ifdim \@linelen <\wd\@linechar \hskip \wd\@linechar \else\@upordown\@clnht\copy\@linechar\fi} %\def\@hline{\ifnum \@xarg <0 \hskip -\@linelen \fi %\vrule \@height \@halfwidth \@depth \@halfwidth \@width %\@linelen \ifnum \@xarg <0 \hskip -\@linelen \fi} \def\@hline{\ifnum \@xarg <0 \hskip -\@linelen \fi \vrule height \@halfwidth depth \@halfwidth width \@linelen \ifnum \@xarg <0 \hskip -\@linelen \fi} \def\@getlinechar(#1,#2){\@tempcnta#1\relax\multiply\@tempcnta 8 \advance\@tempcnta -9 \ifnum #2>0 \advance\@tempcnta #2\relax\else \advance\@tempcnta -#2\relax\advance\@tempcnta 64 \fi \char\@tempcnta} \def\vector(#1,#2)#3{\@xarg #1\relax \@yarg #2\relax \@linelen=#3\unitlength \ifnum\@xarg =0 \@vvector \else \ifnum\@yarg =0 \@hvector \else \@svector\fi \fi} \def\@hvector{\@hline\hbox to 0pt{\@linefnt \ifnum \@xarg <0 \@getlarrow(1,0)\hss\else \hss\@getrarrow(1,0)\fi}} \def\@vvector{\ifnum \@yarg <0 \@downvector \else \@upvector \fi} \def\@svector{\@sline \@tempcnta\@yarg \ifnum\@tempcnta <0 \@tempcnta=-\@tempcnta\fi \ifnum\@tempcnta <5 \hskip -\wd\@linechar \@upordown\@clnht \hbox{\@linefnt \if@negarg \@getlarrow(\@xarg,\@yyarg) \else \@getrarrow(\@xarg,\@yyarg) \fi}% \else\@badlinearg\fi} \def\@getlarrow(#1,#2){\ifnum #2 =\z@ \@tempcnta='33\else \@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta -9 \@tempcntb=#2\relax\multiply\@tempcntb \tw@ \ifnum \@tempcntb >0 \advance\@tempcnta \@tempcntb\relax \else\advance\@tempcnta -\@tempcntb\advance\@tempcnta 64 \fi\fi\char\@tempcnta} \def\@getrarrow(#1,#2){\@tempcntb=#2\relax \ifnum\@tempcntb < 0 \@tempcntb=-\@tempcntb\relax\fi \ifcase \@tempcntb\relax \@tempcnta='55 \or \ifnum #1<3 \@tempcnta=#1\relax\multiply\@tempcnta 24 \advance\@tempcnta -6 \else \ifnum #1=3 \@tempcnta=49 \else\@tempcnta=58 \fi\fi\or \ifnum #1<3 \@tempcnta=#1\relax\multiply\@tempcnta 24 \advance\@tempcnta -3 \else \@tempcnta=51\fi\or \@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta -\tw@ \else \@tempcnta=#1\relax\multiply\@tempcnta \sixt@@n \advance\@tempcnta 7 \fi\ifnum #2<0 \advance\@tempcnta 64 \fi \char\@tempcnta} \def\@vline{\ifnum \@yarg <0 \@downline \else \@upline\fi} \def\@upline{\hbox to \z@{\hskip -\@halfwidth \vrule width \@wholewidth height \@linelen depth \z@\hss}} \def\@downline{\hbox to \z@{\hskip -\@halfwidth \vrule width \@wholewidth height \z@ depth \@linelen \hss}} \def\@upvector{\@upline\setbox\@tempboxa\hbox{\@linefnt\char'66}\raise \@linelen \hbox to\z@{\lower \ht\@tempboxa\box\@tempboxa\hss}} \def\@downvector{\@downline\lower \@linelen \hbox to \z@{\@linefnt\char'77\hss}} %INITIALIZATION \thinlines \newcount\@xarg \newcount\@yarg \newcount\@yyarg \newcount\@multicnt \newdimen\@xdim \newdimen\@ydim \newbox\@linechar \newdimen\@linelen \newdimen\@clnwd \newdimen\@clnht \newdimen\@dashdim \newbox\@dashbox \newcount\@dashcnt \catcode`@=12 % macros supplementaires (J.D.) \newbox\tbox \newbox\tboxa %\def\picbox(#1,#2,#3)#4{\setbox\tbox=\hbox{#4}% % \ht\tbox=#1 \wd\tbox=#2 \dp\tbox=#3 \box\tbox} \def\leftzer#1{\setbox\tbox=\hbox to 0pt{#1\hss}% \ht\tbox=0pt \dp\tbox=0pt \box\tbox} \def\rightzer#1{\setbox\tbox=\hbox to 0pt{\hss #1}% \ht\tbox=0pt \dp\tbox=0pt \box\tbox} \def\centerzer#1{\setbox\tbox=\hbox to 0pt{\hss #1\hss}% \ht\tbox=0pt \dp\tbox=0pt \box\tbox} % sytaxe: \image(hauteur totale reservee, distance % verticale de l'origine par rapport au bas de % la place reservee){materiel a inserer} % L'origine est toujours centre horizontalement % \def\image(#1,#2)#3{\vbox to #1{\offinterlineskip \vss #3 \vskip #2}} % \leftput place le materiel avec son point de reference % en #1,#2 unites \unitlength % De meme pour \centerput et \rightput \def\leftput(#1,#2)#3{\setbox\tboxa=\hbox{% \kern #1\unitlength \raise #2\unitlength\hbox{\leftzer{#3}}}% \ht\tboxa=0pt \wd\tboxa=0pt \dp\tboxa=0pt\box\tboxa} \def\rightput(#1,#2)#3{\setbox\tboxa=\hbox{% \kern #1\unitlength \raise #2\unitlength\hbox{\rightzer{#3}}}% \ht\tboxa=0pt \wd\tboxa=0pt \dp\tboxa=0pt\box\tboxa} \def\centerput(#1,#2)#3{\setbox\tboxa=\hbox{% \kern #1\unitlength \raise #2\unitlength\hbox{\centerzer{#3}}}% \ht\tboxa=0pt \wd\tboxa=0pt \dp\tboxa=0pt\box\tboxa} \unitlength=1mm \def\cput(#1,#2)#3{\noalign{\nointerlineskip\centerput(#1,#2){#3} \nointerlineskip}} %%%%%%%%%%%%% \def\limproj{\mathop{\displaystyle \lim_{\longleftarrow}}} \let\bb=\bboard \let\fle=\longrightarrow \long\def\thc #1|#2\finthc{\medbreak \noindent{#1}\enspace{\it #2}\par \ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi} \def\scal #1|#2|{\langle #1\rangle _{\lower2pt\hbox{$\scriptstyle #2$}}} \long\def\th#1|#2\finth{\medbreak {\petcap #1\pointir}{\it #2}\par \ifdim\lastskip<\medskipamount\removelastskip\penalty55\medskip\fi} \def\cqfd{\penalty 500 \hbox{\qed}\par \goodbreak\medskip} %a macro for grille.tex \def\segment(#1,#2)\dir(#3,#4)\long#5{% \leftput(#1,#2){\lline(#3,#4){#5}}} %%%%%%%%%%%%%%%%%%%% Start of the paper \pageno=131 \let\findem=\cqfd \auteurcourant={I.G. MACDONALD} \titrecourant={A NEW CLASS OF SYMMETRIC FUNCTIONS} \eightpoint \leftline{Publ. I.R.M.A. Strasbourg, $\oldstyle 1988$, 372/S--20} \leftline{Actes 20\emini\ S\'eminaire Lotharingien, p. 131--171} \tenpoint \vskip 1.5cm \centerline{{\bf A NEW CLASS OF SYMMETRIC FUNCTIONS}} \vskip 2.8mm \centerline{\sevenrm BY} \vskip 2.8mm \centerline{I. G. MACDONALD} \vskip 1cm \noindent {\bf Contents.} {\parindent=1cm \item{1.} Introduction \item{2.} The symmetric functions $P_\lambda (q,t)$ \item{3.} Duality \item{4.} Skew $P$ and $Q$ functions \item{5.} Explicit formulas \item{6.} The Kostka matrix \item{7.} Another scalar product \item{8.} Conclusion \item{9.} Appendix } \titre 1. Introduction| I will begin by reviewing briefly some aspects of the theory of symmetric functions. This will serve to fix notation and to provide some motivation for the subject of these lectures. Let $x_1$, \dots, $x_n$ be independent indeterminates. The symmetric group ${\goth S}_n$ acts on the polynomial ring ${\bboard Z}[x_1,\ldots,x_n]$ by permuting the $x$'s, and we shall write $$\Lambda _n={\bb Z}[x_1,\ldots,x_n]^{{\goth S}_n}$$ for the subring of symmetric polynomials in $x_1$, \dots, $x_n$. If $f\in \Lambda _n$, we may write $$f=\sum _{r\geq 0} f^{(r)}$$ where $f^{(r)}$ is the homogeneous component of $f$ of degree~$r$ ; each $f^{(r)}$ is itself symmetric, and so $\Lambda _n$ is a {\it graded} ring : $$\Lambda _n=\bigoplus_{r\geq 0} \Lambda _n^r,$$ where $\Lambda _n^r$ is the additive group of symmetric polynomials of degree~$r$ in $x_1$, \dots, $x_n$. (By convention, $0$ is homogeneous of every degree.) If we now adjoin another indeterminate $x_{n+1}$, we can form $\Lambda _{n+1}={\bb Z}[x_1,\ldots,x_{n+1}]^{{\goth S}_{n+1}}$, and we have a surjective homomorphism (of graded rings) $$\Lambda _{n+1}\rightarrow \Lambda _n$$ defined by setting $x_{n+1}=0$. The mapping $\Lambda _{n+1}^r\rightarrow \Lambda _n^r$ is surjective for all $r\geq 0$, and bijective if and only if $r\leq n$. Often it is convenient to pass to the limit. Let $$\displaylines{\Lambda ^r=\limproj_n \Lambda _n^r\cr \noalign{\hbox{for each $r\geq 0$, and let}} \Lambda =\bigoplus_{r\geq 0} \Lambda ^r.\cr}$$ By the definition of inverse (or projective) limits, an element of $\Lambda _n^r$ is a sequence $(f_n)_{n\geq 0}$ where $f_n\in \Lambda _n^r$ for each~$n$, and $f_n$ is obtained from $f_{n+1}$ by setting $x_{n+1}=0$. We may therefore regard the $f_n$ as the partial sums of an infinite series $f$ of monomials of degree~$r$ in infinitely many indeterminates $x_1$, $x_2$, \dots\ For example, if $f_n=x_1+\cdots+x_n$, then $f=\sum _{i=1}^\infty x_i$. Thus the elements of $\Lambda $ are no longer polynomials, and we call them symmetric {\it functions}. (Of course, they aren't functions either, but they have to be called something !) For each $n$ there is a surjective homomorphism $\Lambda \rightarrow \Lambda _{n}$, obtained by setting $x_{n+1}=x_{n+2}=\cdots=0$. The graded ring $\Lambda $ is the {\it ring of symmetric functions}. If $R$ is any commutative ring, we write $$\Lambda _R=\Lambda \otimes _{\lower 2pt\hbox{$\scriptstyle \bb Z$}} R,\qquad \Lambda _{n,R}=\Lambda _n\otimes _{\lower2pt\hbox{$\scriptstyle \bb Z$}} R$$ for the ring of symmetric functions (resp. symmetric polynomials in $n$ indeterminates) with coefficients in~$R$. There are various ${\bb Z}$-bases of the ring $\Lambda $, some of which we shall review. They all are indexed by {\it partitions}. A partitions~$\lambda $ is a (finite or infinite) sequence $$\lambda =(\lambda _1,\lambda _2,\lambda _3,\ldots\,)$$ of non-negative integers, such that $\lambda _1\geq \lambda _2\geq \cdots$ and $$\left|\lambda \right|=\sum \lambda _i<\infty ,$$ so that from a certain point onwards (if the sequence $\lambda $ is infinite) all the $\lambda _i$ are zero. We shall not distinguish between two such sequences which differ only by a string of zeros at the end. Thus $(2,1)$, $(2,1,0)$, $(2,1,0,0,\ldots\,)$ are all to be regarded as the same partition. The nonzero $\lambda _i$ are called the {\it parts} of $\lambda $, and the number of parts is the {\it length} $\ell(\lambda )$ of~$\lambda $. If $\lambda $ has $m_1$ parts equal to~1, $m_2$ parts equal to~$2$, and so on, we shall occasionally write $\lambda =(1^{m_1}2^{m_2}\ldots\,)$, although strictly speaking we should write this in reverse order. Let $\cal P$ denote the set of all partitions, and ${\cal P}_n$ the set of all partitions of~$n$ (i.e. partitions $\lambda $ such that $\left|\lambda \right|=n$). The {\it natural} (or dominance) {\it partial ordering} in $\cal P$ is defined as follows : $$\lambda \geq \mu \Longleftrightarrow\left|\lambda \right| =\left|\mu \right|\ {\rm and}\ \lambda _1+\cdots+\lambda _r\geq \mu _1+\cdots +\mu _r\ {\rm for\ all}\ r\geq 1.\leqno(1.1)$$ It is a total order on ${\cal P}_n$ for $n\leq 5$, but not for $n\geq 6$. With each partition $\lambda $ we associate a {\it diagram}, consisting of the points $(i,j)\in {\bb Z}^2$ such that $1\leq j\leq \lambda _i$. We adopt the convention (as with matrices) that the first coordinate~$i$ (the row index) increases as one goes downwards, and the second coordinate~$j$ (the column index) increases from left to right. Often it is more convenient to replace the lattice points $(i,j)$ by squares, and then the diagram of $\lambda $ consists of $\lambda _1$ boxes in the top row, $\lambda _2$ boxes in the second row, and so on ; the whole arrangement of boxes being left-justified. If we read the diagram of a partition $\lambda $ by columns, we obtain the {\it conjugate partition}~$\lambda '$. Thus $\lambda '_j$ is the number of boxes in the $j$-th column of~$\lambda $, and hence is equal to the number of parts of~$\lambda $ that are $\geq j$. It is not difficult to show that $$\lambda \geq \mu \Longleftrightarrow \mu '\geq \lambda '.$$ %%%%%%%%% \vskip 5pt \noindent {\bf Bases of $\Lambda $.} \nobreak \section 1. Monomial symmetric functions| Let $\lambda $ be a partition. It defines a monomial $x^\lambda =x_1^{\lambda _1}x_2^{\lambda _2}\ldots$\ The monomial symmetric function $m_\lambda $ is the sum of all {\it distinct} monomials obtainable from $x^\lambda $ by permutations of the $x$'s. For example, $m_{(2,1)}=\sum x_i^2x_j$, summed over all $(i,j)$ such that $i\not=j$. In particular, when $\lambda =(1^r)$ we have $$m_{(1^r)}=e_r=\sum _{i_1<\cdots0$, and define a scalar product $\langle \kern4pt ,\kern3pt \rangle _{\lower2pt\hbox{$\scriptstyle \alpha $}}$ on $\Lambda _{\bb R}$ by $$\langle p_\lambda ,p_\mu \rangle _{\lower2pt\hbox{$\scriptstyle \alpha $}} =\delta _{\lambda \mu }.\alpha ^{\ell(\lambda )}z_\lambda .\leqno(1.11)$$ Then Jack's functions $P_\lambda =P_\lambda (x;\alpha )$ are characterized by the two properties $$\displaylines{\noalign{\vskip -4pt } \kern1cm{\rm (A)}\quad P_\lambda =m_\lambda + {\rm lower\ terms},\hfill\cr \kern1cm{\rm (B)}\quad \langle P_\lambda ,P_\mu \rangle _{\lower2pt\hbox{$\scriptstyle \alpha $}}=0\ {\rm if}\ \lambda \not=\mu .\hfill\cr}$$ The symmetric functions $P_\lambda $ depend rationally on $\alpha $, i.e. they lie in $\Lambda _F$ where $F$ is the field ${\bb Q}(\alpha )$, and we may if we prefer regard the parameter~$\alpha $ as an indeterminate rather than a real number. Clearly when $\alpha =1$ they reduce to the Schur functions $s_\lambda $, and when $\alpha =2$ to the zonal functions~$Z_\lambda $. They also tend to definite limits as $\alpha \rightarrow 0$ and as $\alpha \rightarrow \infty $ (even though the scalar product (1.11) collapses), and in fact $$\eqalign{P_\lambda (x;\alpha )&\rightarrow e_{\lambda '}\quad{\rm as}\ \alpha \rightarrow 0,\cr P_\lambda (x;\alpha )&\rightarrow m_\lambda \quad{\rm as}\ \alpha \rightarrow \infty .\cr}$$ Also when $\alpha ={1\over 2}$ they occur in nature, as zonal spherical functions on the homogeneous space $G/K$, where now $G={\rm GL}_n({\bb H})$ and $K=U(n,{\bb H})$, the quaternionic unitary group of $n\times n$~matrices. \section 8. Hall-Littlewood symmetric functions {\rm [${\rm M}_1$, ch.~III]}| These symmetric functions arose originally in connection with the combinatorial and enumerative lattice properties of finite abelian $p$-groups (where $p$ is a prime number). Let $t$ be an indeterminate, let $F={\bb Q}(t)$ and define a scalar product $\langle \kern4pt ,\kern3pt \rangle _{\lower2pt\hbox{$\scriptstyle (t)$}}$ on $\Lambda _F$ by $$\langle p_\lambda ,p_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (t)$}}=\delta _{\lambda \mu }z_\lambda \prod _{i=1}^{\ell(\lambda )} (1-t^{\lambda _i})^{-1}.\leqno(1.13)$$ Then the Hall-Littlewood symmetric functions $P_\lambda (x;t)$ are characterized by the two properties $$\displaylines{ \kern1cm{\rm (A)}\quad P_\lambda =m_\lambda + {\rm lower\ terms},\hfill\cr \kern1cm{\rm (B)}\quad \langle P_\lambda ,P_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (t)$}}=0\ {\rm if}\ \lambda \not=\mu .\hfill\cr}$$ When $t=0$, the $P_\lambda $ reduce to the Schur functions $s_\lambda $, and when $t=1$ to the monomial symmetric functions~$m_\lambda $. \goodbreak \bigskip \noindent {\bf Remarks.} \nobreak \ssection {\rm 1}| Let $n$ be a positive integer, and arrange the partitions of $n$ in lexicographical order, so that $(1^n)$ comes first and $(n)$ comes last. For example, $$(1^4),\,(2,1^2),\,(2^2),\,(3,1),\,(4),$$ if $n=4$. This is a {\it total} ordering $L_n$ of the set ${\cal P}_n$ of partitions of~$n$, and correspondingly defines a totally ordered basis of $\Lambda ^n$ : $$m_{(1^n)}(=e_n),\ldots, m_\lambda ,\ldots,m_{(n)}(=p_n).$$ Now suppose we are given a (positive definite) scalar product $\langle \kern4pt,\kern3pt\rangle $ on the space $\Lambda _{\bb R}^n$. Then by the Gram-Schmidt process we can derive a unique basis $(u_\lambda )$ of $\Lambda _{\bb R}^n$ with the following two properties : $$\displaylines{ \kern1cm({\rm A}')\quad u_\lambda =m_\lambda +\vtop{\hbox{a linear combination of the $m_\mu $} \hbox{for partitions $\mu $ that precede $\lambda $ in $L_n$ ;}}\hfill\cr \kern1cm({\rm B}')\quad \langle u_\lambda ,u_\mu \rangle _{\lower2pt\hbox{$\scriptstyle \alpha $}}=0\ {\rm if}\ \lambda \not=\mu .\hfill\cr}$$ If we replace $L_n$ by some other total ordering of ${\cal P}_n$, and apply Gram-Schmidt as before, we should expect in general to end up with a different basis $(u_\lambda )$. What in fact happens in each of the cases (5) --- (8) is that, for the appropriate scalar product, the basis obtained is {\it independent} of the total ordering chosen, provided only that it is compatible with the partial ordering~(1.1). (The lexicographical order $L_n$ satisfies this condition : if $\mu <\lambda $ then $\mu $ precedes $\lambda $ in $L_n$.) In other words, the conditions (A) and~(B) (in each of the cases (5)~---~(8)) {\it overdetermine} the corresponding family of symmetric functions. \ssection {\rm 2}| Let $V$ be a vector space (over some field $F$) and let $S=S(V)$ be the symmetric algebra of $V$. (If $x_1$, $x_2$, \dots\ is a basis of $V$ then $S=F[x_1,x_2,\ldots\,]$, the polynomial algebra over $F$ generated by the $x_i$.) Now suppose we are given a scalar product $\langle u,v\rangle $ on $V$, with values in $F$. This scalar product has a natural extension to $S$, defined by $$\langle u_1\ldots u_m,v_1\ldots v_n\rangle =\cases{0,&if $m\not=n$,\cr {\rm per}(\langle u_i,v_j\rangle ),&if $m=n$,\cr}$$ where ${\rm per}(\langle u_i,v_j\rangle )$ is the permanent of the matrix of scalar products $\langle u_i,v_j\rangle $ $(1\leq i,j\leq n)$. Here the $u$'s and $v$'s are arbitrary elements of~$V$. In particular, let $V$ be the vector subspace of $\Lambda _F$ spanned by the power sums $p_r$ $(r\geq 1)$, so that $S=F[p_1,p_2,\ldots\,]=\Lambda _F$. Suppose that $\langle \kern4pt,\kern3pt\rangle $ is a scalar product on $V$ for which the $p_r$ are mutually orthogonal, say $$\langle p_r,p_s\rangle =\delta _{rs}a_r.$$ Then the natural extension of this scalar product to $S=\Lambda _F$ is such that $$\langle p_\lambda ,p_\mu \rangle =\delta _{\lambda \mu }z_\lambda a_\lambda ,$$ where $a_\lambda =a_{\lambda _1}a_{\lambda _2}\ldots$ for any partition~$\lambda $. For the scalar products (5)~---~(8) above, the $a_r$ are respectively 1, 2, $\alpha $, and $(1-t^r)^{-1}$. %%%%%%%%%%%%%%% \let\fled=\Longrightarrow \titre 2. The symmetric functions $P_\lambda (q,t)$| Let $q,t$ be independent indeterminates and let $F={\bb Q}(q,t)$ be the field of rational functions in $q$ and $t$. We shall now change the scalar product yet again, and define $$\displaylines{\rlap{(2.1)}\hfill \langle p_\lambda ,p_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}} =\delta _{\lambda \mu }z_\lambda (q,t)\hfill\cr \noalign{\hbox{where}} \rlap{(2.1)}\hfill z_\lambda (q,t)=z_\lambda \prod _{i=1}^{\ell(\lambda )} {1-q^{\lambda _i} \over 1-t^{\lambda _i}}.\hfill\cr}$$ It is perhaps better to think of the parameters $q$ and $t$ as real variables lying in the interval $(0,1)$ of ${\bb R}$, so that the scalar product (2.1) is positive definite. The main result of this section is the following existence theorem. \th \noindent (2.3) Theorem|For each partition $\lambda $ there is a unique symmetric function $P_\lambda =P_\lambda (q,t)\in \Lambda _F$ such that $$\displaylines{ \kern 1cm{\rm (A)}\quad P_\lambda =m_\lambda +{\sum _{\mu <\lambda } u_{\lambda \mu } m_\mu }\hfill\cr \noalign{\hbox{with coefficients $u_{\lambda \mu }\in F$ ;}} \kern 1cm{\rm (B)}\quad \langle P_\lambda ,P_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}}=0\quad{\rm if}\ \lambda \not=\mu .\hfill\cr}$$ \finth As remarked in the previous section, these two conditions overdetermine the $P_\lambda $, and their existence therefore requires proof. Before embarking on the proof, let us consider some particular cases. (1) When $q=t$, the scalar product (2.1) reduces to the `usual' scalar product (1.9), and hence $P_\lambda (q,q)$ is the Schur function $s_\lambda $. (2) When $q=0$, (2.1) reduces to (1.13), and hence $P_\lambda (0,t)$ is the Hall-Littlewood function $P_\lambda (t)$. (3) Let $q=t^\alpha $ $(\alpha \in {\bb R},\ \alpha >0)$ and let $t\rightarrow 1$, so that $q\rightarrow 1$ also. Then $${1-q^m \over 1-t^m}={1-t^{\alpha m} \over 1-t^m} \rightarrow \alpha ,$$ as $t\rightarrow 1$, for all $m$. Hence the scalar product (2.1) tends to (1.11) as $t\rightarrow 1$ and hence $$\lim_{t\rightarrow 1} P_\lambda (t^\alpha ,t)$$ is the Jack symmetric function $P_\lambda (\alpha )$. (4) When $t=1$ (and $q$ is arbitrary) we have $P_\lambda (q,1)=m_\lambda $. (5) When $q=1$ (and $t$ is arbitrary) we have $P_\lambda (1,t)=e_{\lambda '}$. (6) Finally, it is clear from (2.2) that $$z_\lambda (q^{-1}, t^{-1})=(q^{-1}t)^n z_\lambda (q,t),$$ if $\lambda $ is a partition of $n$. Hence on each $\Lambda _F^n$, the scalar products $\langle \kern4pt,\kern3pt\rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}}$ and $\langle \kern4pt,\kern3pt\rangle _{\lower2pt\hbox{$\scriptstyle q^{-1},t^{-1}$}}$ are proportional, and therefore $$P_\lambda (q^{-1},t^{-1})=P_\lambda (q,t).$$ We may summarize these special cases in the following diagram, in which the point $(q,t)$ represents the basis $(P_\lambda (q,t))$ of $\Lambda _F$ (or of $\Lambda _{\bb R}$, since we are regarding $q$, $t$ as real numbers). $$\vbox{\tabskip=0pt \offinterlineskip \segment(12.5,-51.5)\dir(1,1)\long{36} \newbox\stratbox \setbox\stratbox=\hbox{\vrule height 15pt depth 5pt width 0pt} \def\strat{\relax\ifmmode\copy\stratbox\else\unhcopy\stratbox\fi} \newbox\strotbox \setbox\strotbox=\hbox{\vrule height 1.8cm depth 1.8cm width 0pt} \def\strot{\relax\ifmmode\copy\strotbox\else\unhcopy\strotbox\fi} \def\cle#1{\hbox to 3.6cm{\hfil#1\hfil}} %\def\trait#1{\llap{\strat#1\thinspace}\vrule } \setbox3=\hbox{$\uparrow$} \dimen3=\wd3 \divide\dimen3 by 2 \setbox4=\hbox{$t$} \dimen4=\wd4 \divide\dimen4 by 2 \halign{#&#&#\cr \multispan2\phantom{$\bigl(P_\lambda (t)\bigr)\;$}\kern-\dimen4\kern.2pt\box4\hfil&\cr \noalign{\vskip 3pt } \multispan2\phantom{$\bigl(P_\lambda (t)\bigr)\;$}\kern-\dimen3\kern.3pt\box3\hfil&\cr \noalign{\vskip -3pt } \hfil$(0,1)\;$&\strat\vrule\cle{$(m_\lambda )$}&\vrule$\;(1,1)$\cr \noalign{\hrule} $\bigl(P_\lambda (t)\bigr)\;$&\strot\vrule\hfil\raise12pt\cle{$(s_\lambda )$}\hfil& \vrule$\;(e_{\lambda '})$\hfil\cr \noalign{\vbox to 0pt{\vss \hbox{\hbox to 6cm{\rightarrowfill}\kern5pt\smash{$q$}} \vss}} \hfil$(0,0)\;$&\strat\vrule\hfil&\vrule$\;(1,0)$\hfil\cr}}$$ \thinspace \noindent At each point $(q,q)$ on the diagonal of the square we have the Schur functions $s_\lambda $, at each point on the upper edge $(t=1)$ the monomial symmetric functions $m_\lambda $, and so on. In this scheme the Jack functions $P_\lambda (\alpha )$, for varying $\alpha $, correspond to the points in the infinitesimal neighbourhood of the point $(1,1)$, and more precisely $P_\lambda (\alpha )$ corresponds to the direction through $(1,1)$ with slope $1/\alpha $. Notice that the bottom edge $(t=0)$ of the square remains unmarked ; I do not know if the $P_\lambda (q,0)$ have any reasonable interpretation (except that they can be derived from the Hall-Littlewood functions $P_\lambda (0,t)$ by duality (\S \kern 2pt 3)). \medskip Let $x=(x_1,x_2,\ldots\,)$ and $y=(y_1,y_2,\ldots\,)$ be two sequences of independent indeterminates over $F={\bb Q}(q,t)$, and define $$\displaylines{\rlap{(2.4)}\hfill \Pi =\Pi (x,y;q,t)=\prod _{i,j} {(tx_iy_j;q)_\infty \over (x_iy_j;q)_\infty },\hfill\cr \noalign{\hbox{where as usual}} (a;q)_\infty =\prod _{r=0}^\infty (1-aq^r)\cr}$$ for any $a$ for which the product on the right makes sense. Then we have $$\Pi (x,y;q,t)=\sum _\lambda z_\lambda (q,t)^{-1} p_\lambda (x)p_\lambda (y).\leqno(2.5)$$ {\it Proof}\pointir We compute $\exp(\log \Pi )$ ; first of all, $$\leqalignno{\log \Pi &=\sum _{i,j} \sum _{r=0}^\infty \bigl(\log(1-x_iy_jq^r)^{-1} -\log(1-tx_iy_jq^r)^{-1}\bigr)\cr &=\sum _{i,j} \sum _{r\geq 0} \sum _{n\geq 1} {1\over n}(x_iy_jq^r)^n(1-t^n)\cr &=\sum _{n\geq 1} {1\over n} {1-t^n \over 1-q^n} p_n(x) p_n(y)\cr \noalign{\hbox{and therefore}} \Pi &=\prod _{n\geq 1} \exp\biggl({1\over n} {1-t^n \over 1-q^n} p_n(x)p_n(y)\biggr)\cr &=\prod _{n\geq 1} \sum _{m_n=0}^\infty {1\over m_n!} \biggl({1\over n} {1-t^n \over 1-q^n} p_n(x) p_n(y)\biggr)^{\kern-2pt m_n}\cr}$$ in which the coefficient of $p_\lambda (x)p_\lambda (y)$, where $\lambda =(1^{m_1}2^{m_2}\ldots\,)$ is seen to be $z_\lambda (q,t)^{-1}$.\cqfd \thc (2.6)|For each integer $n\geq 0$ let $(u_\lambda )$, $(v_\lambda )$ be $F$-bases of $\Lambda _F^n$, indexed by the partitions $\lambda $ of~$n$. Then the following statements are equivalent : \thinspace \noindent {\rm (a)}\quad $\langle u_\lambda ,v_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}}=\delta _{\lambda \mu }$ for all $\lambda $, $\mu $ $($i.e., $(u_\lambda )$, $(v_\lambda )$ are dual bases of $\Lambda _F^n$ for the scalar product $(2.1))$ ; \thinspace \noindent {\rm (b)}\quad $\displaystyle \sum _\lambda u_\lambda (x) v_\lambda (y) = \Pi (x,y;q,t)$. \finthc {\it Proof}\pointir Let $p_\lambda ^*=z_\lambda (q,t)^{-1} p_\lambda $, so that $$ \langle p_\lambda ^*,p_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}}=\delta _{\lambda \mu }.$$ Suppose that $$\displaylines{u_\lambda =\sum _\rho a_{\lambda \rho }p_\rho ^*, \qquad v_\lambda =\sum _\sigma a_{\mu \sigma }p_\sigma .\cr \noalign{\hbox{Then we have}} \langle u_\lambda ,v_\mu \rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}}=\sum _\rho a_{\lambda \rho } b_{\mu \rho },\cr \noalign{\hbox{so that (a) is equivalent to}} \rlap{(${\rm a}'$)}\hfill \sum _\rho a_{\lambda \rho } b_{\mu \rho }=\delta _{\lambda \mu }.\hfill\cr}$$ On the other hand, by (2.5), (b) is equivalent to $$\leqalignno{ \sum _\lambda u_\lambda (x) v_\lambda (y)&=\sum _\rho p_\rho ^*(x) p_\rho (y)\cr \noalign{\hbox{and hence to}} \sum _\lambda a_{\lambda \rho } b_{\lambda \sigma }&=\delta _{\rho \sigma }.&({\rm b}')\cr}$$ Since $({\rm a}')$ and $({\rm b}')$ are equivalent ($({\rm a}')$ says that $AB^t=1$, where $A$ is the matrix $(a_{\lambda \mu })$ and $B$ the matrix $(b_{\lambda \mu })$, and $({\rm b}')$ says that $A^tB=1$), it follows that (a) and (b) are equivalent.\cqfd After these preliminaries we can embark on the proof of the existence theorem~(2.3). The idea of the proof is as follows : we shall work initially with a finite set of variables $x=(x_1,\ldots,x_n)$ and construct an $F$-linear map (or operator) $$\leqalignno{D&=D_{q,t}:\Lambda _{n,F}\rightarrow \Lambda _{n,F}\cr \noalign{\hbox{having the following properties :}} Dm_\lambda &=\sum _{\mu \leq \lambda } c_{\lambda \mu } m_\mu &(2.7.1)\cr \noalign{\hbox{{\it for each partition $\lambda $ of length $\leq n$} ;}} \langle Df,g\rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}} &=\langle f,Dg\rangle _{\lower2pt\hbox{$\scriptstyle (q,t)$}}&(2.7.2)\cr \noalign{\hbox{for all $f,g\in \Lambda _F$ ;}} \lambda \not=\mu &\Rightarrow c_{\lambda \lambda }\not=c_{\mu \mu }.&(2.7.3)\cr}$$ These three properties say respectively that the matrix of $D$ relative to the basis $(m_\lambda )$ is triangular (2.7.1) ; that $D$ is self-adjoint (2.7.2) ; and that the eigenvalues of $D$ are distinct (2.7.3). The $P_\lambda $ are then just the eigenfunctions (or eigenvectors) of the operator $D$. Namely we have \thc (2.8)|For each partition $\lambda $ $($of length $\leq n)$ there is a unique symmetric polynomial $P_\lambda \in \Lambda _{n,F}$ satisfying the two conditions $$\displaylines{ \kern1cm {\rm (A)}\quad P_\lambda =\sum _{\mu \leq \lambda } u_{\lambda \mu } m_\mu \hfill\cr \noalign{\hbox{where $u_{\lambda \mu }\in F$ and $u_{\lambda \lambda }=1$ ;}} \kern1cm {\rm (C)}\quad DP_\lambda =c_{\lambda \lambda }P_\lambda .\hfill\cr}$$ \finthc {\it Proof}\pointir From (A) and (2.7.1) we have $$\eqalignno{ DP_\lambda &=\sum _{\mu \leq \lambda } u_{\lambda \mu } Dm_\mu \cr &={\sum _{\nu \leq \mu \leq \lambda } u_{\lambda \mu }c_{\mu \nu }m_\nu }\cr \noalign{\hbox{and}} c_{\lambda \lambda }P_\lambda &=\sum _{\nu \leq \lambda } c_{\lambda \lambda }u_{\lambda \nu }m_{\nu },\cr \noalign{\hbox{so that (A) and (C) are satisfied if and only if}} c_{\lambda \lambda }u_{\lambda \nu }&=\sum _{\nu \leq \mu \leq \lambda } u_{\lambda \mu } c_{\mu \nu },\cr \noalign{\hbox{that is to say, if and only if}} (c_{\lambda \lambda }-c_{\nu \nu })u_{\lambda \nu }&= \sum _{\nu <\mu \leq \lambda } u_{\lambda \mu } c_{\mu \nu }\cr}$$ whenever $\nu <\lambda $. Since $c_{\lambda \lambda }\not=c_{\nu \nu }$ by (2.7.3), this relation determines $u_{\lambda \mu }$ uniquely in terms of the $u_{\lambda \mu }$ such that $\nu <\mu \leq \lambda $. Hence the coefficients $u_{\lambda \mu }$ in (A) are uniquely determined, given that $u_{\lambda \lambda }=1$.\cqfd %%%%%%%%%%%%%%% With the $P_\lambda $ as defined in (2.8) we have, by the self-adjointness of~$D$, $$\eqalign{ c_{\lambda \lambda }\langle P_\lambda ,P_\mu \rangle _{q,t} &=\langle DP_\lambda ,P_\mu \rangle _{q,t}\cr &=\langle P_\lambda ,DP_\mu \rangle _{q,t} =c_{\mu \mu }\langle P_\lambda ,P_\mu \rangle _{q,t}.\cr}$$ But $c_{\lambda \lambda }\not=c_{\mu \mu }$ if $\lambda \not=\mu $, hence $\langle P_\lambda ,P_\mu \rangle _{q,t}=0$ if $\lambda \not=\mu $. This establishes (2.3) when the number of variables is finite (i.e. in $\Lambda _{n,F}$ rather $\Lambda _F$). As explained in \S \kern2pt 1, we may then compute the coefficients $u_{\lambda \mu }$ in (A) by Gram-Schmidt ; they will involve only the scalar products $\langle m_\mu ,m_\nu \rangle _{q,t}$, which are independent of~$n$. Hence the $P_\lambda $ are well-defined as elements of~$\Lambda _F$. The proof of (2.3) therefore reduces to the construction of an operator $D$ satisfying (2.7.1) --- (2.7.3). Let $$\leqalignno{ \Delta &=\prod _{1\leq i\lambda '} a_{\lambda \mu } s_{\mu '}$$ and therefore $P_{\lambda '}(t,q)$ and $\omega P_\lambda (q,t)$ have only $s_{\lambda '}$ in common, so that $$\langle \omega P_\lambda (q,t),P_{\lambda '}(t,q)\rangle =\langle s_{\lambda '},s_{\lambda '}\rangle =1$$ as required. This completes the proof of (3.5).\cqfd Since $(P_\lambda )$, $(Q_\lambda )$ are dual bases of $\Lambda _F$, we have from (2.6) $$\sum _\lambda P_\lambda (x;q,t)Q_\lambda (y;q,t)=\Pi (x,y;q,t).\leqno(3.11)$$ Apply $\omega _{q,t}$ to the $y$-variables ; from (2.5) it is easily computed that $$\omega _{q,t}\Pi (x,y;q,t)=\prod _{i,j} (1+x_iy_j).$$ Hence it follows from (3.5) that $$\sum _\lambda P_\lambda (x;q,t)P_{\lambda '}(y;t,q)=\prod _{i,j} (1+x_iy_j).\leqno(3.12)$$ When $q=t$, (3.11) and (3.12) reduce to the familiar identities $$\displaylines{ \sum _\lambda s_\lambda (x)s_\lambda (y)=\prod _{i,j} (1-x_iy_j)^{-1}\cr \noalign{\hbox{and}} \sum _\lambda s_\lambda (x)s_{\lambda '}(y)=\prod _{i,j} (1+x_iy_j)\cr}$$ respectively. Finally, we can now identify the symmetric functions $P_\lambda (q,t)$ when either $q=1$ or $t=1$. Suppose first that $t=1$. Then from (2.10) we have $$D_{q,1}=\sum _{i=1}^n T_{q,x_i}$$ so that (for any partition $\lambda $ of length $\leq n$) $$D_{q,1}m_\lambda =\biggl(\sum _{i=1}^n q^{\lambda _i}\biggr) m_\lambda .$$ Hence the $m_\lambda $ are the eigenfunctions of the operator $D_{q,1}$, and therefore $$P_\lambda (q,1)=m_\lambda \leqno(3.13)$$ for all partitions $\lambda $. Next, it follows from (3.5) that $$\scal \omega _{q,t}P_\lambda (q,t),P_{\mu '}(t,q)|t,q|=\delta _{\lambda \mu }$$ and hence by (3.4) that $$\langle P_\lambda (q,t),\omega P_{\mu '}(t,q)\rangle =\delta _{\lambda \mu }.$$ By (3.13) this gives $$\langle m_\lambda ,\omega P_{\mu '}(1,q)\rangle =\delta _{\lambda \mu }$$ when $t=1$. But the basis dual to $(m_\lambda )$ for the usual scalar product is $(h_\lambda )$ ($[{\rm M}_1]$, ch.~I). Hence $\omega P_{\mu '}(1,q)=h_\mu $, i.e., $P_{\mu '}(1,q)=\omega h_\mu =e_\mu $. Hence (replacing $q$, $\mu '$ by $t$, $\lambda $) $$P_\lambda (1,t)=e_{\lambda '}.\leqno(3.14)$$ %%%%%%%%%%%%%% \titre 4. Skew $P$ and $Q$ functions| Let $\mu $ and $\nu $ be two partitions. Then the product $P_\mu P_\nu $ is a linear combination of the $P_\lambda $, say $$\displaylines{\rlap{(4.1)}\hfill P_\mu P_\nu =\sum _\lambda f_{\mu \nu }^\lambda P_\mu \hfill\cr \noalign{\hbox{where}} f_{\mu \nu }^\lambda =f_{\mu \nu }^\lambda (q,t) =\scal Q_\lambda ,P_\mu P_\nu |q,t|\in F.\cr}$$ In particular (1) $f_{\mu \nu }^\lambda (t,t)$ is the coefficient $c_{\mu \nu }^\lambda $ of $s_\lambda $ in $s_\mu s_\nu $, which may be calculated by the Littlewood-Richardson rule. \thinspace (2) $f_{\mu \nu }^\lambda (0,t)$ is the {\it Hall polynomial} $f_{\mu \nu }^\lambda (t)$ ($[{\rm M}_1]$, ch.~II). It may be written as a sum $$f_{\mu \nu }^\lambda (t)=\sum _T f_{\lower 2pt\hbox{$\scriptstyle T$}}(t)$$ of monic polynomials, where $T$ runs through the set of LR-tableaux of shape $\lambda '-\mu '$ and weight $\nu '$ ({\it loc. cit.\/}) (3) $f_{\mu \nu }^\lambda (q,1)$ is the coefficient of $m_\lambda $ in $m_\mu m_\nu $, and so is independent of~$q$. (4) $f_{\mu \nu }^\lambda (1,t)=1$ if $\lambda =\mu +\nu $, and is zero otherwise. For $P_\mu (1,t)=e_{\mu '}$, and $e_{\mu '}e_{\nu '}=e_{\mu '\cup \nu '}=e_{(\mu +\nu )'}$. (5) $f_{\mu \nu }^\lambda (q^{-1},t^{-1})=f_{\mu \nu }^\lambda (q,t)$, since $P_\lambda (q^{-1},t^{-1})=P_\lambda (q,t)$ (\S \kern2pt 2). (6) By duality (3.5) we have $$f_{\mu \nu }^\lambda (q,t) =f_{\mu '\nu '}^{\lambda '}(t,q)b_{\mu '}(t,q)b_{\nu '}(t,q)/b_{\lambda '}(t,q).$$ In view of (1) and (2) above, it is natural to ask whether it is possible to attach to each LR-tableau $T$ a non zero rational function $f_{\lower 2pt\hbox{$\scriptstyle T$}}(q,t)$ so that $$f_{\mu \nu }^\lambda (q,t)=\sum _T f_{\lower 2pt\hbox{$\scriptstyle T$}}(q,t),$$ where $T$ runs through the set of LR-tableaux of shape $\lambda -\mu $ and weight~$\nu $. (If so, then by duality ((6) above) there will be likewise a decomposition $f_{\mu \nu }^\lambda $ over the LR-tableaux of shape $\lambda '-\mu '$ and weight $\nu '$.) Again, is it true that $f_{\mu \nu }^\lambda (q,t)\not=0$ if and only if $c_{\mu \nu }^\lambda \not=0$ ? The answers to these questions are not known, at any rate to the author. Clearly we shall have $f_{\mu \nu }^\lambda =0$ unless $\left|\lambda \right|=\left|\mu \right|+\left|\nu \right|$. In fact, more is true : \thc (4.2)|$f_{\mu \nu }^\lambda =0$ unless $\lambda \supset \mu $ and $\lambda \supset \nu $ $($i.e., the diagram of $\lambda $ contains those of $\mu $ and $\nu $$)$. \finthc \pc STANLEY| [S] gives a proof of (4.2) in the context of Jack's symmetric functions. His proof can be transposed to the present context without difficulty. We now define skew $Q$-functions as follows. If $\lambda $, $\mu $ are partitions, define $$\displaylines{\rlap{(4.3)}\hfill Q_{\lambda /\mu }=\sum _\nu f_{\mu \nu }^\lambda Q_\nu \hfill\cr \noalign{\hbox{so that}} \scal Q_{\lambda /\mu },P_\nu |q,t|=\scal Q_\lambda ,P_\mu P_\nu |q,t|.\cr}$$ From (4.2) if follows that $Q_{\lambda /\mu }=0$ unless $\lambda \supset \mu $. Let $x=(x_1,x_2,\ldots\,)$ and $y=(y_1,y_2,\ldots\,)$ be two sequences of independent indeterminates. If $f$ is any symmetric function, $f(x,y)$ shall mean $f(x_1,y_1,x_2,y_2,\ldots\,)$, and likewise we define $f(x,y,z,\ldots\,)$ for three or more sequences $x,y,z,\ldots\,$ Then we have $$Q_{\lambda /\mu }(x,y)=\sum _\nu Q_{\lambda /\nu }(x) Q_{\nu /\mu }(y)\leqno(4.4)$$ {\it summed over partitions $\nu $ such that $\lambda \supset \nu \supset \mu $.} \medskip {\it Proof}\pointir We have $$\leqalignno{\sum _\lambda Q_{\lambda /\mu }(x)P_\lambda (y) &=\sum _{\lambda ,\nu } f_{\mu \nu }^\lambda Q_\nu (x) P_\lambda (y)\cr &=\sum _\nu Q_\nu (x) P_\mu (y) P_\nu (y)\cr &=P_\mu (y) \Pi (x,y)\cr \noalign{\hbox{by (3.11). Hence}} \sum _{\lambda ,\mu } Q_{\lambda /\mu }(x) P_\lambda (y) Q_\mu (z) &=\Pi (x,y)\Pi (z,y)\cr &=\sum _\lambda P_\lambda (y) Q_\lambda (x,z)\cr}$$ by (3.11) again. By comparing the coefficients of $P_\lambda (y)$ on either side, we obtain $$Q_\lambda (x,z)=\sum _\mu Q_{\lambda /\mu }(x) Q_\mu (z).\leqno(4.5)$$ If we now replace $x$ by $x,y$ in (4.5), we have $$\eqalign{\sum _\mu Q_{\lambda /\mu }(x,y) Q_\mu (z) &=Q_\lambda (x,y,z)\cr &=\sum _\nu Q_{\lambda /\nu }(x) Q_\nu (y,z)\cr &=\sum _{\mu ,\nu } Q_{\lambda /\nu }(x) Q_{\nu /\mu }(y)Q_\mu (z)\cr}$$ by two applications of (4.5). If we now equate the coefficients of $Q_\mu (s)$ at either end of this string of equalities, we shall obtain~(4.4).\cqfd The identity (4.4) clearly generalizes to $n$ sets of variables $x^{(1)},\ldots, x^{(n)}$ : if $\lambda $, $\mu $ are partitions, then $$Q_{\lambda /\mu }(x^{(1)},\ldots,x^{(n)}) =\sum _{(\nu )}\prod _{i=1}^n Q_{\nu ^i/\nu ^{i-1}}(x^{(i)})\leqno(4.6)$$ {\it summed over all sequences $(\nu )=(\nu ^0,\nu ^1,\ldots,\nu ^n)$ of partitions such that $\mu =\nu ^0\subset \nu ^1\subset \cdots\subset \nu ^n=\lambda $.} \medskip Let us apply (4.6) in the case where each set of variables $x^{(i)}$ consists of a single element~$x_i$. For a single~$x$, we have $$Q_{\lambda /\mu }(x)=\varphi _{\lambda /\mu }\,x^{|\lambda -\mu |}\leqno(4.7)$$ say, where $\varphi _{\lambda /\mu }=\varphi _{\lambda /\mu }(q,t)\in F$, and in fact \thc (4.8)|$\varphi _{\lambda /\mu }=0$ unless $\lambda \supset \mu $ and $\lambda -\mu $ is a horizontal strip, i.e., unless the partitions $\lambda $, $\mu $ are interlaced : $$\lambda _1\geq \mu _1\geq \lambda _2\geq \mu _2\geq \cdots$$ \finthc Again we refer to [S] for a proof of this. As in the case of (4.2), \pc STANLEY|'s proof (for Jack's functions) can be transposed without difficulty (and indeed (4.2) is a a consequence of (4.8)). From (4.6) and (4.7) we obtain $$Q_{\lambda /\mu }(x_1,\ldots,x_n)=\sum _{(\nu )} \prod _{i=1}^n \varphi _{\nu ^i/\nu ^{i-1}} \,x_i^{|\nu ^i-\nu ^{i-1}|}\leqno(4.9)$$ with $(\nu )$ as in (4.6) and each skew diagram $\nu ^i-\nu ^{i-1}$ a horizontal strip. The sequence $(\nu )$ of partitions determines a (column strict) {\it tableau} $T$ of shape $\lambda -\mu $, in which the symbol $i$ occurs in each square of $\nu ^i-\nu ^{i-1}$, for $1\leq i\leq n$. If we now define $$\displaylines{ \rlap{(4.10)}\hfill \varphi _{\lower2pt\hbox{$\scriptstyle T$}}=\prod _{i\geq 1} \varphi _{\nu ^i/\nu ^{i-1}}\hfill\cr \noalign{\hbox{then we shall have}} \rlap{(4.11)}\hfill Q_{\lambda /\mu }=\sum _T \varphi _{\lower2pt\hbox{$\scriptstyle T$}}\, x^T\hfill\cr}$$ summed over all tableaux $T$ of shape $\lambda -\mu $, where $x^T$ is the monomial determined by the tableau $T$, i.e., $x^T=x^\alpha $, where $\alpha $ is the weight of~$T$. Later (\S \kern2pt 5) we shall derive an explicit formula for $\varphi _{\lambda /\mu }$ and hence also for $\varphi _{\lower2pt\hbox{$\scriptstyle T$}}$, and then (4.11) will provide an explicit (if complicated) expression for $Q_{\lambda /\mu }$ as a sum of monomials. Finally, one can define skew $P$-functions $P_{\lambda /\mu }$ by interchanging the roles of the $P$'s and the $Q$'s throughout. The relation between the two is $$P_{\lambda /\mu }=b_\lambda ^{-1} b_\mu Q_{\lambda /\mu }\leqno(4.12)$$ with $b_\lambda $ as defined by (3.1). %%%%%%%%%%%%%%% \titre 5. Explicit formulas| We have introduced various scalars $b_\lambda $, $\varphi _{\lambda /\mu }$, $\varphi _{\lower 2pt\hbox{$\scriptstyle T$}}$ in the preceding sections, but so far we have no way of computing them explicitly. The key to this is a specialization theorem, which will be stated in a moment. Let $u$ be a new indeterminate, and define a homomorphism (or specialization) $$\displaylines{ \epsilon _{u,t}:\Lambda _F\rightarrow F[u]\cr \noalign{\hbox{by}} \rlap{(5.1)}\hfill \epsilon _{u,t}(p_r)={1-u^r \over 1-t^r}\hfill\cr}$$ for each $r\geq 1$. To motivate this definition, suppose that $u=t^n$, $n$ a positive integer. Then $$\leqalignno{ \epsilon _{t^n,t}(p_r)&={1-t^{nr} \over 1-t^r}\cr &=1+t^r+\cdots+t^{(n-1)r}\cr &=p_r(1,t,\ldots,t^{n-1})\cr}$$ so that $$\epsilon _{t^n,t}(f)=f(1,t,\ldots,t^{n-1})$$ for any symmetric function $f$ : i.e., the effect of $\epsilon _{t^n,t}$ is to evaluate at $(x_1,\ldots,x_n,x_{n+1},\ldots\,) =(1,t,\ldots,t^{n-1},0,0,\ldots\,)$. There is a nice formula for $\epsilon _{u,t}(P_\lambda (q,t))$. In order to state it in a convenient form, let us introduce the following notation. For each square $s=(i,j)$ in the diagram of a partition~$\lambda $, let $$\left\{ \vcenter{\halign{$#\hfil$&\quad$#$\hfil\cr a(s)=\lambda _i-j,&a'(s)=j-1,\cr l(s)=\lambda '_j-i,&l'(s)=i-1,\cr}}\right.\leqno(5.2)$$ so that $l'(s)$, $l(s)$, $a(s)$ and $a'(s)$ are respectively the numbers of squares in the diagram of $\lambda $ to the north, south, east and west of the square~$s$. The numbers $a(s)$ and $a'(s)$ may be called respectively the {\it arm-length} and the {\it arm-colength} of~$s$, and $l(s)$, $l'(s)$ the {\it leg-length} and {\it leg-colength}. The {\it hook-length} at~$s$ is $a(s)+l(s)+1$. \th \noindent (5.3) Theorem|We have $$\epsilon _{u,t}\bigl(P_\lambda (q,t)\bigr)= \prod _{s\in \lambda } {q^{a'(s)}u-t^{l'(s)} \over q^{a(s)} t^{l(s)+1} - 1}.$$ \finth {\it Proof}\pointir Since by (5.1) $\epsilon _{u,t}(p_r)$ is a polynomial of degree~$r$ in~$u$ with coefficients in~$F$, it follows that $\epsilon _{u,t}(P_\lambda )$ is a polynomial of degree $\leq \left|\lambda \right|$, say $$\epsilon _{u,t}\bigl(P_\lambda (q,t)\bigr)=\Phi _\lambda (u;q,t)\in F[u].$$ The idea of the proof is first to locate the zeros of this polynomial, which will give the numerator of the expression in (5.3). For this we require two relations. The first comes from the formula (4.5) (with $Q$'s replaced by $P$'s), which gives $$P_\lambda (x_1,\ldots,x_n)=\sum _\mu P_{\lambda /\mu }(x_1)P_\mu (x_2,\ldots,x_n).\leqno(5.4)$$ Set $(x_1,\ldots,x_n)=(1,t,\ldots,t^{n-1})$ and let $$\psi _{\lambda /\mu }=P_{\lambda /\mu }(1;q,t).\leqno(5.5)$$ Then (5.4) becomes $$\Phi _\lambda (t^n;q,t)=\sum _\mu \psi _{\lambda /\mu }(q,t) t^{|\mu |} \Phi _\mu (t^{n-1};q,t)$$ for all $n\geq 1$, and hence we have $$\Phi _\lambda (u;q,t)=\sum _\mu \psi _{\lambda /\mu }(q,t) t^{|\mu |} \Phi _\mu (ut^{-1};q,t).\leqno({\rm A})$$ Next we have $$\epsilon _{u,t} \omega _{t,q}(f)=(-q)^{-n} \epsilon _{u,q^{-1}}(f)\leqno(5.6)$$ if $f\in \Lambda _F^n$. (Since both $\epsilon $ and $\omega $ are ring homomorphisms, it is enough to check (5.6) when $f=p_n$.) Hence by duality (3.5) we have $$\eqalign{\epsilon _{u,t}P_\lambda (q,t)&=\epsilon _{u,t} \omega _{t,q} Q_{\lambda '}(t,q)\cr &=(-q)^{-|\lambda |} \epsilon _{u,q^{-1}} Q_\lambda (t,q)\cr &=(-q)^{-|\lambda |} b_{\lambda '}(t,q) \epsilon _{u,q^{-1}} P_{\lambda '}(t^{-1},q^{-1})\cr}$$ (since $P_{\lambda '}(t,q)=P_{\lambda '}(t^{-1},q^{-1})$) and therefore $$\Phi _\lambda (u;q,t)=(-q)^{-|\lambda |} b_{\lambda '}(t,q) \Phi _{\lambda '}(u;t^{-1},q^{-1}).\leqno({\rm B})$$ We observe next that \thc (5.7)|$P_\lambda (x_1,\ldots,x_n)=0$ if $n<\ell(\lambda )$. \finthc \noindent For $P_\lambda $ is a linear combination of the $m_\mu $ such that $\mu \leq \lambda $, and \smallskip \centerline{$\mu \leq \lambda \Rightarrow \mu '\geq \lambda '\Rightarrow \ell(\mu )=\mu '_1\geq \lambda '_1=\ell(\lambda )>n$} \centerline{${}\Rightarrow \ell(\mu )>n\Rightarrow m_\mu (x_1,\ldots,x_n)=0$.\qed} \medskip It follows from (5.7) that $$\Phi _\lambda (u;q,t)=0\quad {\rm for}\ u=1,t,\ldots,t^{\ell(\lambda )-1}.$$ Hence by (B) the polynomial $\Phi _{\lambda '}(u;t^{-1},q^{-1})$ vanishes also for these values of~$u$. By replacing $(\lambda ,q,t)$ by $(\lambda ',t^{-1},q^{-1})$, we see that $$\Phi _\lambda (u;q,t)=0\quad {\rm for}\ u=1,q^{-1},\ldots,q^{1-\lambda _1}$$ and therefore $$\Phi _\lambda (u;q,t)\ {\it is\ divisible\ in}\ F[u]\ {\it by}\ \prod _{j=1}^{\lambda _1} (q^{j-1}u -1).\leqno({\rm C})$$ Now we consider the relation (A) above. By (C), each term on the right of (A) is divisible by $$\prod _{j=1}^{\lambda _2} (q^{j-1} u - t)$$ (because the sum in (A) is over partitions $\mu $ that interlace~$\lambda $). Hence $\Phi _\lambda (u;q,t)$ is divisible by this product. We now repeat the argument : each term on the right-hand side of~(A) is divisible by $$\prod _{j=1}^{\lambda _3} (q^{j-1} u - t^2)$$ (since $\mu _2\geq \lambda _3$) and therefore $\Phi _\lambda (u;q,t)$ is also divisible by this product. Thus finally $\Phi _\lambda (u;q,t)$ is divisible in $F[u]$ by $$\prod _{i=1}^{\ell(\lambda )} \prod _{j=1}^{\lambda _i} (q^{j-1} u - t^{i-1}) =\prod _{s\in \lambda } (q^{a'(s)} u - t^{l'(s)}).$$ (Observe that all these linear factors are distinct.) But we know that $\Phi _\lambda $ has degree at most $|\lambda |$ in~$u$. Hence we have $$\Phi _\lambda (u;q,t)=v_\lambda (q,t) \prod _{s\in \lambda } (q^{a'(s)} u - t^{l'(s)}),\leqno({\rm D})$$ and it remains to identify the scalar factor $v_\lambda (q,t)$. For this purpose we require \thc (5.8)|Let $\lambda $ be a partition of length $n$, and let $\lambda =(\lambda _1-1,\ldots,\lambda _n-1)$. Then $$P_\lambda (x_1,\ldots,x_n)=x_1\ldots x_nP_\mu (x_1,\ldots,x_n).$$ \finthc \noindent (Compute $D_{q,t}(x_1\ldots x_nP_\mu )$ and show that $x_1\ldots x_n P_\mu $ is an eigenfunction of $D_{q,t}$ with eigenvalue $\sum q^{\lambda _i} t^{n-i}$.) \medskip From (5.8) it follows that $$\Phi _\lambda (t^n;q,t)= t^{n(n-1)/2} \Phi _\mu (t^n;q,t)$$ if $\ell(\lambda )=n$, and hence from (D) that $$\eqalign{ {v_\lambda (q,t) \over v_\mu (q,t)} &=t^{n(n-1)/2} \prod _{s\in \lambda -\mu } \bigl(q^{a'(s)} t^n - t^{l'(s)}\bigr)^{-1}\cr &=\prod _{i=1}^n \bigl(q^{\lambda _i-1} t^{n-i+1} -1\bigr)^{-1}.\cr}$$ Now the factors in this product are precisely $\bigl(q^{a(s)} t^{l(s)+1}-1\bigr)^{-1}$ for $s$ in the first column of~$\lambda $. Hence by induction on the number of columns of~$\lambda $ we conclude that $$v_\lambda (q,t)=\prod _{s\in \lambda }\bigl(q^{a(s)} t^{l(s)+1}-1\bigr)^{-1}$$ and the proof of (5.3) is complete.\cqfd %%%%%%%%%%%% From (B) we have $$b_{\lambda '}(t,q)=(-q)^{|\lambda |} {\Phi _\lambda (u;q,t) \over \Phi _{\lambda '}(u;t^{-1},q^{-1})}$$ which together with (5.3) leads to the formula $$b_\lambda (q,t)= \prod _{s\in \lambda } {1-q^{a(s)}t^{l(s)+1} \over 1-q^{a(s)+1} t^{l(s)}}.$$ So we know now the value of $\scal P_\lambda ,P_\mu |q,t|=b_\lambda (q,t)^{-1}$. Knowing the $b_\lambda $, it turns out that it is now quite straightforward to calculate the scalars $$\varphi _{\lambda /\mu }=Q_{\lambda /\mu }(1)$$ defined in \S \kern 2pt 4. I will omit the details and merely state the result. For each square $s$ and each partition~$\lambda $, define $$b_\lambda (s)=b_\lambda (s;q,t)={1-q^{a(s)} t^{l(s)+1} \over 1-q^{a(s)+1}t^{l(s)}} \leqno(5.10)$$ if $s\in \lambda $, and $b_\lambda (s)=1$ if $s\notin \lambda $. Next, if $S$ is any set of squares (contained in the diagram of $\lambda $ or not), let $$b_\lambda (S)=\prod _{s\in S} b_\lambda (s).\leqno(5.11)$$ Now let $\lambda $, $\mu $ be partitions such that $\lambda _1\geq \mu _1\geq \lambda _2\geq \mu _2\geq \cdots$ or equivalently such that $\lambda \supset \mu $ and $\lambda -\mu $ is a horizontal strip. Let $C_{\lambda /\mu }$ denote the union of the columns that contain squares of $\lambda -\mu $. Then the formula for $\varphi _{\lambda /\mu }$ is $$\leqalignno{\varphi _{\lambda /\mu }&=b_\lambda (C_{\lambda /\mu })/b_\mu (C_{\lambda /\mu })&(5.12)\cr \noalign{\hbox{and for a tableau $T$}} \varphi _{\lower2pt\hbox{$\scriptstyle T$}} &=\prod _{i\geq 0} b_{\lambda ^i}(C_i)/b_{\lambda ^i}(C_{i+1}),&(5.13)\cr}$$ where $\lambda ^0\subset \lambda ^1\subset \cdots$ is the sequence of partitions defined by the tableau, and $C_i$ is the union of the columns that contain a symbol~$i$ (so that $C_0$ is empty). In the case of Jack's symmetric functions, the results of this section are all due to Richard \pc STANLEY|~[S], and the transposition of his proofs into the present context is quite straightforward. Also when $q=0$ the formulas (5.9), (5.12) and (5.13) reduce respectively to $[{\rm M}_1]$, ch.~III, (2.12), (5.8) and~(5.9). Finally we may remark that the identities of Schur and Littlewood $$\eqalign{\sum _\lambda s_\lambda &=\prod (1-x_i)^{-1} \prod _{i