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\topmatter 
\title On bideterminantal formulas for characters of classical
groups
\endtitle 
\author C.~Krattenthaler
\endauthor 
\affil 
Institut f\"ur Mathematik der Universit\"at Wien,\\
Strudlhofgasse 4, A-1090 Wien, Austria.\\
e-mail: KRATT\@Pap.Univie.Ac.At
\endaffil
\address Institut f\"ur Mathematik der Universit\"at Wien,
Strudlhofgasse 4, A-1090 Wien, Austria.
\endaddress
\subjclass Primary 20G05;
 Secondary 05E05, 05E10, 05E15, 20C30, 22E45
\endsubjclass
\keywords Schur functions, general linear characters, symplectic
character, orthogonal character, characters of Lie groups, 
Jacobi--Trudi identity\endkeywords
\abstract New bideterminantal formulas for the irreducible symplectic
and orthogonal characters are given that generalize the classical
bideterminantal formulas.
These formulas are analogous to Regev's
(Israel J. Math. 80 (1992), 155--160) bideterminantal formulas for
Schur functions, the irreducible general linear characters.
Also, new
bideterminantal formulas for Proctor's intermediate symplectic
characters are derived.
\endabstract
\endtopmatter
\document
\TagsOnRight
\rightheadtext{Characters for classical groups}

\subhead 1. Introduction and statement of results \endsubhead
We prove formulas that express the irreducible symplectic and orthogonal
characters (cf\. \cite{1, 2, 3}) as ratios of certain determinants whose
entries involve complete homogeneous, respectively elementary
symmetric functions (see (1.8)--(1.13) below).
These new identities
generalize the classical bideterminantal expressions for irreducible
symplectic and orthogonal characters (cf\. \cite{1, \S~24.2}). 
We were motivated by work of Regev \cite{8} who proved
analogous formulas for Schur functions, the irreducible general linear
characters.

Let us recall
the classical character formulas.
An $n$-tupel 
$\lambda =(\lambda _1,\lambda_2,
\dots ,\lambda _n)$ of integers with $\lambda_1\ge \lambda_2\ge \dots\ge
\lambda_n\ge 0$ is called a {\bf partition}. 
The components of $\lambda$ are called the {\bf parts} of $\lambda$.
The Schur function $s_\lambda (x_1, x_2,\dots ,x_n)$ is given by
(cf\. \cite{4, I.(3.1); 1, p.~403, (A.4)})
$$
s_\lambda (x_1, x_2,\dots ,x_n)=\frac { 
 \det\limits _{1\le i,j\le n}(x_j^{\lambda _i+n-i})} { 
\det\limits_{1\le i,j\le n}(x_j^{n-i})}.\tag 1.1
$$

The irreducible (even) symplectic character $sp_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n)$ is given by (cf\. \cite{1,
(24.18)})
$$
sp_\lambda (x^{\pm1}_1,x^{\pm1}_2,\dots ,x^{\pm1}_n)=\frac { 
\det\limits _{1\le i,j\le n} (x_j^{\lambda _i+n-i+1}-x_j^{-(\lambda _i+n-i+1)})}
{\det\limits_{1\le i,j\le n} (x_j^{n-i+1}-x_j^{-(n-i+1)})}.
\tag 1.2
$$

Odd symplectic characters have been defined by Proctor \cite{5,
6, 7}.
These correspond to indecomposable representations
of odd symplectic groups.
A bideterminantal formula for the odd
symplectic character $sp_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,1)$ appeared in \cite{5,
Theorem~2.2}.
These characters are indexed by partitions
$\lambda=(\lambda_1,\lambda_2,\dots,\lambda_{n+1})$ with $n+1$ parts.
Proctor's
formula may be written as
$$
sp_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,1)=\left.\frac { 
\det\limits _{1\le i,j\le n+1} (x_j^{\lambda _i+n-i+ {3} /{2}}
+x_j^{-(\lambda _i+n-i+ {3} /{2})})}
{\det\limits_{1\le i,j\le n+1} (x_j^{n-i+ {3} /{2}}+x_j^{-(n-i+
{3} /{2})})}
\right|_{x_{n+1}=1}.
\tag 1.3
$$

Now let $\lambda=(\lambda_1,\dots,\lambda_n)$ again be a partition with $n$
parts.
For the even orthogonal groups, the irreducible orthogonal
character $o_\lambda (x^{\pm1} _{1} ,x^{\pm1} _{2} ,\dots ,x^{\pm1} _{n}
)$ is given by (cf\. \cite{1, (24.40) + first paragraph on
p.~411, 7, Appendix~A2})
$$
o_\lambda (x^{\pm1}_1,x^{\pm1}_2,\dots ,x^{\pm1}_n)=\frac { 
\det\limits _{1\le i,j\le n} (x_j^{\lambda _i+n-i}+x_j^{-(\lambda _i+n-i)})}
{ \det\limits_{1\le i,j\le n}(x_j^{n-i}+x_j^{-(n-i)})}.
\tag 1.4
$$

Finally, for the odd orthogonal groups, the irreducible orthogonal
character $o_\lambda (x^{\pm1} _{1} ,x^{\pm1} _{2} ,
\dots ,x^{\pm1} _{n},1 )$ is given by (cf\. \cite{1, (24.28)})
$$
o_\lambda (x^{\pm1}_1,x^{\pm1}_2,\dots ,x^{\pm1}_n,1)=\frac {\det\limits_{ 1\le i,j\le n }
(x_j^{\lambda _i+n-i+1/2}-x_j^{-(\lambda _i+n-i+1/2)})}
{\det\limits_{ 1\le i,j\le n } (x_j^{n-i+1/2}-x_j^{-(n-i+1/2)})}.
\tag1.5
$$
Let $h_m(x_1,\dots ,x_r)$ denote the $m$-th
complete homogeneous symmetric function in the variables $x_1,x_2,\dots ,x_r$,
and let $e_m(x_1,\dots ,x_r)$ denote the  $m$-th elementary
symmetric function in $x_1,x_2,\dots ,x_r$,
(cf\. \cite{4, pp.~12--15}).
Then Regev's formulas for Schur functions
$s_\lambda $ read as follows, \cite{8, Theorem~1.(b)}
$$
s_\lambda (x_1,\dots ,x_{n+r})=\frac {\det\limits_{ 1\le i,j\le n }
(h_{\lambda _i+n-i} (x_j,\dots ,x _{j+r}
))} {\det\limits_{ 1\le i,j\le n } (
h_{n-i} (x_j,\dots ,x_{j+r}))}, \tag1.6
$$
and if $n-2\le r$ \cite{8, Theorem~1'.(b), corrected}
$$
s_{\lambda '}(x_n,\dots ,x_{r+1})=\frac {\det\limits_{ 1\le i,j\le n }
(e_{\lambda _i+n-i} (x_j,\dots ,x _{j+r}
))} {\det\limits_{ 1\le i,j\le n } (
e_{n-i} (x_j,\dots ,x_{j+r}))}.
\tag 1.7
$$
Here $\lambda '$ denotes the partition conjugate to $\lambda $
(cf\. \cite{4, p.~2}).
Clearly, setting $r=0$ in (1.6) yields
(1.1).

The analogous identities for $sp_\lambda $ and $o_\lambda $ which we
are going to prove in the next section, are the following.
For the
symplectic characters we have
$$
sp_\lambda (x^{\pm1}_1,\dots ,x^{\pm1}_{n+r}(,1))=\frac {\det\limits_{ 1\le i,j\le n }
(h_{\lambda _i+n-i} (x_j^{\pm 1},\dots ,x _{j+r}
^{\pm1}(,1)))} {\det\limits_{ 1\le i,j\le n } (
h_{n-i} (x_j^{\pm 1},\dots ,x_{j+r} ^{\pm 1}(,1)))}, \tag 1.8
$$
and if $n-2\le r$,
$$
sp_{\lambda '}(x^{\pm1}_n,\dots ,x^{\pm1}_{r+1}(,1))=\frac {\det\limits_{ 1\le i,j\le n }
(e'_{\lambda _i+n-i} (x_j^{\pm 1},\dots ,x _{j+r}
^{\pm1}(,1)))} {\det\limits_{ 1\le i,j\le n } (
e'_{n-i} (x_j^{\pm 1},\dots ,x_{j+r} ^{\pm 1}(,1)))}, \tag 1.9
$$
where $e'_m (
):=e_m ()-e_{m-2}()$.
By $h_m(x_1^{\pm1}, \dots,
x_r^{\pm 1}(,1))$ we mean $h_m(x_1,x_1^{-1}, \dots,\mathbreak 
x_r,x_r^{-1}(,1))$, etc.
The notation $(,1)$ has
to be understood in the sense that this group has to be omitted in
case of even symplectic characters, while in case of odd symplectic
characters it is meant that 1 is an additional argument in the
symplectic character and the complete homogenous and the elementary
symmetric functions in (1.8) and (1.9).
Obviously, the identity (1.2) for the even symplectic character 
comes out of the even case of (1.8) by 
setting $r=0$ and multiplying the $j$-th column in both determinants
by $x_j-x_j^{-1}$, for $j=1, 2,\dots ,n$. 
The identity (1.3) cannot be derived
in full generality from (1.8).
The reason is that in (1.3) the
partition $\lambda$ is allowed to have $n+1$ parts which is not true for
(1.8).
However, if $\lambda_{n+1}=0$ then (1.3) and the odd case of (1.8)
with $r=0$ are completely equivalent.
This will be shown at the end of
section~2.

Similarly, for the
orthogonal characters we shall prove
$$
o_\lambda (x^{\pm1}_1,\dots ,x^{\pm1}_{n+r}(,1))=\frac {\det\limits_{ 1\le i,j\le n }
(h'_{\lambda _i+n-i} (x_j^{\pm 1},\dots ,x _{j+r} ^{\pm 1} (,1)))}
{\det\limits_{ 1\le i,j\le n } (h'_{n-i} (x_j^{\pm 1},\dots ,
x_{j+r} ^{\pm 1} (,1)))}, \tag 1.10 
$$
where $h'_m():=h_m()-h_{m-2}()$, and, if $n-2\le r$,
$$
o_{\lambda '}(x^{\pm1}_n,\dots ,x^{\pm1}_{r+1}(,1))=\frac {\det\limits_{ 1\le i,j\le n }
(e_{\lambda _i+n-i} (x_j^{\pm 1},\dots ,x _{j+r} ^{\pm 1} (,1)))}
{\det\limits_{ 1\le i,j\le n } (e_{n-i} (x_j^{\pm 1},\dots ,
x_{j+r} ^{\pm 1} (,1)))}.
\tag 1.11 
$$
Also here the terms $(,1)$ in (1.10) and (1.11) are
optional, depending on whether $o_\lambda$ is meant to be the even or
odd orthogonal character.
Again, (1.4) (the even case) comes out of (1.10) by setting $r=0$
while (1.5) (the odd case) comes out of (1.10) by setting $r=0$ and
multiplying the $j$-th column in both determinants by $x_j^{1/2}-
x_j^{-1/2}$, for $j=1,2,\dots , n$.

Identities (1.6)--(1.11) allow a unified formulation, which we give in
the following theorem.
\proclaim{Theorem~1}
Let $\lambda =(\lambda_1,\lambda_2,\dots,\lambda_n)$ 
be a partition and let $\chi_\lambda(x_1,x_2,\dots,x_m) $ be any one of the
characters $s_\lambda(x_1,x_2,\dots,x_m) ,sp_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_m(,1))$, or $o_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots , x^{\pm1}_m(,1))$.
Then there hold
$$
\chi _\lambda (x_1,x_2,\dots ,x_{n+r})=\frac {\det\limits_{ 1\le i,j\le n }
(\chi _{(\lambda _i+n-i)} (x_j,x_{j+1},\dots ,x _{j+r}
))} {\det\limits_{ 1\le i,j\le n } (
\chi _{(n-i)} (x_j,x_{j+1},\dots ,x_{j+r}))}, \tag 1.12 
$$
and, if $n-2\le r$,
$$
\chi _{\lambda '}(x_n,x_{n+1},\dots ,x_{r+1})=\frac {\det\limits_{ 1\le i,j\le n }
(\chi _{(1^{\lambda _i+n-i})} (x_j,x_{j+1},\dots ,x _{j+r}
))} {\det\limits_{ 1\le i,j\le n } (
\chi _{(1^{n-i})} (x_j,x_{j+1},\dots ,x_{j+r}))}.
\tag1.13 
$$
\endproclaim
Regev notes that the determinants in the denominators of (1.6) and (1.7)
factor.
Namely, \cite{8, Theorem~1.(b)}
$$
 \det\limits_{ 1\le i,j\le n }  (h_{n-i}(x_j,\dots ,
x_{j+r}))= \prod _{1\le i<j\le n}(x_i-x_{j+r})\tag 1.14 
$$
and \cite{8, Theorem~1'.(b)}
$$
\det\limits_{ 1\le i,j\le n }  (e_{n-i}(x_j,\dots ,
x_{j+r}))= \prod _{1\le j\le i\le n-1} (x_i-x_{i-j+r+2}).
\tag 1.15 
$$
Also the determinants in the denominators of (1.8)--(1.11) factor.
Namely we have 
$$
\multline 
\det\limits_{ 1\le i,j\le n }  (h_{n-i}(x_j
^{\pm 1},\dots ,x_{j+r}^{\pm 1}))=\det\limits_{ 1\le i,j\le n }
(h'_{n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1}(,1)))\\
=\prod _{ 1\le i<j\le n }(x_i-x_{j+r})(1-\frac {1}
{x_ix_{j+r}})\endmultline\tag 1.16
$$
and
$$
\multline
\det\limits_{ 1\le i,j\le n }  (e'_{n-i}(x_j
^{\pm 1},\dots ,x_{j+r}^{\pm 1}))=\det\limits_{ 1\le i,j\le n }
(e_{n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1}(,1)))\\
=\prod_{ 1\le j\le i\le n-1 } (x_i-x_{i-j+r+2})(1-\frac {1}
{x_ix_{i-j+r+2}}).
\endmultline\tag 1.17
$$

In the next section we first give the corrected proof of (1.7) (which
is completely analogous to the proof of (1.6) in \cite{8}).
Subsequently, by following the same idea, we prove (1.8)--(1.11), and
thus (1.12) and (1.13), in a uniform fashion.
The proofs of (1.16) and (1.17) are also contained in the next
section.
In section~3 we give a generalization of Theorem~1, which for the
special case of Schur functions already appeared in Regev's paper
\cite{8, p.~159}.
From this generalization (Theorem~2) we
derive more bideterminantal formulas for odd orthogonal
characters and bideterminantal formulas for Proctor's \cite{5,
6} intermediate symplectic characters that interpolate between
symplectic characters and Schur functions. 

\subhead 2. The proofs \endsubhead
For a proof of (1.7) we start by defining three $n\times n$ matrices,
$H_\lambda ,B_\lambda $, and $M^{(r)}$,
$$
\align 
H_\lambda &:=(e_{\lambda _i-i+j}(x_n,\dots ,x_{r+1}))
_{1\le i,j\le n}\\
B_\lambda &:=(e_{\lambda _i+n-i}(x_j,\dots ,x_{j+r}))
_{1\le i,j\le n}\\
M^{(r)} &:=(e_{n-i}(x_j,\dots ,x_{n-1},x_{r+2},\dots ,x_{j+r}))
_{1\le i,j\le n}.
\endalign
$$
We claim that there holds
$$
B_\lambda =H_\lambda \cdot M^{(r)}.\tag 2.1
$$
In order to show this identity, we have to verify
$$
\multline
e_{\lambda _i+n-i}(x_j,\dots ,x_{j+r})\\
=\sum _{k=1} ^{n}e_{\lambda _i
-i+k}(x_n,\dots ,x_{r+1})\, e_{n-k}(x_j,\dots ,x_{n-1},x_{r+2},
\dots ,x_{j+r}).
\endmultline
\tag 2.2
$$
Now, the generating function for elementary symmetric functions is
$$
\sum _{m=0} ^{\infty }e_m(y_1,\dots ,y_s)z^m=\prod _{i=1} ^{s}
(1+y_iz).\tag 2.3
$$
Therefore, by comparing the coefficients of $z^m$ on both sides of
$$
\prod _{i=j} ^{j+r}(1+x_iz)=\prod _{i=n} ^{r+1}(1+x_iz)\cdot 
\bigg(\prod _{i=j} ^{n-1}(1+x_iz)\prod _{i=r+2} ^{j+r}(1+x_iz)\bigg)
$$
we obtain
$$
e_m(x_j,\dots ,x_{j+r})=\sum _{p=0} ^{m}e_{m-p}(x_n,\dots ,x_{r+1})
\, e_p(x_j,\dots ,x_{n-1},x_{r+2},\dots ,x_{j+r}).\tag 2.4
$$
Since the number of arguments in $e_p(x_j,\dots ,x_{n-1},x_{r+2},
\dots ,x_{j+r})$ is $n-1$, the summands at the right-hand side of
(2.4) vanish for $p\ge n$.
Hence, we may replace $p$ by $n-k$, where
$k$ runs from 1 to $n$, thus obtaining (2.2) when we also set
$m=\lambda _i+n-i$.
This establishes (2.1). 

Next, on both sides of
(2.1) we take determinants.
By the dual Jacobi--Trudi identity for Schur
functions (cf\. \cite{4, I.(3.5); 1, (24.11), (A.6)}) we have $\det\limits
H_\lambda =s_{\lambda'}(x_n,\dots ,x_{r+1})$.
By setting $\lambda =0$
in (2.1) we deduce $\det\limits M^{(r)}=\det\limits B_0=\det\limits (e_{n-i}(x_j,\dots
,x_{j+r}))$.
Hence, (1.7) is established. 
\qed
\smallskip

The proofs of (1.8)--(1.11)
are analogous, only the choice of the matrices $H_\lambda ,
B_\lambda ,M^{(r)}$ and the Jacobi--Trudi type identity that is used
differ.
So it will suffice to give a detailed proof only for the even
case of (1.8).
For the odd case of (1.8) and for (1.9)--(1.11) 
we only state the choice of the matrices $H_\lambda ,
B_\lambda ,M^{(r)}$ and give the respective references for the
Jacobi--Trudi type identities.
\smallskip

In order to prove the even case of (1.8) we choose
$$
\align 
H_\lambda &=(h_{\lambda _i-i+1}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1})
\quad \vdots\quad  h_{\lambda _i-i+j}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1})\\
&\hskip5.9cm + h_{\lambda _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1}))
_{1\le i,j\le n}\\
B_\lambda &=(h_{\lambda _i+n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1}))
_{1\le i,j\le n}\\
M^{(r)} &=((-1)^{n-i}e_{n-i}(x_1^{\pm 1},\dots ,x_{j-1}^{\pm
1},x_{j+r+1}^{\pm 1},\dots ,x_{n+r}^{\pm 1}))_{1\le i,j\le n}.
\endalign
$$
The notation for $H_\lambda $ means that the first expression gives
the entries of the first column while the second expression gives the
entries of the remaining columns, $j\ge 2$.
Again we claim that
$$
B_\lambda =H_\lambda \cdot M^{(r)}.\tag 2.5
$$
Here we have to verify
$$
\align 
h_{\lambda _i+n-i}(x_j^{\pm 1},\dots &,x_{j+r}^{\pm 1})=h_{\lambda_i-i+1}
(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1})\cdot (-1)^{n-1}\\
&\hskip2cm \cdot e_{n-1}(x_1^{\pm 1},\dots ,x_{j-1}^{\pm 1},x_{j+r+1}^{\pm 1},
\dots ,x_{n+r}^{\pm 1})\\
&+\sum _{k=2} ^{n}(h_{\lambda _i-i+k}(x_1^{\pm 1},\dots ,x_{n+r}
^{\pm 1})+h_{\lambda _i-i-k+2}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1}))\\
&\hskip2cm\cdot (-1)^{n-k}e_{n-k}(x_1^{\pm 1},\dots ,x_{j-1}^{\pm 1},x_{j+r+1}
^{\pm 1},\dots ,x_{n+r}^{\pm 1}).
\tag  2.6 \endalign
$$
The generating function for the complete homogeneous symmetric
functions is
$$
\sum _{m=0} ^{\infty}h_m(y_1,\dots ,y_s)z^m=\prod _{i=1} ^{s}
\frac {1} {(1-y_iz)}.
\tag 2.7
$$
Therefore, by comparing the coefficients of $z^m$ on both sides of
$$
\multline 
\prod _{i=j} ^{j+r}\frac {1} {(1-x_iz)(1-x_i^{-1}z)}=\prod _{i=1}
^{n+r}\frac {1} {(1-x_iz)(1-x_i^{-1}z)} \\
\bigg( \prod _{i=1} ^{j-1}
(1-x_iz)(1-x_i^{-1}z)\prod _{i=j+r+1} ^{n+r}(1-x_iz)(1-x_i^{-1}z)
\bigg) \endmultline
$$
we get
$$
\multline 
h_m(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1})=\sum _{p=0} ^{m}h_{m-p}
(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1})(-1)^p \\
\cdot e_p(x_1^{\pm 1},\dots ,x_{j-1}^{\pm 1},x_{j+r+1}^{\pm 1},\dots ,
x_{n+r}^{\pm 1}).\endmultline \tag 2.8
$$
The number of arguments in $e_p(x_1^{\pm 1},\dots ,x_{j-1}^{\pm 1},
x_{j+r+1}^{\pm 1},\dots ,x_{n+r}^{\pm 1})$ is $2n-2$.
Hence the
summands at the right-hand side of (2.8) vanish for $p\ge 2n-1$.
Besides, we have
$$
 e_p(x_1^{\pm 1},\dots ,x_{j-1}^{\pm 1},x_{j+r+1}^{\pm 1},
\dots ,x_{n+r}^{\pm 1})
=e_{2n-2-p}(x_1^{\pm 1},\dots ,x_{j-1}^{\pm 1},x_{j+r+1}^{\pm 1},\dots ,
x_{n+r}^{\pm 1}), 
$$
since the product of all $2n-2$ arguments equals 1 and the set of
arguments is invariant under taking inverses.
So, if we reorder the
summands at the right-hand side of (2.8) by singling out 
the $(n-1)$-st summand
and pairing the $p$-th with the $(2n-2-p)$-th summand, for $p=n-2,n-3,
\dots ,1,0$,
after replacing $m$ by $\lambda _i+n-i$ we obtain (2.6). 
This establishes (2.5).
The remaining steps are the same as before in the
proof of (1.7).
This time $\det\limits H_\lambda $ is identified to
be the
symplectic character $sp_\lambda (x^{\pm1}_1,\dots ,x^{\pm1}_{n+r})$ by virtue of
the symplectic Jacobi--Trudi identity (cf\. \cite{1,
Prop.~24.22}). 
\qed
\smallskip

The odd case of (1.8) is established in the same manner.
We choose
$$
\align 
H_\lambda &=(h_{\lambda _i-i+1}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1},1)
\quad \vdots\quad  h_{\lambda _i-i+j}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm
1},1)\\
&\hskip5.5cm + h_{\lambda _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm
1},1))
_{1\le i,j\le n}\\
B_\lambda &=(h_{\lambda _i+n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1},1))
_{1\le i,j\le n}\\
M^{(r)} &=((-1)^{n-i}e_{n-i}(x_1^{\pm 1},\dots ,x_{j-1}^{\pm
1},x_{j+r+1}^{\pm 1},\dots ,x_{n+r}^{\pm 1}))_{1\le i,j\le n}.
\endalign
$$
The Jacobi--Trudi type identity for the odd symplectic character that
has to be used in this instance is \cite{5, p.~317}
$$
\align 
sp_\lambda(x^{\pm1} _{1} ,&
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,1)\\
&=\det_{1\le i,j\le n}
(h_{\lambda _i-i+1}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},1)
\quad \vdots\quad  h_{\lambda _i-i+j}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},1)\\
&\hskip6cm + h_{\lambda _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n}^{\pm
1},1)).
\endalign
$$
\qed
\smallskip

In order to prove (1.9) we choose
$$
\align 
H_\lambda &=(e'_{\lambda _i-i+1}(x_n^{\pm 1},\dots ,x_{r+1}^{\pm
1}(,1))
\quad\vdots\quad  e'_{\lambda _i-i+j}(x_n^{\pm 1},\dots ,x_{r+1}^{\pm
1}(,1))\\
&\hskip5.4cm + e'_{\lambda _i-i-j+2}(x_n^{\pm 1},\dots ,x_{r+1}^{\pm
1}(,1)))
_{1\le i,j\le n}\\
B_\lambda &=(e'_{\lambda _i+n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm
1}(,1)))
_{1\le i,j\le n}\\
M^{(r)} &=(e_{n-i}(x_j^{\pm 1},\dots ,x_{n-1}^{\pm1},x_{r+2}^{\pm 1},
\dots ,x_{j+r}^{\pm 1}))_{1\le i,j\le n}.\endalign
$$
 That $\det\limits H_\lambda $ equals
$sp_{\lambda '}(x^{\pm1}_n,\dots ,x^{\pm1}_{r+1}(,1))$ is due to the dual symplectic
Jacobi--Trudi identities (cf\. \cite{1, Cor.~24.24} for the
even case and \cite{7, Appendix~A2} for the odd case). 
\qed
\smallskip

For the proof of (1.10) we choose
$$
\align 
H_\lambda &=(h'_{\lambda _i-i+1}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1}
(,1))\quad\vdots\quad  h'_{\lambda _i-i+j}(x_1^{\pm 1},\dots 
,x_{n+r}^{\pm 1}(,1)) \\
&\hskip5.3cm + h'_{\lambda _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n+r}^{\pm 1}(,1)))
_{1\le i,j\le n}\\
B_\lambda &=(h'_{\lambda _i+n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1}
(,1)))_{1\le i,j\le n}\\
M^{(r)} &=((-1)^{n-i} e_{n-i}(x_1^{\pm 1},\dots ,x_{j-1}^{\pm1},
x_{j+r+1}^{\pm 1},\dots ,x_{n+r}^{\pm 1}))_{1\le i,j\le n}.
\endalign
$$
The orthogonal Jacobi--Trudi
identities (cf\. \cite{1, Prop.~24.44, Prop.~24.33}) guarantee that
$\det\limits H_\lambda $ equals $o_\lambda (x^{\pm1}_1,\dots
,x^{\pm1}_{n+r}(,1))$. 
\qed
\smallskip

Finally, for proving (1.11) we choose
$$
\align 
H_\lambda &=(e_{\lambda _i-i+1}(x_n^{\pm 1},\dots ,x_{r+1}^{\pm 1}
(,1))\quad\vdots\quad  e_{\lambda _i-i+j}(x_n^{\pm 1},\dots 
,x_{r+1}^{\pm 1}(,1))\\
&\hskip5.4cm + e_{\lambda _i-i-j+2}(x_n^{\pm 1},\dots ,x_{r+1}^{\pm 1}(,1)))
_{1\le i,j\le r}\\
B_\lambda &=(e_{\lambda _i+n-i}(x_j^{\pm 1},\dots ,x_{j+r}^{\pm 1}
(,1)))_{1\le i,j\le r}\\
M^{(r)} &=(e_{n-i}(x_j^{\pm 1},\dots ,x_{n-1}^{\pm1},x_{r+2}^{\pm 1},
\dots ,x_{j+r}^{\pm 1}))_{1\le i,j\le r}.\endalign
$$
Here we use the dual orthogonal Jacobi--Trudi identities
(cf\. \cite{1, Cor.~24.45, Cor.~24.35}). 
\qed

\smallskip
Now we turn to the determinant factorizations (1.16) and (1.17). 
What we do is to change
Regev's proofs of (1.14) and (1.15) into proofs of (1.16) and (1.17).
First we have to find analogues for
Lemmas~5 and 5' (bottom line of p.158) in Regev's paper.
These
analogues can be given in a unified form.
They read
$$
\multline \tilde h_s(x_1^{\pm1},\dots,x_v^{\pm1}(,1))-
\tilde h_s(x_2^{\pm1},\dots,x_{v+1}^{\pm1}(,1))\\
=(x_1-x_{v+1})(1-\frac {1} {x_1x_{v+1}})
\tilde h_{s-1}(x_1^{\pm1},\dots,x_{v+1}^{\pm1}(,1)),
\endmultline\tag2.9
$$
where $\tilde h_m$ is any of $h_m$ or $h'_m$, and
$$
\multline \tilde e_s(x_1^{\pm1},\dots,x_v^{\pm1}(,1))-
\tilde e_s(x_2^{\pm1},\dots,x_{v+1}^{\pm1}(,1))\\
=(x_1-x_{v+1})(1-\frac {1} {x_1x_{v+1}})
\tilde e_{s-1}(x_2^{\pm1},\dots,x_{v}^{\pm1}(,1)).
\endmultline\tag2.10
$$
where $\tilde e_m$ is any of $e_m$ or $e'_m$.
These two identities
are easily proved by means of generating functions.
For a proof of (1.16), let $\det{(n,r)}$
denote the determinant
$$
\det\limits_{ 1\le i,j\le n }  (\tilde h_{n-i}(x_j
^{\pm 1},\dots ,x_{j+r}^{\pm 1}(,1))).
$$
The last row of this determinant consists of all 1's.
Subtract the
$j$-th column from the $(j-1)$-st column, $j=2,\dots,n$, and then
expand in the last row.
By using the
relation (2.9) we obtain
$$
\det{(n,r)}=\det{(n-1,r+1)}\prod _{j=1} ^{n-1}(x_j-x_{1+j+r})(1-\frac {1}
{x_jx_{j+1+r}}).
$$
By induction on $n$ we are done.
The proof of (1.17) is similar.
\qed

\smallskip
At the end of this section we show that (1.3) with $\lambda_{n+1}=0$ 
and the odd case of
(1.8) with $r=0$ are equivalent.
In order to see this, first observe that
because of $x_{n+1}=1$ in both determinants in (1.3) the last column
consists of all 2's.
Hence, if we subtract the last row from all the
other rows in both determinants in (1.3), the last column will
consist of only zeros with the exception of a 2 in the last row.
Expansion in the last column gives
$$
\multline
sp_\lambda (x^{\pm1}_1,x^{\pm1}_2,\dots ,x^{\pm1}_n,1)\\
=\frac { 
\det\limits _{1\le i,j\le n} (x_j^{\lambda _i+n-i+ 3 /2}+
x_j^{-(\lambda _i+n-i+ 3 /2)}-x_j^{ {1} /{2}}-x_j^{- {1}
/{2}})}
{\det\limits_{1\le i,j\le n} (x_j^{n-i+ 3 /2}+x_j^{-(n-i+ 3 /2)}
-x_j^{ {1} /{2}}-x_j^{- {1} /{2}})}.
\endmultline
$$
Now in both determinants we divide the $j$-th column by $x_j^{
{3} /{2}}+x_j^{- {3} /{2}}-x_j^{ {1} /{2}}-x_j^{- {1} /{2}}$
and obtain (1.8) with $r=0$ by using the identity
$$
h_m(x,x^{-1},1)=\frac {x^{m+ {3} /{2}}+x^{-(m+ {3} /{2})}-
x^{ {1} /{2}}-x^{- {1} /{2}}} 
{x^{ {3} /{2}}+x^{- {3} /{2}}-
x^{ {1} /{2}}-x^{- {1} /{2}}}.
$$
That this identity actually holds is easily verified by using
generating functions.
\qed


\subhead 3. A generalization\endsubhead
Here we are going to state a generalization of Theorem~1.
Let $\lambda=(\lambda_1,\dots,\lambda_n)$ be a partition and let
$\z$ denote the sequence $z_1,z_2,\dots,z_m$ of variables.
We
formally define
$$
\align sp_\lambda(x^{\pm1} _{1} &,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,z_1,\dots,z_m)=sp_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,\z)\\
&=\det_{1\le i,j\le n}
(h_{\lambda _i-i+1}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z)
\quad \vdots\quad  h_{\lambda _i-i+j}(x_1^{\pm 1},\dots ,x_{n}^{\pm
1},\z)\\
&\hskip6cm + h_{\lambda _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z))
\tag3.1a\\
&=\det_{1\le i,j\le n}
(e'_{\lambda' _i-i+1}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z)
\quad \vdots\quad  e'_{\lambda' _i-i+j}(x_1^{\pm 1},\dots ,x_{n}^{\pm
1},\z)\\
&\hskip6cm + e'_{\lambda' _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z))
\tag3.1b
\endalign
$$
and 
$$
\align o_\lambda(x^{\pm1} _{1} &,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,z_1,\dots,z_m)=o_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,\z)\\
&=\det_{1\le i,j\le n}
(h'_{\lambda _i-i+1}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z)
\quad \vdots\quad  h'_{\lambda _i-i+j}(x_1^{\pm 1},\dots ,x_{n}^{\pm
1},\z)\\
&\hskip6cm + h'_{\lambda _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z))
\tag3.2a\\
&=\det_{1\le i,j\le n}
(e_{\lambda' _i-i+1}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z)
\quad \vdots\quad  e_{\lambda' _i-i+j}(x_1^{\pm 1},\dots ,x_{n}^{\pm
1},\z)\\
&\hskip6cm + e_{\lambda' _i-i-j+2}(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z))
.\tag3.2b
\endalign
$$
The short notation $h_k(x_1^{\pm 1},\dots ,x_{n}^{\pm 1},\z)$ means 
$h_k(x_1,x_1^{-1},\dots ,x_{n},x_n^{-1},z_1,\dots,\mathbreak z_m)$, and
similarly for $e_k$, $h'_k$, and $e'_k$.
That the determinants in (3.1) agree and that 
the determinants in (3.2) agree is
well-known (see \cite{1, Prop.~A.44, Cor.~A.46} for a proof). 
The expressions $sp_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_m,\z)$ and $o_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,\mathbreak x^{\pm1}_m,\z)$ are sometimes called {\bf
formal} characters.

Now we are able to
formulate the announced generalization of Theorem~1.
\proclaim{Theorem~2}
Let $\lambda =(\lambda_1,\lambda_2,\dots,\lambda_n)$ 
be a partition and let $\chi_\lambda(x_1,x_2,\dots,x_m) $ be any one of the
characters $s_\lambda(x_1,x_2,\dots,x_m,\z) ,sp_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_m,\z)$, or $o_\lambda (x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots , x^{\pm1}_m,\z)$.
Then there hold
$$
\chi _\lambda (x_1,x_2,\dots ,x_{n+r})=\frac {\det\limits_{ 1\le i,j\le n }
(\chi _{(\lambda _i+n-i)} (x_j,x_{j+1},\dots ,x _{j+r}
))} {\det\limits_{ 1\le i,j\le n } (
\chi _{(n-i)} (x_j,x_{j+1},\dots ,x_{j+r}))}, \tag 3.3 
$$
and, if $n-2\le r$,
$$
\chi _{\lambda '}(x_n,x_{n+1},\dots ,x_{r+1})=\frac {\det\limits_{ 1\le i,j\le n }
(\chi _{(1^{\lambda _i+n-i})} (x_j,x_{j+1},\dots ,x _{j+r}
))} {\det\limits_{ 1\le i,j\le n } (
\chi _{(1^{n-i})} (x_j,x_{j+1},\dots ,x_{j+r}))}.
\tag3.4 
$$
\endproclaim
The proof of Theorem~2 is identical with the proof of Theorem~1 and
is therefore omitted.
Clearly, Theorem~1 is just a special case of
Theorem~2.
\qed
\smallskip


The $sp_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,\z)$ which were formally defined in (3.1)
are {\bf actual} characters.
They are characters of ``trace-free"
representations of intermediate symplectic groups \cite{5} that
interpolate between the symplectic groups and the general linear
groups.
Actually, these trace-free characters are indexed by
partitions with up to $n+m$ parts.
So our Theorem~2 only gives
bideterminantal expressions for these characters for partitions $\lambda$
with at most $n$ parts.
It can be shown in a similar manner as at the
end of section~2 that the $\chi_\lambda=sp_\lambda$, $r=0$, $\z=-1$ special case of (3.3) is
equivalent to the $\lambda_{n+1}$-case of the
following bideterminantal formula for the odd
symplectic character evaluated at $x_1^{\pm1},\dots,x_n^{\pm1},-1$,
$$
\multline
sp_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,-1)\\
=\left.\frac { 
\det\limits _{1\le i,j\le n+1} (x_j^{\lambda _i+n-i+ {3} /{2}}
-x_j^{-(\lambda _i+n-i+ {3} /{2})})}
{\det\limits_{1\le i,j\le n+1} (x_j^{n-i+ {3} /{2}}-x_j^{-(n-i+
{3} /{2})})}
\right|_{x_{n+1}=-1}.
\endmultline
\tag 3.5
$$
This identity does not appear in \cite{5} but can be derived
completely analogously as (1.3) was derived in \cite{5, sec.~8}.

A representation-theoretic meaning for the $o_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,\z)$, formally defined in (3.2), is
not so immediate.
Perhaps, if intermediate orthogonal representations
would be defined in analogy with the definition of intermediate
symplectic representations in \cite{5}, then the expressions in
(3.2) would turn out to be characters for these representations.
We
confine ourselves with considering (3.2) with $\z=-1$.
This has a
representation-theoretic meaning, since $o_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,-1)$ is the usual odd orthogonal
character evaluated at the ``negative" part of the orthogonal group.
In
particular, the $\chi_\lambda=o_\lambda$, $r=0$, $\z=-1$ special case of (3.3)
is equivalent to the formula (cf\. \cite{7, proof of A2.1(d)})
$$
o_\lambda(x^{\pm1} _{1} ,
x^{\pm1} _{2} ,\dots ,x^{\pm1}_n,-1)=\frac { 
\det\limits _{1\le i,j\le n} (x_j^{\lambda _i+n-i+ {1} /{2}}
+x_j^{-(\lambda _i+n-i+ {1} /{2})})}
{\det\limits_{1\le i,j\le n} (x_j^{n-i+ {1} /{2}}+x_j^{-(n-i+
{1} /{2})})}.
\tag 3.6
$$

\Refs

\ref\no 1\by W.    Fulton and J. Harris \yr 1991 
\book Representation Theory
\publ Springer--Verlag
\publaddr New York\endref

\ref\no 2\by R. C. King \yr 1971 
\paper Modification rules and products of irreducible representations of the unitary, orthogonal, and symplectic groups
\jour J. Math\. Phys\.\vol 12
\pages 1588--1598\endref

\ref\no 3\by K.    Koike and I. Terada \yr 1987 
\paper Young--diagrammatic methods for the representation theory of the classical groups of type $B_n$, $C_n$, $D_n$
\jour J. Algebra\vol 107
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\ref\no 4\by I. G. Macdonald \yr 1979 \book Symmetric Functions and 
Hall Polynomials \publ Oxford University Press\publaddr New 
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\ref\no 5\by R. A. Proctor \yr 1988 
\paper Odd symplectic groups 
\jour Invent\. Math\.\vol 92
\pages 307--332\endref

\ref\no 6\by R. A. Proctor \yr 1991 
\paper A generalized Berele--Schensted algorithm and conjectured Young tableaux for intermediate symplectic groups
\jour Trans\. Amer\. Math\. Soc\.\vol 324
\pages 655--692\endref

\ref\no 7\by R. A. Proctor \yr 19?? 
\paper Young tableaux, Gelfand patterns, and branching rules for classical Lie groups
\paperinfo to appear in J. Algebra\jour \vol 
\pages \endref

\ref\no 8\by A.    Regev \yr 1992 
\paper A note on Schur functions 
\jour Israel J. Math\. \vol 80 
\pages 155--160\endref

\endRefs
\enddocument
