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%        Makros zum Formatieren des Entwurfs fuer Christian:               %
%        Begonnen am 3. November 1993                                      %
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% 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

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                   % Symmetric group
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                   % symplectic Schur function
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%               GRIDS.TEX: Begonnen am 16. Juli 1993                       %
% Dieses kleine Macropaket soll einfache Befehle zum Zeichnen von Gittern  %
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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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% Ob ich wohl je verstehe, warum	das nicht einfacher funktioniert? Siehe   %
% TeX-Book, p.352. -- Also ehrlich gesagt: Die folgenden Macros sind durch	%
% trial-and-error geworden, wie sie jetzt sind; immerhin funktioniert's|   %
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{\obeylines\gdef\getrowlist
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}
{\obeylines\gdef\getrow #1
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	} %
}
 
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% Die folgenden Macros sind dem Buch ``Einfuehrung in TeX'' von N. Schwarz %
% entnommen, zumindest im wesentlichen: Siehe dort p.115.						%
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\begin{document}
\abovedisplayskip=6pt plus2pt minus 4pt
\belowdisplayskip=6pt plus2pt minus 4pt


\hbox{}\nopagebreak\vskip-1cm
\nopagebreak\hbox{}

\newbox\fullversion
\setbox\fullversion\hbox{{\rm\small (Shortened version. The full-length article will appear
elsewhere)}}
 
\title[Characters of symplectic and orthogonal groups]
{\uppercase
{{\bf Lattice path proofs for determinantal formulas for symplectic and
orthogonal characters}}\\
\unhbox\fullversion} 
 
\author{Markus Fulmek}
\author{Christian Krattenthaler}
\address{Institut f\"ur Mathematik\\Universit\"at Wien\\Strudlhofgasse~4\\
	A-1090 Vienna, Austria}
 
% \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
% 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}
% Franzoesische Uebersetzung von Rodica Simion.
\begin{abstract}
We give bijective proofs for Jacobi--Trudi-type and Giambelli-type
identities for symplectic and orthogonal characters. These proofs
% Kleine Aenderung von Rodica Simion: ``are based'' statt ``base''.
are based on interpreting King and El-Sharkaway's symplectic tableaux,
Proctor's odd and intermediate symplectic tableaux, Proctor'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.

\noindent{\sc R\'esum\'e.}
% Quelle: Comptes Rendus Math. 9 (87), Instructions auf p. 176
On donne des d\'emonstrations bijectives des identit\'es de type
Jacobi--Trudi et Giambelli pour les caract\`eres symplectiques et
orthogonaux.   Ces d\'emonstrations sont bas\'ees sur une interpr\'etation 
des tableaux symplectiques de King et  El-Sharkaway,
% markus.
% Gefuehlsmaeesig sollte es doch eher heissen:
% des tableaux symplectiques impairs et interm\'ediaires de Proctor,
% oder?
% markus.
% Ausgebessert gemaess Deinem Vorschlag.
des tableaux symplectiques impairs et symplectiques interm\'ediaires
de Proctor,
% des tableaux impairs et symplectiques interm\'ediaires de Proctor,
des tableaux  orthogonaux de Proctor,
et des tableaux orthogonaux impairs de Sundaram en termes de certaines
familles de chemins  qui ne s'entrecoupent pas.
Le but de ce travail est de proposer,
dans le cas des caract\`eres symplectiques et orthogonaux,
 des d\'emonstrations  analogues \`a la preuve de
Gessel et Viennot des identit\'es de Jacobi--Trudi pour
les fonctions de Schur.
\end{abstract}
\maketitle

\vskip-1cm


\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.                                         %
%                                                                             %
%	Definitionen schon in Einleitung bringen? Momentan geschieht das nicht|    %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\message{introduction: what will happen here...} 
\ Schur functions, the irreducible general linear characters, can be
combinatorially defined by means of semistandard Young tableaux (see
\cite[I, (5.12)]{Macdonald:Symmetric-Functions}).
There are several
determinant formulas for Schur functions. Those which are relevant
for this paper are the Jacobi--Trudi identity and its dual form,
and the Giambelli identity 
(see \cite[I, (3.4), (3.5), p.~30,
Ex.~9]{Macdonald:Symmetric-Functions}).
In their
well-known (yet unpublished) paper \cite{Gessel-Viennot:Determinants-paths}
Gessel and Viennot give a beautiful bijective proof for the
Jacobi--Trudi identities for Schur functions
%Christian.
(see also \cite[sec.~7+1]{Stembridge:Nonintersecting-paths}). 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.

There are Jacobi--Trudi-type and Giambelli-type determinant formulas
for irreducible symplectic and orthogonal characters 
(see \cite[Prop.~24.22, Cor.~24.24, Prop.~24.44, Prop.~24.33, 
(24.47)]{Fulton-Harris:Representation-Theory}), too. 
Since there are also tableaux
descriptions for symplectic and orthogonal characters, it is natural
to ask for bijective proofs of the symplectic and orthogonal
Jacobi--Trudi and Giambelli identities. 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
%Christian.
(but are interesting in their own right).
Additional algebraic steps were necessary to prove the Jacobi--Trudi
identities themselves.		

We solve the problem completely for the symplectic case
and partially 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}. Particularly nice is the bijective
proof for the {\em dual} symplectic Jacobi--Trudi identity, which
combines the Gessel--Viennot method with a modified reflection
principle. The bijective proof for the ``ordinary" symplectic
Jacobi--Trudi identity is more elaborate. On the other hand, the
bijective proof of the symplectic Giam\-bel\-li identity is almost
trivial. In addition, we are able to 
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.

There are several candidates for orthogonal tableaux. We show that
Proctor's orthogonal tableaux \cite{Proctor:Young-tableaux,Proctor:A-Schensted} are the
``right" tableaux for proving the dual orthogonal Jacobi--Trudi
identity. As might be surprising at first sight, these tableaux have a 
very natural lattice path interpretation. The proof itself is very
illustrative. It also employs the Gessel--Viennot method and some
kind of reflection argument. Unfortunately, we were not able to use
the same tableaux in order to bijectively prove the ``ordinary"
orthogonal Jacobi--Trudi identity and the orthogonal Giambelli
identity. As a substitute, by using Sundaram's tableaux
\cite{Sundaram:Orthogonal} at least we are able to give 
bijective proofs for the odd
orthogonal Giambelli identity and for determinant formulas that are
only slight variations  of the odd orthogonal Jacobi--Trudi
identities.

\section{Some Definitions}\label{Defs}\ 
By {\em paths} we always mean lattice paths in the plane integer
lattice $\mathbb Z^2$ 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 $\mathbf u=(u_1,\dots,u_m)$
and $\mathbf v=(v_1,\dots,v_m)$ 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
$\fpath\of{\mathbf u,\mathbf v}$. 
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 $\mathbb Z^2$ 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 $w(\mathbf\path)$ of an $m$-tuple
$\mathbf\path=(\path_1,\dots,\path_m)$ 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 $\mathcal A$,
by the generating function $\GF{\mathcal A}$ we mean $\sum
_{x\in\mathcal A} ^{}w(x)$. 



\section{Bijective proofs for symplectic
identities}\label{symplectic}\
{\sc Bijective proof of the symplectic dual Jacobi--Trudi identity}. 
Let $\lambda = \parof{\lambda_1,\dots,\lambda_r}$ be a partition
of length $r\le n$. A
%Christian.
(semistandard)
tableau $\tableau$ of shape $\lambda$ with entries from $\{1,2,\dots\}$ is called a {\em symplectic\/}
({\em semistandard}) {\em tableau} (see \cite{Proctor:Odd,Sundaram:Tableaux-in}) 
if it obeys the
additional constraint
\begin{equation}
\label{eq:symplectic-constraint}
\tableau_{i,j} \ge 2i-1.
\end{equation}
Let 
$\mathbf{x} = \seqof{x_1, x_1^{-1},x_2,x_2^{-1},\dots,x_n,x_n^{-1}}$. 
The weight of a symplectic tableau $\tableau$ is given by
\begin{equation}\label{weight:symplectic}
\mathbf{x}^\tableau = \prod_{i}x_i^{\numof{\setof{\tableau_{j,k} = 2i-1}} -
	\numof{\setof{\tableau_{j,k} = 2i}}}.
\end{equation}
The {\em symplectic character\/} 
associated to $\lambda$ is combinatorially defined by (see 
\cite{Proctor:Odd,Sundaram:Tableaux-in})\break 
$ \sschf{2n}{\lambda} =
	\sum_{\tableau}\mathbf{x}^{\tableau}, $
where the sum is over all symplectic tableaux $\tableau$ of shape $\lambda$
%Christian.
with entries $\le 2n$.

We are going to sketch a bijective proof of the symplectic
``$e$-formula" (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{\mathbf{x}} -
		\esf_{\lambda_j^\prime-j-i}\of{\mathbf{x}}}. 
\end{equation} 
% Christian.
Here, $e_m(\mathbf x)$ denotes the elementary symmetric function of
order $m$ in the variables $\mathbf x$. As usual, $\lambda'$ denotes the
partition conjugate to $\lambda$.
Formula \eqref{thm:e-symplectic} is 
the symplectic analogue of the dual form of the
Jacobi--Trudi identity for Schur functions.
\begin{Figure}\label{FF}
\begin{picture}(20,6)(0,0)
%\put(0,0){\line(1,0){20}}
%\put(0,7){\line(1,0){20}}
 
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	\begingrid{0}{0}{5}{3}
	\makeaxes
	\makexscale
	\makeyscale
	\put(0.5,4){$h$-labelling:}
        \thinlines
        %points
        \gridpoint{1}{1}
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        \gridpoint{1}{3}
        \gridpoint{2}{3}
        \gridpoint{3}{3}
        \gridpoint{4}{3}
        \gridpoint{5}{3}
%Christian. Habe Figure verkleinert. Hier fehlen aber einige Fonts,
%sodass ich nicht wirklich ueberpruefen kann, ob alles in Ordnung ist.
%Daher bitte ich Dich, nachzusehen, ob alle Linien und strichlierten
%Linien ordentlich gesetzt sind.
	\hpath{0}{1}{5}
	\hpath{0}{2}{5}
	\hpath{0}{3}{5}
        \hlabel{0}{1}{1}
        \hlabel{1}{1}{1}
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        \hlabel{3}{1}{1}
        \hlabel{4}{1}{1}
        \hlabel{0}{2}{2}
        \hlabel{1}{2}{2}
        \hlabel{2}{2}{2}
        \hlabel{3}{2}{2}
        \hlabel{4}{2}{2}
        \hlabel{0}{3}{3}
        \hlabel{1}{3}{3}
        \hlabel{2}{3}{3}
        \hlabel{3}{3}{3}
        \hlabel{4}{3}{3}
	\endgrid
}
 
\put(9.2,2){level 1}
\put(9.2,3){level 2}
\put(9.2,4){level 3}
 
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	\begingrid{-1}{0}{4}{3}
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        \hlabel{2}{2}{5}
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        \hlabel{0}{3}{4}
        \hlabel{1}{3}{5}
        \hlabel{2}{3}{6}
        \hlabel{3}{3}{7}
	\skewline{-1}{1}{1}
	\skewline{-1}{2}{2}
	\skewline{-1}{3}{3}
	\skewline{0}{3}{3}
	\skewline{1}{3}{3}
	\skewline{2}{3}{3}
	\skewline{3}{3}{2}
	\skewline{4}{3}{1}
	\endgrid
}
\end{picture}
\end{Figure}

First we interpret symplectic tableaux in terms of lattice paths.
Let $\tableau$ be a symplectic tableau of shape $\lambda$ 
%Christian.
with entries
$\le 2n$, where $\la$
is a partition of length $r\le n$. 
With $\tableau$ we associate a
$\lambda_1$-tuple $(\path_1,\dots,\path_{\lambda_1})$ of
nonintersecting lattice paths, where $\path_j$ runs from $u_j=(-j+1,j-1)$
to $v_j=(\lambda_j'-j+1,
%Christian.
2n-\lambda'_j+j-1)$, by reading $\path_j$ off the $j$-th
column. Here we use the $e$-labelling of horizontal edges
which is e.g\@. explained in \cite[ch.~4]{Sagan:The-Symmetric} (see
Figures~\ref{FF} and \ref{FI}).

Obviously, $\tableau$
obeys the symplectic constraint if and only if the first path of the associated
$\la_1$-tuple of paths does not cross the line $y=x-1$.  
% (See \figref{fig:associated-symplectic-paths}.)
The following figure gives an example for % $n=6$
% markus.
$n=3$
and $\lambda=(4,3,2)$.
% \begin{figure}
% \caption{Illustration of paths associated to symplectic tableau:}
% \label{fig:associated-symplectic-paths}
\begin{Figure}\label{FI}
\begin{picture}(20,10)(0,0)
\put(16,3){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
		\put(-1.9,1.5){$T$ =}
	\endtableau
}
\put(1,1){
	\begingrid{-4}{0}{5}{8}
		\makexaxes
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		% Anfangspunkte
		\gridpoint{0}{0}
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		% Endpunkte
		\gridpoint{3}{3}
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		% levels auszeichnen
		\skewline{-4}{4}{4}
		% Gerade y=x-1
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                \thicklines
		% Pfad 1, entsprechend der ersten Spalte 
		\vedge{0}{0}
		\hedge{0}{1}
		\hlabel{0}{1}{2}
		\vedge{1}{1}
		\hpath{1}{2}{2}
		\hlabel{1}{2}{4}
		\hlabel{2}{2}{5}
		\vedge{3}{2}
		% Pfad 2, entsprechend der zweiten Spalte 
		\vpath{-1}{1}{2}
		\hpath{-1}{3}{2}
		\hlabel{-1}{3}{3}
		\hlabel{0}{3}{4}
		\vedge{1}{3}
		\hedge{1}{4}
		\hlabel{1}{4}{6}
		% Pfad 3, entsprechend der dritten Spalte 
		\vpath{-2}{2}{3}
		\hedge{-2}{5}
		\hlabel{-2}{5}{4}
		\vedge{-1}{5}
		\hedge{-1}{6}
		\hlabel{-1}{6}{6}
		% Pfad 4, entsprechend der vierten Spalte 
		\vpath{-3}{3}{5}
		\hedge{-3}{8}
		\hlabel{-3}{8}{6}
	\endgrid
}
\end{picture}
\end{Figure}
If we define the weight of a
horizontal edge with $e$-label (cf\@. Figure~\ref{FF}) 
$2i-1$ to be $x_i$, and the weight of
a horizontal edge 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 the generating
function for all $\lambda_1$-tuples $(\path_1,\dots,\path_{\lambda_1})$ of
nonintersecting lattice paths, where $\path_j$ runs from $(-j+1,j-1)$
to $(\lambda_j'-j+1,n-\lambda'_j+j-1)$ and does not cross $y=x-1$.

Next we give the lattice path interpretation of the right side of
(\ref{thm:e-symplectic}). 
Let $\reflect$ denote the reflection in the line $y=x-2$. 
For a permutation $\si\in\SG{\la_1}$ denote 
$(v_{\sigma\of{1}},\dots,v_{\sigma\of{\la_1}})$ by $\mathbf v_\si$.
For $\epsilon\in\setof{1,-1}^{\lambda_1}$ denote by
${\mathbf{u}}^{(\epsilon)}$
the $\lambda_1$-tuple of points whose $j$-th component is ${u}_j$ if
$\epsilon_j=1$ and $\reflect\of{{u}_j}$ if $\ep_j=-1$. 
We consider the set 
$\bigcup _{\si\in\SG{\la_1},\,\ep\in\{1,-1\}^{\la_1}}
^{}\fpath(\mathbf u^{(\ep)},\mathbf v_\si)$. It is easy to see that the
determinant in (\ref{thm:e-symplectic}) can be written as
%Christian.
a certain generating function for this set, 
$$ \detof{\lambda_1}{\esf_{\lambda_j^\prime-j+i}\of{\mathbf{x}} -
		\esf_{\lambda_j^\prime-j-i}\of{\mathbf{x}}} =
		{\sum_{\sigma\in \SG{\lambda_1},\,
                \epsilon\in\{1,-1\}^{\lambda_1}}}
	\sgn\of{\sigma}
	\sgn\of{\epsilon}
	\GF{\fpath\of{{\mathbf{u}}^{(\epsilon)},{\mathbf{v}}_\sigma}},
$$
%Christian.
where $\sgn(\epsilon)=\prod _{i=1} ^{\lambda_1}\epsilon_i$.
In order to establish (\ref{thm:e-symplectic}),
we have to give an involution that 
%Christian.
in the right-hand side sum cancels all
%Christian
contributions of
 $\lambda_1$-tuples of
paths in $\bigcup _{\si\in\SG{\la_1},\,\ep\in\{1,-1\}^{\la_1}}
^{}\fpath({\mathbf{u}}^{(\epsilon)},\break {\mathbf{v}}_\sigma)$ 
that are either intersecting or contain a path that crosses the
line $y=x-1$.

This is done by a combination of the Gessel--Viennot method 
\cite[sec.~1]{Gessel-Viennot:Determinants-paths,%
%Christian.
Stembridge:Nonintersecting-paths}
and a modified reflection
principle.
A path $\path$ crossing the line $y=x-1$ must meet the line $y=x-2$. Let $s$ be the
last meeting point. Now, to $\path$'s initial portion up to $s$
apply the following modified reflection in $y=x-2$,
% markus. Der Verweis kommt am Ende des Absatzes noch einmal!?
% Christian. Dann behalten wir den frueheren.
which is exemplified in 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., $(x,y)\mapsto(y+2,x-2)$,
and so are all {\em odd\/} points ($x+y\equiv1\pmod2$) whose adjacent
steps are both vertical or both horizontal.
The remaining case is a ``kink'' in an odd point $p$, i.e\@., a horizontal step
of the path meets a vertical one in $p$. Here the whole kink is shifted
until its even points have reached their new (reflected) positions.

Now we are able to describe the desired weight-preserving and
sign-reversing involution.
For a $\lambda_1$-tuple $(\path_1,\dots,\path_{\la_1})$ 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 easy to see that this mapping is
a weight-preserving and sign-reversing involution. Thus only those
$\lambda_1$-tuples remain that are nonintersecting and none of their
paths cross the line $y=x-1$. But as was exhibited above,
 these $\lambda_1$-tuples
correspond to symplectic tableaux. 
\QED
% \begin{figure}
% \caption{Illustration of modified reflection in $y=x-2$:}
% \label{fig:modified-reflection}
\begin{Figure}\label{FJ}
% \begin{picture}(20,7)(0,1)
% \put(1,0){
% 	\begingrid{-1}{-3}{8}{5}
% markus.
\begin{picture}(20,7)(0,1)
\put(1,1){
	\begingrid{-1}{-2}{8}{5}
		\makexaxes
		\makexscale
		% gerade Punkte und ihre Spiegelbilder
		\gridpoint{0}{0}		\gridrpoint{2}{-2}
		\gridpoint{0}{2}		\gridrpoint{4}{-2}
		\gridpoint{2}{2}		\gridrpoint{4}{0}
		\gridpoint{3}{3}		\gridrpoint{5}{1}
		\gridpoint{4}{4}		\gridrpoint{6}{2}
		\gridpoint{6}{4}
%Christian. Habe die Bezeichnung des Schnittpunktes eingefuegt. Auch
%habe ich einige Verkleinerungen vorgenoomen. Wiederum, da hier einige
%Fonts fehlen, kann ich nicht ueberpruefen, ob alles in Ordnung ist
		\put(6.3,3.7){s}
		\skewline{0}{0}{2}
		\skewline{0}{2}{4}
		\skewline{2}{2}{2}
		\skewline{3}{3}{2}
		\skewline{4}{4}{2}
		% Gerade y=x-2
                \thinlines
		\put(0,-2){\line(1,1){7}}
                \thicklines
		% Ungespiegelter Pfad
		\vpath{0}{0}{2}
		\hpath{0}{2}{2}
		\vedge{2}{2}
		\hpath{2}{3}{2}
		\vedge{4}{3}
		\hpath{4}{4}{2}
		\put(6,4){\vector(0,1){2}}
		% Gespiegelter Pfad
		\hrpath{2}{-2}{2}
		\vrpath{4}{-2}{3}
		\hrpath{4}{1}{2}
		\vrpath{6}{1}{3}
	\endgrid
}
\end{picture}
\end{Figure}



{\sc Bijective proof of the symplectic Jacobi--Trudi identity}. 
Let $\lambda = \parof{\lambda_1,\dots,\lambda_r}$ be a partition
of length $r\le n$.
A bijective proof for the symplectic ``$h$-formula", 
the symplectic analogue of the ``ordinary"
Jacobi--Trudi identity for Schur functions,
(see \cite[Prop.~24.22]{Fulton-Harris:Representation-Theory})
\begin{equation} \label{thm:h-symplectic}
\sschf{2n}{\lambda} =
\detof{r}{
	\firstcolumn{\hsf_{\lambda_i-i+1}\of{\mathbf{x}}}
	\hsf_{\lambda_i-i+j}\of{\mathbf{x}} +
		\hsf_{\lambda_i-i-j+2}\of{\mathbf{x}}} ,
\end{equation} 
\noindent
% markus. Sollten hier nicht - ganz kurz - die homogen symmetrischen
% Funktionen erwaehnt werden?
%Christian. Sicherlich.
is more difficult. Here, $h_m(\mathbf x)$ denotes the complete
homogeneous symmetric function of order $m$ in the variables $\mathbf x$. 
We shall give only a rough sketch of the algorithm without
rigorous argumentation. This time we encode symplectic tableaux of shape
$\la$ in terms of $r$-tuples $(\path_1,\dots,\path_{r})$ of 
nonintersecting lattice paths where $\path_i$ runs from $u_i=(r-i,1)$
to $v_i=(\lambda_i+r-i,
%Christian
2n)$, 
by reading $\path_i$ from the
$i$-th row of the tableau. Here we use the usual $h$-labelling of
horizontal edges
\cite[ch.~4]{Sagan:The-Symmetric} (see Figure~\ref{FF}). 
%Christian. Hier eingefuegt, dafuer weiter unten gestrichen.
The symplectic constraint simply says that $\path_i$ must not contain
a horizontal step strictly below the line $y = 2i-1$.
We define the weight of a
horizontal edge with $h$-label 
$2i-1$ to be $x_i$, and the weight of
a horizontal edge with $h$-label $2i$ to be $x_i^{-1}$ in order that
this correspondence is
weight-preserving with respect to the weight (\ref{weight:symplectic}). 


Let now $\reflect$
denote the reflection in the line $x=r-1$.
We shall consider the set $\bigcup
_{\si\in\SG{r},\,\ep\in\{1\}\times\{1,-1\}^{r-1}}
^{}\fpath({\mathbf u^{(\ep)},\mathbf v_\si})$. The notation $\mathbf 
u^{(\ep)}$ has
to be understood in the same sense as above, only the reflection
$\reflect$ has a different meaning now.
The determinant in (\ref{thm:h-symplectic}) can be written as 
$$\detof{r}{
	\firstcolumn{\hsf_{\lambda_i-i+1}\of{\mathbf{x}}}
	\hsf_{\lambda_i-i+j}\of{\mathbf{x}} +
		\hsf_{\lambda_i-i-j+2}\of{\mathbf{x}}}=
		{\sum_{\sigma\in \SG{r},\,
                \epsilon\in\{1\}\times\{1,-1\}^{r-1}}} % \in\SG{\lambda_1}}
	\negthickspace\negthickspace\negthickspace\negthickspace\sgn\of{\sigma}
	\GF{\fpath\of{{\mathbf{u}}^{(\epsilon)},{\mathbf{v}}_\sigma}}.
$$
%Christian.
The following
algorithm performs a cancellation in the set $\bigcup
_{\si\in\SG{r},\,\ep\in\{1\}\times\{1,-1\}^{r-1}}
^{}\fpath({\mathbf u^{(\ep)},\mathbf v_\si})$ such
that only those $r$-tuples of paths
survive for which $\ep=(1,\dots,1)$, and where $\path_i$ does
not contain any horizontal step strictly below the line $y = 2i-1$.
In the final step we shall apply Gessel--Viennot involution to cancel 
all intersecting $r$-tuples. Thus only those $r$-tuples will remain 
which correspond to symplectic tableaux.

Let $(\path_1,\dots,\path_r)$ be an $r$-tuple of paths
where there is either a path $\path_i$ containing a horizontal step
below $y=2i-1$
or where the associated $\epsilon$ differs from
$\parof{1,1,\dots,1}$.

At first, we adjoin to each path $\path_{i}$
a set $\pathset_i$
of variables. $\pathset_i$ is defined by
\begin{equation}\label{setassign}
\pathset_i =
	\begin{cases}
		\emptyset &  \hbox{ if } \epsilon_{i} = 1 \\
		% \setof{x_1,x_1^{-1},\dots,
		%			x_{\sigma\of{i}-1},x_{\sigma\of{i}-1}^{-1}} &:
		\{x_1,x_1^{-1},\dots,x_{i-1},x_{i-1}^{-1}\} & 
			\hbox{ if } \epsilon_{i} = -1.
	\end{cases}
\end{equation}
We are thus concerned with objects of the form
\begin{equation}\label{initialobject}
\brkof{\parof{\path_{1},\pathset_1},\dots,
	\parof{\path_r,\pathset_r},\sigma,\epsilon},
\end{equation}
where $(\path_1,\dots,\path_r)$ is an $r$-tuple of paths
with associated $\sigma$ and $\ep$,
and the $\pathset_i$'s are subsets from the
sequence $\mathbf x$. The values $\epsilon_i$ actually are encoded in the sets
$\pathset_i$.
Note that for the above object the $x$-coordinate $u_{i}^x$ of
the starting point of
$\path_{i}$ can be expressed in $\pathset_i$'s cardinality,
$ u_{i}^x = \numof{\pathset_i} + r - i. $

The algorithmic involution consists of several steps, each yielding
``intermediate" objects. This set of intermediate objects, which we
denote by $\mathcal O$, is the set of all objects of the form
\begin{equation}\label{objects}
\brkof{\parof{\path_{1},\pathset_1},\dots,
	\parof{\path_r,\pathset_r},\sigma},
\end{equation}
subject to
\begin{align}
\path_{i} &\text{ runs from }(u_i^x,1)\text { to }
({\lambda_{\sigma\of{i}}+r-\sigma\of{i},2n})
	\label{cond:0} \\
\pathset_i & \subseteq\{x_1,x_1^{-1},\dots,x_{i-1},x_{i-1}^{-1}\}
	\label{cond:A} \\
\numof{\pathset_i} &= u_{i}^x - r + i.
	\label{cond:B}
\end{align}
Together, conditions (\ref{cond:A}) and (\ref{cond:B})
imply
%Christian
 $r-i\leq u_{i}^x\leq r+i-2$.
 
We define the {\em weight} of an intermediate object in $\mathcal O$ to be the weight of
the $r$-tuple of the paths times
the product of all variables contained in the multiset union of the sets. 
 
On the set $\mathcal O$ we define two operations, $\mathbf A$ and $\mathbf
B$.
First we define operation $\mathbf{A}$. Let $k$ be minimal such that
either $\path_{k}$
contains a horizontal step at height $h<2k-1$ or its adjoined
set $\pathset_k$ contains the $h$-th component of $\mathbf{x}$
(%Christian.
$x_{(h+1)/2}$ if $h$ is odd and $x_{h/2}^{-1}$ if $h$ is even); 
let $h$ be minimal, too. Now replace
$\parof{\path_{k},\pathset_k}$ by
$$
% \parof{\path_k,\pathset_k}\mapsto
\begin{cases}
\parof{\path_{k}\cup h, \pathset_k\setminus\setof{h}} &
	\hbox{ if } h\in\pathset_k,\\
\parof{\path_{k}\setminus h, \pathset_k\cup\setof{h}} &
	\hbox{ if } h\not\in\pathset_k.
\end{cases}
$$
This has to be read in the following sense.
If the $h$-th component of $\mathbf x$ is in $\pathset_k$,
remove it
and insert a horizontal step in $\path_{k}$ at height $h$,
fixing the end point (this shifts the starting point one unit to the left).
If not, remove one horizontal step at height $h$ in $\path_{k}$,
fixing the end point (this
shifts the starting point one unit to the right) and adjoin the
$h$-th component
of $\mathbf{x}$ to $\pathset_k$. 

Two paths in an intermediate object may
have a common starting point, or the {\em reflection\/} of a
starting point in the line $x=r-1$ 
may coincide with the starting point of another path.
For these two cases, we define operation $\mathbf{B}$.
Let $\parof{k,l}$ be a (lexicographic) minimal pair of distinct indices
so that either $u_{k}^x=u_{l}^x$
(starting points of $\path_{k}$,
$\path_{l}$ coincide)
or $u_{k}^x+u_{l}^x=2r-2$ (starting points of $\path_{k}$, $\path_{l}$ lie
symmetrically with respect to $x=r-1$).
 
In the first case, simply interchange $\path_k$ and
$\path_l$, and multiply
$\sigma$ with the transposition $\parof{l,k}$. This results in
\begin{equation}\label{B1}
\brkof{\cdots,\parof{\path_k^\prime,\pathset_k},\dots,
\parof{\path_l^\prime,\pathset_l},\dots,\sigma\circ\parof{l,k}},
\end{equation}
where $\path_k^\prime=\path_l$ and $\path_l^\prime=\path_k$.
Clearly, conditions (\ref{cond:0}), (\ref{cond:A}) and (\ref{cond:B}) are satisfied.
 
In the second case, again interchange paths $\path_k$ and
$\path_l$, and multiply $\sigma$ with the transposition $\parof{l,k}$; in addition, replace $\pathset_k,\pathset_l$ by their
``complements"
$\pathset^\prime_k,\pathset^\prime_l$, resulting in
\begin{equation}\label{B2}
\brkof{\cdots,\parof{\path_k^\prime,\pathset^\prime_k},\dots,
	\parof{\path_l^\prime,\pathset^\prime_l},\dots,\sigma\circ\parof{l,k}},
\end{equation}
where $\path_k^\prime=\path_l$, $\path_l^\prime=\path_k$ and
$\pathset_k^\prime\defeq\{x_1,x_1^{-1},\dots,x_{k-1},x_{k-1}^{-1}\}
	\setminus\pathset_k^{-1}$,
$\pathset_l^\prime\defeq\{x_1,x_1^{-1},\dots,x_{l-1},\break 
x_{l-1}^{-1}\}	\setminus\pathset_l^{-1}
$.
Here, $\pathset_k^{-1}$ means the set $\{y^{-1}:y\in\pathset_k\}$,
and the same for $\pathset_l^{-1}$.

Now we start with an object defined by (\ref{initialobject}) and
(\ref{setassign}), and forget the $\epsilon$, thus obtaining an
object 
$[{({\path_{1},\pathset_1}),\dots,
	({\path_r,\pathset_r}),\sigma}]$,
satisfying
(\ref{cond:0})--(\ref{cond:B}). Hence, it is an element of $\mathcal O$. 
%Christian. Sparmassnahme.
We define a weight-preserving mapping by applying operation $\mathbf{A}$
to this object, then
$\mathbf{B}$, then $\mathbf{A}$ again, etc\@., 
(formally: $\mathbf{B}^{[0\text { or }1]}\circ
\parof{\mathbf{A}\circ\mathbf{B}}^n\circ\mathbf{A}$),
as long as the particular operation is applicable. 
 
It is not difficult to see that the algorithm terminates with an application of
$\mathbf A$.
Let $[{({\bar\path_{1},\bar\pathset_1}),\dots,\break
	({\bar\path_r,\bar\pathset_r}),\bar\sigma}]$, be the
terminal object.
It is easy to show that $\bar u_i^x\in\{r-i,\break r+i-2\}$. 
($\bar
u_i^x$ denotes the $x$-coordinate of the starting point of $\bar\path_i$.)
Besides,
if $\bar u_i^x=r-i$ we have $\bar \pathset_i=\emptyset$, and
if $\bar u_i^x=r+i-2$ we have $\bar \pathset_i=
\{x_1,x_1^{-1},\dots,x_{i-1},x_{i-1}^{-1}\}$. In the first case we
set $\bar\epsilon_i=1$, in the latter $\bar\epsilon_i=-1$.
We thus obtain an object
\begin{equation}\label{terminalobject}
[{({\bar\path_{1},\bar\pathset_1}),\dots,
	({\bar\path_r,\bar\pathset_r}),\bar\sigma,\bar\epsilon}].
\end{equation}
Thus $(\path_1,\dots,\path_r)$ is mapped to 
$(\bar\path_1,\dots,\bar\path_r)$.
It is true that the permutation $\bar\sigma$ exactly corresponds to
the permutation of the end points of $\bar\path_1,\dots,\bar\path_r$,
and $\bar\epsilon$ corresponds to the ``right" reflections of the
starting points of $\bar\path_1,\dots,\bar\path_r$.
It can be shown that this algorithm is weight-preserving and
sign-reversing (i.e\@. that $\si$ and $\bar\si$ differ in sign).

Among the remaining $r$-tuples we apply the Gessel--Viennot
involution thus cancelling all intersecting $r$-tuples. It can be
shown that, altogether, this defines a weight-preserving and
sign-reversing involution.
\QED


{\sc Bijective proof of the symplectic Giambelli identity}.
The symplectic analogue of the Giambelli identity for Schur functions 
reads (see \cite[(24.47)]{Fulton-Harris:Representation-Theory})
\begin{equation}
\label{thm:Giambelli-symplectic}
\sschf{2n}{\parof{\alpha|\beta}} = \detof{m}{\sschf{2n}{\parof{\alpha_i|\beta_j}}}.
\end{equation}
Here, $\parof{\alpha|\beta}$ is the Frobenius notation for partitions
(cf\@. \cite{Macdonald:Symmetric-Functions})
%Christian.
and $m$ is the 
% markus. Wie waer's mit der Ergaenzung:
% rank of the partition, i.e., the
number of cells in the main diagonal of $(\alpha,\beta)$.
Again, we consider
paths in the integer lattice consisting of horizontal steps in the
positive direction. 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 {\em does not\/} lie strictly to the right of the
$y$-axis.

To the left of the horizontal axis we consider a
``shifted'' $h$-labelling by assigning label $i+1$ to a step at height $i$,
while we consider the 
%Christian.
``usual'' $e$-labelling to the right.
Given a tableau of shape $\parof{\alpha|\beta}$ with entries from
%Christian. Ueberall n->2n
$\setof{1,2,\dots, 2n}$, 
we associate to its
$i$-th principal hook a path $\path_i$ from $\parof{-\alpha_i,2n-1}$ to
$\parof{\beta_i+1,2n-\beta_i-1}$ by interpreting the entries of the hook
(read from ``right to bottom'') as labels of the corresponding
steps of $\path_i$.

Figure~\ref{FR} shows an example
(with $n=3$) of this
correspondence between nonintersecting lattice paths and symplectic
tableaux.
\begin{Figure} \label{FR}
\begin{picture}(20,8)(0,0)
\put(14,4){
	\begintableau{3}{4}
		\row{2346}
		\row{446}
		\row{56}
	\endtableau
}
\put(1,0){
	\begingrid{-4}{-1}{4}{6}
		\makexaxes
		\makexscale
		\makeyaxes
		\skewline{0}{2}{2}
		\skewline{0}{4}{4}
		\skewline{0}{6}{4}
		% Anfangspunkte
		\gridpoint{-3}{5}
		\gridpoint{-1}{5}
		% Endpunkte
		\gridpoint{3}{3}
		\gridpoint{2}{4}
		% Gerade y=x-1
                \thinlines
		\put(1,0){\line(1,1){5}}
                \thicklines
		% Pfad 1, entsprechend dem ersten Haken:
		\hedge{-3}{5}\hlabel{-3}{5}{6}
		\vpath{-2}{3}{2}
		\hedge{-2}{3}\hlabel{-2}{3}{4}
		\vedge{-1}{2}
		\hedge{-1}{2}\hlabel{-1}{2}{3}
		\vedge{0}{1}
		\hedge{0}{1}\hlabel{0}{1}{2}
		\vedge{1}{1}
		\hpath{1}{2}{2}\hlabel{1}{2}{4}\hlabel{2}{2}{5}
		\vedge{3}{2}
		% Pfad 2, entsprechend dem zweiten Haken:
		\hedge{-1}{5}\hlabel{-1}{5}{6}
		\vpath{0}{3}{2}
		\hedge{0}{3}\hlabel{0}{3}{4}
		\vedge{1}{3}
		\hedge{1}{4}\hlabel{1}{4}{6}
	\endgrid
}
\end{picture}
\end{Figure}
As is clear from the picture, the symplectic constraint 
translates into the condition that the first path must not cross the
line $y=x-1$. Obviously, the set of all $m$-tuples of lattice paths
subject to his condition (where $m$ denotes the rank of the partition
$\lambda$) is invariant under Gessel--Viennot involution. Therefore
the same arguments as in Stembridge's
%Christian.
\cite[sec.~9]{Stembridge:Nonintersecting-paths} proof of the ``ordinary"
Giambelli identity establish (\ref{thm:Giambelli-symplectic}).
\QED


{\sc Bijective proofs of Jacobi--Trudi type identities for Proctor's 
odd and intermediate symplectic characters}.
In \cite{Proctor:Odd,Proctor:Intermediate,Proctor:Young-tableaux}
Proctor introduces odd symplectic characters, and, more generally,
intermediate symplectic characters that interpolate between Schur
functions and (even) symplectic characters. These characters can also
be described combinatorially % be
% markus.
by means of tableaux which (roughly
speaking) obey the symplectic constraint (\ref{eq:symplectic-constraint}) 
for the first few rows and
an additional rather intricate constraint involving the jeu de taquin. In certain
special cases, he gives Jacobi--Trudi identities for these characters 
\cite[p.~317]{Proctor:Odd}, \cite[Prop.~8.1]{Proctor:Intermediate},
\cite[App.~A.2]{Proctor:Young-tableaux}. It comes as no surprise
that, because in these cases the second constraint is either
superfluous or very simple, the above proof methods for the symplectic
$e$-formula (\ref{thm:e-symplectic}) and the symplectic $h$-formula 
(\ref{thm:h-symplectic})
also suffice to yield bijective proofs of Proctor's Jacobi--Trudi identities.


\section{Bijective proofs for orthogonal
identities}\label{orthogonal}\
In \cite{Proctor:Young-tableaux,Proctor:A-Schensted} Proctor gave
tableaux interpretations for orthogonal characters. 
In fact, Proctor defined two slightly different types of orthogonal
tableaux, {\em coarse orthogonal tableaux}
%Christian.
(to be defined below)
 and {\em fine 
orthogonal tableaux}. The number of coarse tableaux of shape $\lambda$ 
equals the
dimension of the irreducible representation of the orthogonal group
indexed by $\lambda$.
However, the coarse tableaux are not very well
suited for describing the irreducible characters. This task is better
performed by the {\em fine} orthogonal tableaux. 
Orthogonal characters are indexed by $N$-orthogonal partitions 
\cite{Proctor:Young-tableaux,Proctor:A-Schensted}. These are
partitions with $\la_1'+\la_2'\le N$. They obey the
orthogonal ``$e$-formula", the orthogonal analogue of the dual form
of the Jacobi--Trudi identity,
\begin{equation}\label{thm:e-orthogonal}
\oschf{N}{\lambda} = 
\detof{\lambda_1}{
	\firstrow{\esf_{\lambda^\prime_j-j+1}\of{\mathbf{x}}}
	\esf_{\lambda_j^\prime-j+i}\of{\mathbf{x}} +
		\esf_{\lambda_j^\prime-j-i+2}\of{\mathbf{x}}}. 
\end{equation}
Here, $\mathbf{x} = \seqof{x_1, x_1^{-1},\dots x_n,x_n^{-1}}$ if $N$ is
even, and
$\mathbf{x} = \seqof{x_1, x_1^{-1},\dots x_n,x_n^{-1},1}$ if $N$ is odd.
Setting all $x_i$ equal to 1, one obtains 
\begin{equation}\label{thm:weak-e-orthogonal}
\oschftriv{N}{\lambda}  =
\detof{\lambda_1}{
	\firstrow{\esf_{\lambda^\prime_j-j+1}({(1^N)})}
	\esf_{\lambda_j^\prime-j+i}{((1^N))} +
		\esf_{\lambda_j^\prime-j-i+2}{((1^N))}}.
\end{equation}
($(1^N)$ is the
abbreviation for the
$N$-tuple $(1,1,\dots,1)$.) As is well-known,
when we replace in $\oschf{N}\lambda$ each $x_i$ by 1 we obtain
the dimension of the corresponding irreducible representation. 
As mentioned above, this dimension is equinumerous with 
the coarse orthogonal tableaux of the shape $\lambda$
%Christian.
with entries $\le N$. Therefore, we may
write
\begin{equation}\label{thm:equinumerous}
\oschftriv{N}{\lambda} =
\numof{\setof{\tableau:\;\tableau\hbox{ coarse orthogonal, }\tableau_{i,j}\leq
N}}.
\end{equation}
 
We are able to give a bijective proof for (\ref{thm:e-orthogonal}) by using
Proctor's fine orthogonal tableaux. However, there is not enough
space to describe it here. Instead, we describe a bijective proof for
the weaker (\ref{thm:weak-e-orthogonal}) by using coarse orthogonal
tableaux and the interpretation
(\ref{thm:equinumerous}).
The proof idea for (\ref{thm:e-orthogonal}) is
the same, however the argumentation has to be more careful because we have
to take care of weights which is not the case for our proof of 
(\ref{thm:weak-e-orthogonal}). 

\medskip
{\sc Bijective proof of the orthogonal dimension formula
(\ref{thm:weak-e-orthogonal})}.
Let $\lambda=(\lambda_1,\dots,\break 
\lambda_r)$ be an $N$-orthogonal
partition of length $r$. 
A tableau is called {\em coarse orthogonal\/} if it satisfies the value $p$ 
case of the coarse orthogonal condition
for all $p\in\mathbb N$.
A tableau
$\tableau$ of shape $\lambda$ is said to satisfy the {\em value $p$ case
of the coarse orthogonal condition\/} if the number of entries $\leq p$ in
$\tableau$'s first two columns does not exceed $p$. 

By reading $\path_j$ from the $j$-th column of the tableau
%Christian.
and using the $e$-labelling,
a coarse orthogonal tableau of shape $\la$ corresponds to
a $\la_1$-tuple $(\path_1,\dots,\path_{\la_1})$ of nonintersecting 
lattice paths, where $\path_j$
runs from $u_j=(-j+1,j-1)$ to $v_j=(\lambda_j'-j+1,N-\lambda'_j+j-1)$, and
where the reflection of $\path_1$ in the line $y=x$ does not intersect
$\path_2$. 
This is exemplified in Figure~\ref{FS}.
\begin{Figure}\label{FS}
\begin{picture}(20,11)(0,0)
\put(12,5){
	\begintableau{4}{4}
	\row{1445}
	\row{266}
	\row{47}
        \row{6}
	\endtableau
}
\put(1,1){
\begingrid{-3}{0}{5}{11}
\makexaxes
\makexscale
%\makeyscale
 
% Anfangspunkte
\gridpoint{0}{0}
\gridpoint{-1}{1}
\gridpoint{-2}{2}
\gridpoint{-3}{3}
 
% Endpunkte
\gridpoint{4}{5}
\gridpoint{2}{7}
\gridpoint{0}{9}
\gridpoint{-2}{11}
 
% gespiegelte Punkte
\gridpoint{5}{4}


		% Gerade y=x
                \thinlines
		\put(-1,-1){\line(1,1){7}}
                \thicklines

% Pfad 1, entsprechend der ersten Zeile 
\hlabel{3}{1}{$\path_1$}
\vedge{2}{0}
\hedge{0}{0}\hlabel{0}{0}{1}
\vedge{3}{1}
\hedge{1}{0}\hlabel{1}{0}{2}
\vpath{4}{2}{3}
\hedge{2}{1}\hlabel{2}{1}{4}
\hedge{3}{2}\hlabel{3}{2}{6}
%\vpath{6}{5}{3}
%\put(6,5){\vector(0,1){4}}
 
% Pfad 2, entsprechend der zweiten Zeile 
\hlabel{2}{5}{$\path_2$}
\vpath{-1}{1}{3}
\hedge{-1}{4}\hlabel{-1}{4}{4}
\vedge{0}{4}
\hedge{0}{5}\hlabel{0}{5}{6}
\hedge{1}{5}\hlabel{1}{5}{7}
\vpath{2}{5}{2}
 
 
% Pfad 3, entsprechend der dritten Zeile 
\hlabel{0}{7}{$\path_3$}
\vpath{-2}{2}{3}
\hedge{-2}{5}\hlabel{-2}{5}{4}
\vedge{-1}{5}
\hedge{-1}{6}\hlabel{-1}{6}{6}
\vpath{0}{6}{3}
 
% Pfad 4, entsprechend der vierten Zeile 
\hlabel{-2}{7}{$\path_4$}
\vpath{-3}{3}{4}
\hedge{-3}{7}\hlabel{-3}{7}{5}
\vpath{-2}{7}{4}

%gespiegelter erster Pfad
\vrpath{0}{0}{2}
\hrpath{0}{2}{1}
\vrpath{1}{2}{1}
\hrpath{1}{3}{1}
\vrpath{2}{3}{1}
\hrpath{2}{4}{3}
\endgrid
}
\end{picture}
\end{Figure} 
Let us denote the reflection in the line $y=x$ by $\reflect$.
For the bijective proof of (\ref{thm:weak-e-orthogonal}) we consider
the set $\bigcup
_{\si\in\SG{\la_1}\,\ep\in\{1\}\times\{1,-1\}^{\la_1-1}}
^{}\fpath({\mathbf u^{(\ep)},\mathbf v_\si})$. It is easy to see that the
determinant in (\ref{thm:weak-e-orthogonal}) equals the weighted sum 
$\sum _{\mathbf\path} ^{}\sgn\of\si$ where the sum is over all
$\la_1$-tuples 
%Christian.
$\mathbf\path$ of paths in this set.

We shall give a sign-reversing involution that cancels all ${\la_1}$-tuples
$(\path_1,\dots,\path_{\la_1})$ 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
% markus. Perfekt genuegt doch, oder?
% Christian. Das ist nicht Praeperfekt sondern Perfekt-Konjunktiv
been done. The remaining
${\la_1}$-tuples are nonintersecting and for all pairs $\path_i,\path_j$
the reflected path $\path_i$ and the 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
($v_{j+1}$ lies in the
North-West
of $v_{j}$, $j=1,2,\dots,{\la_1}-1$, and in addition $v_2$ lies in the
North-West of $\reflect\of{v_1}$; the latter is true because
$\la$ is an $N$-orthogonal partition), this implies that the
end point of $\path_1$ must be $v_1$. Since for $l\ge 2$ the end
points $v_l$ lie in the North-West of $v_1$ and $\reflect\of{v_1}$, 
the complete paths
$\path_i$, $i\ge 2$, must lie in the North-West of $\path_1$ and
$\reflect\of{\path_1}$, not intersecting any of them. Since also
$\path_2,\dots,\path_{\la_1}$ are nonintersecting, because of the order of
the starting and end points, for $i\ge 2$
$\path_i$ must run from $u_i$ to $v_i$.
In other words, the remaining ${\la_1}$-tuples are those for which $\si=\text
{id}$ and $\ep=(1,\dots,1)$, 
which are nonintersecting, and where
$\reflect\of{\path_1}$ for $i\ge 2$ does not meet $\path_i$. This would prove the
assertion.
 
Now we define the involution. First cancel all intersecting
${\la_1}$-tuples by the Gessel--Viennot involution.
For a remaining
${\la_1}$-tuple, look for the lowest level $x+y=c$ containing a point
$p$ of intersection {\em above $y=x$\/}
of $\path_i$ and $\reflect\of{\path_j}$. Choose $(i,j)$ to be minimal
in lexicographic order.
%Christian. Sparmassnahme.
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 $\path_j'$. The following figure illustrates this
operation.
\begin{Figure}\label{FK}
\begin{picture}(20, 7)(0,0)
\put(1,0){
	\begingrid{-1}{0}{7}{7}
                \thinlines
		\put(0,0){\line(1,1){7}}
                \thicklines
                %labels
                \put(-1.8,4.6){$u_i$}
                \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}{$\hskip3pt p$}
		% Unterhalb y=x:
		\put(2,5){\circle{0.4}}
		\gridpoint{3}{1}
		\hedge{3}{1}
		\vedge{4}{1}
		\hedge{4}{2}
		\gridrpoint{1}{3}
		\vredge{1}{3}
		\hredge{1}{4}
		\vredge{2}{4}
		% Oberhalb y=x:
		\hpath{-1}{5}{3}
		\gridpoint{-1}{5}
		\vedge{2}{5}
		\hpath{2}{6}{3}
		\gridpoint{5}{6}
		\vpath{5}{2}{2}
		\hpath{5}{4}{2}
		\gridpoint{7}{4}
		\hrpath{2}{5}{2}
		\vredge{4}{5}
		\vredge{4}{6}
		\gridrpoint{4}{7}
	\endgrid
}
 
\put(11,3){$\longleftrightarrow$}

\put(14,0){
	\begingrid{-1}{-1}{7}{7}
                \thinlines
		\put(0,0){\line(1,1){7}}
                \thicklines
                %labels
                \put(-1.8,4.6){$u_i$}
                \put(2.3,-0.8){$\reflect\of{u_i}$}
                \put(5.2,2){$\reflect\of{\path_i'}$}
                \hlabel{0}{2}{$\reflect\of{u_j}$}
                \put(2,6.5){$\path_j'$}
                \hlabel{3}{4}{$\hskip-4pt{\path_i'}$}
                \hlabel{1}{5}{$\hskip3pt p$}
		% Unterhalb y=x:
		\put(2,5){\circle{0.4}}
		\gridpoint{5}{-1}
		\gridpoint{1}{3}
		\vedge{1}{3}
		\hedge{1}{4}
		\vedge{2}{4}
		% Oberhalb y=x:
		\hrpath{-1}{5}{3}
		\gridrpoint{-1}{5}
		\vedge{2}{5}
		\hpath{2}{6}{3}
		\gridpoint{5}{6}
		\vpath{5}{-1}{5}
		\hpath{5}{4}{2}
		\gridpoint{7}{4}
		\hrpath{2}{5}{2}
		\vredge{4}{5}
		\vredge{4}{6}
		\gridrpoint{4}{7}
	\endgrid
}
\end{picture}
\end{Figure}
\noindent
Thus, from the original ${\la_1}$-tuple of paths we obtain a new ${\la_1}$-tuple
by replacing $\path_i$ by $\reflect\of{\path_i'}$ and $\path_j$ by
$\path_j'$.
It does not introduce a new
point of (ordinary) intersection in the resulting ${\la_1}$-tuple. In fact, there
cannot be such a point above level $x+y=c$; and there is no such point below,
because $c$ was chosen to be minimal. This mapping is sign-reversing since
the associated permutations differ by the transposition $(i,j)$. And
it is straight-forward to verify that it is an involution.
\QED 


{\sc Bijective proofs for odd orthogonal identities by means of
Sundaram's odd orthogonal tableaux}.
Sundaram's odd orthogonal tableaux 
\cite{Sundaram:Orthogonal,Sundaram:Tableaux-in} 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\@., the symbol
$\infty$ may occur {\em more than once in a column\/}), which obey the
symplectic constraint (\ref{eq:symplectic-constraint}),
and in addition have at most one entry $\infty$ in each {\em row\/}. 
The weight of a Sundaram tableaux is defined by
(\ref{weight:symplectic}), 
entries $\infty$ do not contribute to the weight.
Since the difference with symplectic tableaux is not too big, slight
modifications of the above proofs for the symplectic  Jacobi--Trudi
identities (\ref{thm:e-symplectic}) and (\ref{thm:e-symplectic}) can
be used to prove the odd orthogonal Jacobi--Trudi type
identities
\begin{equation}
\oschf{2n+1}{\lambda} =
\Big\vert{\sum_{k\geq 0}\esf_{\lambda'_i-i+j-k}\of
        {x_1,x_1^{-1},\dots,x_n,x_n^{-1}} -
	\sum_{k\geq 0}\esf_{\lambda'_i-i-j-k}\of
        {x_1,x_1^{-1},\dots,x_n,x_n^{-1}}
}\Big\vert_{\lambda_1\times \lambda_1}
\end{equation}
and 
\begin{equation}
\oschf{2n+1}{\lambda} =
%Christian.
\detof{r}{\firstcolumn{h'_{\lambda_i-i+1}}
h'_{\lambda_i-i+j}\of{\mathbf{x}} +
	h'_{\lambda_i-i-j+2}\of{\mathbf{x}}
},
\end{equation}
where $h_k'=h_k-h_{k-2}$ and $\mathbf
x=\seqof{x_1,x_1^{-1},\dots,x_n,x_n^{-1},1}$.
The proof of the odd orthogonal Giambelli identity
(see \cite[(24.47)]{Fulton-Harris:Representation-Theory})
\begin{equation}
\label{thm:Giambelli-orthogonal}
\oschf{2n+1}{\parof{\alpha|\beta}} = \detof{m}{\oschf{2n+1}{\parof{\alpha_i|\beta_j}}} ,
\end{equation}
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. This correspondence 
maps Sundaram's tableaux to sets of nonintersecting
lattice paths that contain at
most one diagonal step (with label $\infty$) in the left half-plane,
and that do not cross the line $y=x-1$. We give an
example for $n=4$ in Figure~\ref{FZ}.
Application of the Gessel--Viennot method immediately implies
(\ref{thm:Giambelli-orthogonal}).
\begin{Figure}\label{FZ}
\begin{picture}(20,11)(0,0)
\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}
		\makexaxes
		\makexscale
		\makeyaxes

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

                \thinlines
		\put(1,0){\line(1,1){4}}
                \thicklines
		\put(2,7){\line(1,-1){2}}
                %labels
                \put(2.5,6.7){$\infty$}
                \put(3.5,5.7){$\infty$}
		\gridpoint{4}{5}
		\gridpoint{1}{8}
		\gridpoint{-2}{8}
		\gridpoint{-3}{8}
                %Pfad 1
		\put(-3,8){\line(1,-1){1}}
		\put(-3.4,7.3){$\infty$}
		\vpath{-2}{5}{2}
		\hedge{-2}{5}
		\hlabel{-2}{5}{6}
		\vpath{-1}{2}{3}
		\hedge{-1}{2}
		\hlabel{-1}{2}{3}
		\vedge{0}{1}
		\hedge{0}{1}
		\hlabel{0}{1}{2}
		\hedge{1}{1}
		\hlabel{1}{1}{3}
		\vpath{2}{1}{6}
                %Pfad 2
		\put(-2,8){\line(1,-1){1}}
		\put(-1.2,7.3){$\infty$}
		\vpath{-1}{6}{1}
		\hedge{-1}{6}
		\hlabel{-1}{6}{7}
		\vpath{0}{3}{3}
		\hedge{0}{3}
		\hlabel{0}{3}{4}
		\vpath{1}{3}{5}
	\endgrid
}
 
\end{picture}
\end{Figure}

\ifx\undefined\bysame
\newcommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\,}
\fi
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\end{document}