%% Referee Punkt 4:
The \lgt\bindestrich method \cite{gessel-viennot:1998,Gessel-Viennot:1985,karlin:1988,lindstroem:1973} is well\bindestrich known:
But since we want to present a ``\nilp''\bindestrich proof of the
Cauchy\bindestrich Binet formula \eqref{eq:CauchyBinet} which involves a slight generalization, we repeat
this beautiful idea here in some detail, using the Jacobi\bindestrich Trudi\bindestrich determinant as an
illustrating example.

\subsection{Lattice paths}

Consider the square lattice $\mathbb Z^2$, i.e., the directed
graph with vertices $\mathbb Z\times\mathbb Z$ (we shall call them \EM{points}),
where the set of arcs consists of
\bit
\item horizontal arcs $\harc{\pas{j,k}}$ from $(j, k)$ to $(j+1, k)$ for $j, k\in\Z$,  and
\item vertical arcs $\varc{\pas{j,k}}$ from $(j, k)$ to $(j, k+1)$ for $j, k\in\Z$.
\eit
Assign to these arcs the following weights:
\begin{align*}
\weight\of{\varc{\pas{j,k}}} &\defeq 1\text{ (i.e., vertical arcs have weight $1$)}, \\
\weight\of{\harc{\pas{j,k}}} &\defeq x_k\text{ (i.e., horizontal arcs have weight $x_{\text{height of the arc}}$)}.
\end{align*}

A \EM{lattice path} $p$ of length $k$ connecting starting point $v$ to ending point $w$ is a sequence
of points $\pas{v=v_0,v_1,\dots,v_k=w}$, such that $\pas{v_{i-1},v_{i}}$ is a (horizontal
or  vertical) arc $a_i$ for $i=1,2,\dots,k$.
We say that all these arcs $a_i$ and points $v_i$ \EM{belong} to the path $p$ and write
$a_i\in p$ and $v_i\in p$. 
% Not needed!?
% For some set $S$ of points (arcs) we denote by $p\cap S$ the set of points (arcs) in $S$
% that belong to $p$.
The weight of $p$ is defined as the product of the weights of the arcs belonging to $p$
\begin{equation}
\label{eq:pathweight}
\weight\of p=\prod_{a\in p}\weight\of a.
\end{equation}

Denote by $\paths{v}{w}$ the set of \EM{all} lattice paths connecting starting point
$v$ to ending point $w$.
Observe that we may view the complete homogeneous symmetric function
$$
\csymf_m\of{x_j,x_{j+1},\dots,x_k}, k\geq j
$$
as the \EM{generating function} of $\paths{v}{w}$, where
% all lattice paths connecting the points
$v=\pas{t,j}$ and $w=\pas{t+m,k}$ and $t\in\Z$ is an arbitrary \EM{shift parameter}:
%can be chosen arbitrarily:
$$
\gfpaths{v}{w}=\sum_{p\in\paths{v}{w}}\weight\of p.
$$
I.e., the sum is over all paths $p$ connecting $v$ and $w$.
(See \figref{fig:hsymgflp}.)

Note that the ending point of some path $p$ never lies below (or to the left of) its
starting point: From now on, we shall call
\bit
\item the starting point $p$ its \EM{lower} point,
\item and the ending point $p$ its \EM{upper} point.
\eit
This change of nomenclature seems pointless here, but 
later we shall also consider \EM{trails} in the \EM{undirected} graph
$\mathbb Z^2$ which connect starting and ending points of \EM{paths}: A \EM{trail} $p^\prime$ of length $k$ connecting point $v$ to point $w$ is a sequence
of points $\pas{v=v_0,v_1,\dots,v_k=w}$, such that $\pas{v_{i-1},v_{i}}$ \EM{or}
$\pas{v_{i},v_{i-1}}$ is a (horizontal
or  vertical) arc $\arca_i$ for $i=1,2,\dots,k$ (i.e., $p^\prime$ may use arcs ``in the wrong direction''). For such trail, it is
not clear whether $v$ or $w$ is the starting or ending point.
%, so in order to
%avoid confusion, for some lattice path $p$ we shall call
%\bit
%\item the starting point $p$ its \EM{lower} point,
%\item and the ending point $p$ its \EM{upper} point
%\eit
%from now on.

\begin{figure}
\caption{Complete homogeneous symmetric functions may be viewed as generating functions of
lattice paths with fixed starting (lower) and ending (upper) point. The picture below illustrates this
for $\csymf_8\of{x_5,x_6,\dots,x_{13}}$, which appears as the sum of the weights of all
lattice paths connecting starting point $\pas{-4,5}$ to ending point $\pas{4,13}$, showing
the lattice paths associated to the two monomials
$\pas{x_5\cdot x_6^4\cdot  x_7\cdot  x_9^2}$ and
$\pas{x_{10}\cdot  x_{11}^2\cdot  x_{12}^4\cdot  x_{13}}$
in $\csymf_8\of{x_5,x_6,\dots,x_{13}}$. Note that we may shift the picture horizontally
by an arbitrary vector $\pas{t,0}$, $t\in\Z$, without changing the generating function
of the lattice paths.
}
\label{fig:hsymgflp}
\begin{center}
\input graphics/hsymgflp
\end{center}
\end{figure}



\subsection{Young tableaux and \nilp s}
%% Referee Punkt 4:
The following weight-preserving bijection gives an equivalent
description of a semistandard Young tableau $\tableau$ of shape $\lambda/\mu$ as an
$m$\bindestrich tuple $P=\pas{p_1,\dots,p_{m}}$ of \nilp s, where $m\defeq\length{\lambda/\mu}$. %, as follows.
%Fix some (arbitrary) integer \EM{shift}
%$t$ and consider $m$ paths in the lattice $\mathbb Z^2$.
%(i.e., in the directed
%graph with vertices $\mathbb Z\times\mathbb Z$ and arcs from $(j, k)$ to $(j+1, k)$ and
%from $(j, k)$ to $(j, k+1)$ for all $j, k$).
The $i$\bindestrich th path $p_i$ starts at lower point $(\mu_i-i, 1)$ and ends at upper point
$(\lambda_{i}-i, \n)$:
%(where $t\in\Z$ is an arbitrary \EM{shift parameter}):
We call these points the lower/upper points associated to the
skew shape $\lambda/\mu$ (%with shift $t$, 
and we count these points always \EM{from the right}).
The $k$\bindestrich th \EM{horizontal} step in $p_i$ goes from
$\pas{\mu_i-i+k-1,x}$ to $\pas{\mu_i-i+k,x}$, where $x$ is the $k$\bindestrich th entry in row $i$ of
$\tableau$ (i.e., the heights of the horizontal steps in path $p_i$ are read off the $i$\bindestrich th
row of $\tableau$). 

Note that the conditions on the entries of $\tableau$ imply that
no two paths $p_i$ and $p_j$ thus defined have a lattice point in common if $i\neq j$: Such an
$m$-tuple of paths is called \EM{nonintersecting},
see the right picture of Figure~\ref{fig:skew-Young-tableau} for an illustration.
An $m$\bindestrich tuple of paths which is \EM{not} nonintersecting is called \EM{intersecting}.

In fact, this translation of tableaux to nonintersecting lattice paths is
a \EM{bijection} between the set of all $\n$\bindestrich semistandard Young tableaux of shape $\lambda/\mu$
and the set of all $\length{\lambda/\mu}$\bindestrich tuples of nonintersecting lattice paths with lower (starting) and
upper (ending) points as defined above.

%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\caption{The left picture presents a semistandard Young tableau $\tableau$ of shape $\lambda/\mu$, where
$\lambda=\pas{9, 6, 6, 5, 3, 3, 3}$ and $\mu=\pas{5, 4, 4, 3, 2, 2, 2}$. Assuming that the
entries of $\tableau$ %this Young tableau
are chosen from $\setof{1,2,\dots,8}$ (i.e.: $\tableau$ is an $8$\bindestrich semistandard Young tableau), the right picture
shows the corresponding family of $7=\length{\lambda/\mu}$ nonintersecting lattice paths: Note that the height
of the $k$\bindestrich th horizontal step in the $i$\bindestrich th path (the paths are counted from right to left) is equal
to the $k$\bindestrich th entry in row $i$ of $\tableau$.}
\label{fig:skew-Young-tableau}
\begin{center}
\input graphics/ferrers
\end{center}
\end{figure}

This bijection clearly is \EM{weight preserving} if we define
the weight of an $m$\bindestrich tuple $P=\pas{p_1,\dots,p_m}$ of lattice paths in the obvious way, i.e., as
\begin{equation}
\label{eq:dfnweightlp}
\weight\of P \defeq\prod_{k = 1}^{m} \weight\of{p_k} = \prod_{k = 1}^{\n} x_k^{\nofentries{P}{k}},
\end{equation}
where $\nofentries{P}{k}$ is the number of horizontal steps at height $k$ in $P$. So in definition
\eqref{eq:skewSchur} we could equivalently replace symbol ``$\tableau$'' by symbol ``$P$'',
and sum over $m$\bindestrich tuples $P$ of nonintersecting lattice
paths with prescribed lower and upper points instead of tableaux with prescribed shape:
\begin{equation}
\label{eq:skewSchurNILP}
\schurf_{\lambda/\mu}%\of{x_1, x_2, x_3, \dots, x_\n}
\defeq \sum_{{P}}\weight\of{P}.
\end{equation}


Note that the horizontal coordinates of lower and upper points determine uniquely the
\EM{shape} $\lambda/\mu$ of the tableau, and the vertical coordinates (we shall call the 
vertical coordinate of points the \EM{level} in the following) of the ending points determines
uniquely the \EM{set of entries} $\setof{1, 2,\dots,\n}$ of the tableau: More generally,
lower points at level $\alpha$ and upper points at level $\omega\geq\alpha$ would give rise to entries
$\setof{\alpha,{\alpha+1},\dots, \omega}$. 
%The %choice of the
%shift parameter $t$ does influence neither the shape nor the set of entries.

Observe that the operation of deleting part $i$ in $\mu=\pas{\mu_1,\dots,\mu_m}$ and part $j$ in
$\lambda=\pas{\lambda_1,\dots,\lambda_m}$ (defined in \eqref{eq:deletepart}) translates
to the \EM{removal} of the $i$\bindestrich th lower point and the $j$\bindestrich th upper point associated to
$\lambda/\mu$,
% (by \eqref{eq:skewshifts}, the skew Schur function
%$\schurf_{\addtoparts{\lambda}{t}/\addtoparts{\mu}{t}}$ is the same as $\schurf_{\lambda/\mu}$
%for arbitrary $t\in\N$, hence deleting parts does not lead to negative entries in the
%partition $\addtoparts{\mu}{1}$).
%\alert{Wird nun einfacher!!!}
and by \eqref{eq:minorspartitions}, this removal of lower/upper points translates to deleting row $i$ and column $j$ in $\jtmat{\lambda/\mu}$. Clearly,
this generalizes to the following observation:
\begin{obs}
\label{obs:deletepointminors}
Minors of $\jtmat{\lambda/\mu}$
consisting of rows $\setof{i_1,\dots,i_k}$ and columns $\setof{j_1,\dots,j_l}$
are in one\bindestrich to\bindestrich one correspondence to the selection of
\bit
\item lower points with indices in $\setof{i_1,\dots,i_k}$
\item and upper points with indices in $\setof{j_1,\dots,j_l}$
\eit
from the points associated to the shape $\lambda/\mu$.
\end{obs}
Together, Observations~\ref{obs:genericSchurIdentity} and \ref{obs:deletepointminors}
explain the proper meaning of the statement given in \secref{sec:intro}.

\subsection{The \lgt\bindestrich proof of the Jacobi\bindestrich Trudi identity}
Let $\lambda/\mu$ be some skew shape, let $m=\length{\lambda/\mu}$, and consider
the lower/upper points of the corresponding lattice paths. 
Denote the $i$\bindestrich th lower point $(\mu_i-i, 1)$ by $s_i$, and the $j$\bindestrich th upper point
$(\lambda_{j}-j, \n)$ by $t_j$.
Observe that the entry $\pas{i,j}$ in the matrix $\jtmat{\lambda/\mu}$ is the
\EM{generating function} % of all lattice paths
$$
\gfpaths{s_i}{t_j}\defeq\sum_{p}\weight\of{p}=\csymf_{\lambda_j-\mu_i-j+i},
$$
where the sum is over all lattice paths $p$ that run from $s_i$
to $t_j$,
and where the weight $\weight\of{p}$ is defined as in \eqref{eq:pathweight}.
By expanding the determinant in
\eqref{eq:JacobiTrudiSkew}, we thus obtain
% \alert{Allgemeiner: $\csymf_r\of{x_k,\dots,x_m}$ als GF of lattice paths!}
\begin{equation}
\label{eq:detgf}
\detof{\csymf_{\lambda_j-\mu_i-j+i}}_{i,j=1}^{m}
=
\sum_{\pi\in\symm_m}\sgn\of{\pi}\cdot
\prod_{j=1}^m
\pas{\gfpaths{s_{\pi_j}}{t_j}}.%\csymf_{\lambda_i-\mu_{\pi_i}-i+\pi_i}
\end{equation}
% where $m=\length{\lambda/\mu}$. % Fix shift parameter $t=0$ here for sake of simplicity:
% Then the determinant itself appears as the generating function of the following set:
Consider the following set
\begin{equation}
\label{eq:detobjects}
\alljt_{\lambda/\mu}
\defeq
\bigcup_{\pi\in\symm_m}
\paths{s_{\pi_1}}{t_1}\times\paths{s_{\pi_2}}{t_2}\times\cdots\times\paths{s_{\pi_m}}{t_m}
\end{equation}
%\begin{equation*}
%\alljt_{\lambda/\mu}
%\defeq
%\bigcup_{\pi\in\symm_m}
%%\bigl\{
%\setof{
%	P:P \text{ is an $m$\bindestrich tuple of paths }
%	s_{\pi_j}\to t_j, j=1,2,\dots,m
%},
%\bigr\}.
%\end{equation*}
%all $m$-tuples of lattice paths $P$ connecting $(\mu_{\pi_i}-\pi_i+t, 1)$
%to $(\lambda_{i}-i+t, \n)$ for $\pi\in\symm_m$, which i
of $m$\bindestrich tuples $P$ of lattice paths connecting the \EM{permuted} lower points
$\pas{s_{\pi_1},s_{\pi_2},\dots,s_{\pi_m}}$
with the
upper points
$\pas{t_1,t_2,\dots,t_m}$,
where in addition to the weight $\weight\of P$, the elements of $\alljt_{\lambda/\mu}$
also carry a \EM{sign} which equals the sign of the corresponding permutation $\pi$:
$$\sgn\of P = \sgn\of\pi.$$
%Denote by $\alljt_{\lambda/\mu}$ the set of these $m$\bindestrich tuples $P$ of lattice paths connecting
%\EM{permuted} lower points to end points.
Then \eqref{eq:detgf} can be rewritten equivalently as the generating function of 
$\alljt_{\lambda/\mu}$:
\begin{equation}
\label{eq:detgf2}
\detof{\csymf_{\lambda_j-\mu_i-j+i}}_{i,j=1}^{m}
=
\sum_{P\in{\alljt_{\lambda/\mu}}}\sgn\of P\cdot \weight\of P.
\end{equation}
Denote by $\nijt_{\lambda/\mu}$ the subset of \EM{nonintersecting} $m$-tuples of lattice
paths in $\alljt_{\lambda/\mu}$. In order to prove \eqref{eq:JacobiTrudiSkew}, we only
need to show that we do in fact have
\begin{equation}
\label{eq:detisgf}
\detof{\csymf_{\lambda_j-\mu_i-j+i}}_{i,j=1}^{m}
=
\sum_{P\in{\nijt_{\lambda/\mu}}}\sgn\of P\cdot \weight\of P, % =\schurf_{\lambda/\mu}.
\end{equation}
since $P\in{\nijt_{\lambda/\mu}}$ implies $\pi=\idperm$ (i.e., all these $m$\bindestrich tuples $P$
objects have sign $+1$) and $\schurf_{\lambda/\mu}=\sum_{P\in{\nijt_{\lambda/\mu}}}\sgn\of P\cdot \weight\of P$
by definition \eqref{eq:skewSchurNILP}.
This certainly would be achieved by showing that all the signed weights of $m$\bindestrich tuples
$P$ in $\ijt_{\lambda/\mu}\defeq\alljt_{\lambda/\mu}\setminus\nijt_{\lambda/\mu}$
\EM{cancel} in \eqref{eq:detgf2}. To this end,
we shall present an
\EM{involution} (a self\bindestrich inverse bijective mapping) on \EM{intersecting} $m$\bindestrich tuples
of lattice paths
$$
\invol:
\pas{\ijt_{\lambda/\mu}}
\to
\pas{\ijt_{\lambda/\mu}}
$$
which is
\bit
\item \EM{weight\bindestrich preserving}, i.e., $\weight\of{\invol\of{P}} = \weight\of P$
\item and \EM{sign\bindestrich reversing}, i.e., $\sgn\of{\invol\of{P}} = -\sgn\of P$.
\eit
%for all $P\in{\mathfrak D}\setminus{\mathfrak N}$.
The construction of $\invol$ is simple: Let $P$ be an \EM{intersecting} $m$\bindestrich tuple in
% some path $p$ in
$\ijt_{\lambda/\mu}$. Consider the \EM{smallest} point of
intersection $q$ in $P$, in lexicographic order:
$$
\pas{a,b}\leq\pas{c,d}:\Leftrightarrow a\leq b \vee
\pas{a=b\wedge b\leq c}.
$$
Observe that (by the minimality of $q$) there are \EM{precisely two} paths $p_k$ and $p_l$ meeting in $q$: $\invol\of P$ is obtained from $P$ by \EM{interchanging}
the initial segments (from lower points up to $q$) of $p_k$ and $p_l$, see
\figref{fig:lgt-involution} for an illustration.

\begin{figure}[h]
\caption{Illustration of the \lgt\bindestrich involution $\invol$.}
\label{fig:lgt-involution}
Both pictures show quadruples of
lattice paths belonging to $\ijt_{\lambda/\mu}$, where $\lambda=\pas{8,6,4,3}$ and
$\mu=\pas0$. There are $4$ points of intersections at positions
%\begin{equation*}
%\newline\newline\centerline{
$$\pas{1,2} < \pas{1,5} < \pas{4,5} < \pas{4,6},$$
%}
%\newline\newline
%\end{equation*}
written in lexicographic order. %These lattice paths belong to $\lambda=\pas{8,6,4,3}$.
(Note that the lattice paths are drawn with rounded corners
and small offsets here, just to make obvious the course of the paths, which is not clear for
intersecting paths). The smallest such point in lexicographic order is
$q\defeq\pas{1,2}$ (indicated by a circle). The right picture is obtained from the left picture by interchanging
the initial segments (from lower points up to $q$) of the two paths intersecting in $q$, and
vice versa. To the left picture, the identity permutation (i.e., sign $+1$) is associated, while to the right picture, the transposition $\pas{3,4}$ (i.e., sign $-1$) is associated. This illustrates that in the determinantal
expansion of $\schurf_\lambda=\detof{\csymf_{\lambda_j-j+i}}$, the following terms cancel:
%\newline\newline\centerline{
$$
\pas{x_7^3}\pas{x_5^4}\pas{x_2^3 x_5^3}\pas{x_1^2 x_2 x_3^2 x_6^3}
-
\pas{x_7^3}\pas{x_5^4}\pas{x_2^4 x_3^2 x_6^3}\pas{x_1^2 x_5^3}
=0.
$$
%}
%\newline
\begin{center}
\input graphics/lgt-involution
\end{center}
\end{figure}

It is immediately clear that $\invol$
is an involution which is weight\bindestrich preserving and sign\bindestrich reversing (since it modifies the
original permutation associated to $P$ by the transposition corresponding to the swapping
of the lower points of $p_k$ and $p_l$).

Clearly, $\invol$
describes the pairwise cancellation
$$\sgn\of{P}\cdot\weight\of{P}+\invol\of{\sgn\of{P}\cdot\weight\of{P}} = 0$$
of all objects from $\ijt_{\lambda/\mu}$ in the
sum \eqref{eq:detgf2}.
Stated otherwise: Only the \EM{nonintersecting} objects in $\nijt_{\lambda/\mu}\subseteq\alljt_{\lambda/\mu}$
``survive'', which proves \eqref{eq:detisgf}.
\hfill\qedsymbol



\subsection{Viewing determinants as nonintersecting lattice paths} Our considerations
so far showed that
the \EM{generic} determinant $\detof{\mata_{i,j}}_{i,j=1}^m$ (``generic'' means that the entries $\mata_{i,j}$ are
independent variables) may be viewed as a skew Schur function of \EM{appropriate} shape
$\lambda/\mu$ with $\length{\lambda/\mu}=m$ (``appropriate'' means that all entries in $\jtmat{\lambda/\mu}$ are distinct). Together with the the \lgt\bindestrich view, the determinant
$\detof{\mata_{i,j}}_{i,j=1}^m$appears
as
\bit
\item the \EM{generating function of $m$\bindestrich tuples of \nilp s},
\item where the rows (from top to bottom) correspond to the starting points of the lattice paths 	(from right to left),
\item and where the columns (from left to right) correspond to the starting points of the
	lattice paths (from right to left).
\eit

We shall demonstrate the power of this point of view by giving a simple proof of the
Cauchy\bindestrich Binet formula~\eqref{eq:CauchyBinet}, which becomes even more transparent
if we prove the multiplicativity of the determinant function as a preparatory step.
(A similar proof in an easily comprehensible exposition is given in \cite{Aigner:2001}.)

%First note that the \lgt\bindestrich idea is not restricted to the case of determi\-nants
%$\detof{a}$, where the entry $a_{i,j}$ is
%% a  complete homogeneous symmetric function (i.e.,
%the generating function of ``standard'' lattice paths $\paths{s_i}{t_j}$: $a_{i,j}$
% might be the generating function of ``constrained'' lattice paths
%$\pathsmod{s_i}{t_j}{\prime}$,
%%There might be some \EM{modifications} or
%%\EM{constraints} on the corresponding lattice paths % $\pathsmod{s_i}{t_j}{\prime}$,
%as long as we can still find a % \lgt\bindestrich mapping is a
%weight\bindestrich preserving and sign\bindestrich reversing involution on the
%%$\invol$ maps the
%set of intersecting $m$\bindestrich tuples of such ``constrained'' paths: The meaning of this
%statement will become clear in the following.
%%We shall clarify the meaning of this somewhat obscure statement by two simple
%%examples.

\subsubsection{Multiplicativity of the determinant function: A proof ``by example''.}
\label{sec:multiplicativity}
Given some arc $\arca$ in the lattice $\Z^2$, we say that a path $p$ \EM{starts} in $\arca$
(has $\arca$ as its \EM{lower} arc), if $p$ starts in the upper (if $\arca$ is vertical) or
right (if $\arca$ is horizontal) point of $\arca$. Likewise, we say that $p$ \EM{ends} in
$\arca$ (has $\arca$ as its \EM{upper} arc), if $p$ ends in the lower (if $\arca$ is vertical) or
left (if $\arca$ is horizontal) point of $\arca$.

We want to illustrate the multiplicativity of the determinant function
\begin{equation}
\label{eq:detmultfunc}
\detof{a\cdot b} = \detof{a}\cdot\detof{b}
\end{equation}
by considering the special case
$$
a\defeq\pas{\csymf_{\lambda_j-j+i}\of{x_1,x_2,\dots x_7}}_{i,j=1}^4\text{ and }
b\defeq\pas{\csymf_{\sigma_j-\lambda_i-j+i}\of{x_8,x_9\dots x_{13}}}_{i,j=1}^4,
$$
where $\lambda=\pas{8,6,4,3}$ and $\sigma=\pas{18,16,13,11}$ .

%% Referee Punkt 4:
By the \lgti, we may view
$\detof{a}$ as the generating function of quadruples of nonintersecting
lattice paths connecting lower points $\asvector{r}=\pas{r_1,r_2,r_3,r_4}$ with upper arcs
$\asvector{s}=\pas{s_1,s_2,s_3,s_4}$, where
\begin{align*}
\asvector{r} &= \pas{{%\scriptstyle
	\pas{-1,1},\;\pas{-2,1},\;\pas{-3,1},\;\pas{-4,1}}},\\
\asvector{s} &= \pas{{%\scriptstyle
	\pas{\pas{7,7},\pas{7,8}},\;\pas{\pas{4,7},\pas{4,8}},\;\pas{\pas{1,7},\pas{1,8}},\;\pas{\pas{-1,7},\pas{-1,8}}}}.
\end{align*}
% see \figref{fig:DeterminantMultiplicative} for an illustration.
% (here, we chose $t=0$ as the shift parameter).

Likewise, we may view $\detof{b}$ as the generating function of quadruples of nonintersecting
lattice paths connecting lower arcs $\asvector{s}$ (as above: Note that the set of variables involved in $b$ is $\setof{x_8,\dots,x_{13}}$!) with ending
points $\asvector{t}=\pas{t_1,t_2,t_3,t_4}$, where
%where the lower points coincide with the ending points associated to $a$
%(note that the set of variables involved in $b$ is $\setof{x_7,\dots,x_{13}}$!), and where the
%ending points are given as
$$
\asvector{t} = \pas{{%\scriptstyle
	\pas{17,13},\;\pas{14,13},\;\pas{10,13},\;\pas{7,13}}}.
$$
See \figref{fig:DeterminantMultiplicative}
for an illustration.

\begin{figure}
\label{fig:DeterminantMultiplicative}
\caption{Illustration: Multiplicativity of the determinant.}
The picture shows the
lattice paths corresponding to the monomial
$$
\scriptstyle
\pas{\pas{x_7^3}\cdot\pas{x_9^2 x_{11} x_{13}^5}}
\cdot
\pas{\pas{x_5^4}\cdot\pas{x_8^2 x_{11}^5 x_{12}^2}}
\cdot
\pas{\pas{x_2^3 x_4^3}\cdot\pas{x_9^5 x_{11}^5}}
\cdot
\pas{\pas{x_1^3 x_3^4 x_5}\cdot\pas{x_8^4 x_{10}^6}},\hskip5em
$$
which appears in the expansion of the product of skew Schur functions
$$\schurf_\lambda\of{x_1,x_2,\dots x_7}\cdot
\schurf_{\sigma/\lambda}\of{x_8,x_9,\dots x_{13}},$$
where $\lambda=\pas{8,6,4,3}$ and $\sigma=\pas{18,16,13,11}$.
\begin{center}
\input graphics/DeterminantMultiplicative
\end{center}
\end{figure}

So the $\pas{i,j}$\bindestrich entry in $a\cdot b$ is
$$
\sum_{k=1}^4\gfpaths{r_i}{s_k}\cdot\gfpaths{s_k}{t_j},
$$
which may be viewed as the generating function $\gfof{\pathsmod{r_i}{t_j}{\prime}}$ of
the following set of \EM{constrained} paths (see
\figref{fig:DeterminantMultiplicative} %{fig:lgt-modified}
for an illustration):
% of objects
% \EM{labeled and constrained} lattice paths, 
% i.e., as % $\gfpaths{r_i}{t_j}^\prime$ the sum of the weights of objects in the union
% where
$$
\pathsmod{r_i}{t_j}{\prime}\defeq
\bigcup_{k=1}^4 \setof{p:\text{ $p$ is a lattice path connecting $r_i$ to $t_j$ \EM{passing through} $s_k$}}.
$$
%where the weight of the pair $\pas{s_k,p}$ is simply the weight of $p$:
%$\weight\of{\pas{s_k,p}}\defeq\weight\of p$. Such pair $\pas{s_k,p}$ may be viewed
%as a \EM{marked} path, i.e., as an \EM{ordinary} path $p$
%% with $p\cap\setof{s_1,s_2,s_3,s_4}\neq\emptyset$, and
%where point $s_k\in p$ is \EM{marked} (in particular,
%$p\cap\setof{s_1,s_2,s_3,s_4}\neq\emptyset$; note
%that path $p$ appears $\cardof{p\cap\setof{s_1,s_2,s_3,s_4}}$\bindestrich times in the set
%${\pathsmod{r_i}{t_j}{\prime}}$.)


\begin{figure}[h]
\caption{Illustration: The \lgt\bindestrich involution for the ``constrained'' lattice paths
$\gfof{\pathsmod{r_i}{t_j}{\prime}}$: The fact that \EM{every} constrained path must pass through
one of the arcs $s_1,s_2,s_3$ or $s_4$ is indicated %in the picture
by omitting
all \EM{other} arcs connecting level $7$ and level $8$.
The picture shows a path connecting $r_1$ to $t_1$ which passes through the arc $s_2$,
and a path connecting $r_4$ to $t_3$,
which passes through the arc $s_3$. The smallest point of intersection in lexicographic order is
$q\defeq\pas{-4,9}$ (indicated by a circle;
as in \figref{fig:lgt-involution}, the lattice paths are drawn with rounded corners
here, just to make obvious the run of the paths.) Clearly, the \lgt\bindestrich involution gives
a path connecting $r_1$ to $t_3$ which passes through $s_2$, and a path connecting $r_4$ to $t_1$
which passes through $s_3$.
}
\label{fig:lgt-modified}
\begin{center}
\input graphics/lgt-modified
\end{center}
\end{figure}


So as in the above proof of the Jacobi\bindestrich Trudi identity, $\detof{a\cdot b}$ can be rewritten
equivalently as the generating function of the set ${\mathfrak D}^\prime$ of quadruples of
\EM{constrained} paths,
\begin{equation*}
{\mathfrak D}^\prime\defeq
\bigcup_{\pi\in\symm_4}
\pathsmod{s_{\pi_1}}{t_1}{\prime}\times
\pathsmod{s_{\pi_2}}{t_2}{\prime}\times
\pathsmod{s_{\pi_3}}{t_3}{\prime}\times
\pathsmod{s_{\pi_4}}{t_4}{\prime},
\end{equation*}
i.e., as
\begin{equation}
\label{eq:detmultgf}
\detof{a\cdot b}
=
\gfof{\alljt^\prime}=
\sum_{P\in{\alljt^\prime}}\sgn\of P\cdot \weight\of P.
\end{equation}
Denote by $\nijt^\prime$ the subset of \EM{nonintersecting} $m$-tuples of constrained lattice
paths in $\alljt^\prime$. Clearly, the \lgt\bindestrich operation $\invol$ can be
applied %without modification
to \EM{intersecting} $m$-tuples of constrained lattice paths (see
\figref{fig:lgt-modified}
for an illustration)
%to objects % quadruples of marked paths
%in $\ijt^\prime\defeq\alljt^\prime\setminus\nijt^\prime$, with the following additional specification:
%If the minimal point of intersection $q$ of marked paths $\pas{s_i,p_a}$, $\pas{s_j,p_b}$
%happens to be one of the marked points, say $s_i$ which is marked in $p_a$, then after application of $\invol$
%\bit
%\item $s_i$ is marked in $\invol\of{p_a}$ and $s_j$ is marked in $\invol\of{p_b}$
%	if $s_j\geq s_i$ in lexicographic order,
%\item $s_i$ is marked in $\invol\of{p_b}$ and $s_j$ is marked in $\invol\of{p_a}$
%	if $s_j\leq s_i$ in lexicographic order.
%\eit
%(This modification is just to avoid that $\invol$ attributes two marked points to one path and none to the other. Here, by ``abuse of notation'', $\invol\of{p}$ denotes
%the effect of $\invol$ on a single path $p$ in some intersectiong $m$-tuple $P$.)
% It is easy to see that $\invol$
and establishes a weight\bindestrich preserving and sign\bindestrich reversing
involution, i.e.,
$$\detof{a\cdot b}=\gfof{\nijt^\prime}.$$
On the other hand, $\nijt^\prime$ appears (just \EM{look} at \figref{fig:DeterminantMultiplicative}!)
as the Cartesian product of
\bit
\item the set
of nonintersecting quadruples of lattice paths connecting $\asvector{r}$ and $\asvector{s}$
\item and the set of nonintersecting quadruples of lattice paths connecting $\asvector{s}$ and $\asvector{t}$,
\eit
i.e.,
$$
\gfof{\nijt^\prime}=\detof a\cdot \detof b.
$$

We presented our argument for a specific choice of partitions $\lambda\subpart\sigma$
to make it more tangible, but it is clear that it holds for arbitrary choices of (semi)partitions.
%
%In particular, for fixed positive integer $m$ we may choose
%$\lambda=\pas{}$  for
%arbitrary positive integer $m$, and Proposition~\ref{pro:hsym-alg-ind}so in the light of Proposition ... we proved \eqref{eq:detmultfunc}.
%
%Note that the range of subscripts in column $j$ of $\jtmat{\lambda}$ is $\brk{\lambda_j-j+1,\lambda_j-j+m}$, so
%
%$(i,j)$\bindestrich entry in $\jtmat{\lambda}$ is
%$\pas{\lambda_j - j + i}$:
%
So fix $m>0$ and consider the following partitions $\lambda$ and $\sigma$,
$\length{\lambda}=\length{\sigma}=m$:
\begin{align*}
\lambda&=\pas{(m\cdot\pas{m-1},\pas{m-1}\cdot\pas{m-1},\dots,\pas{m-1}} 
\text{ (i.e., } \lambda_j=\pas{m-j+1}\cdot\pas{m-1}\text{),}\\
\sigma &=\constpart{\pas{m\cdot m}}{m}=\pas{m^2,m^2,\dots,m^2}
\text{ (i.e., } \sigma_i\equiv m^2\text{).}
\end{align*}
% $\lambda=\pas{\lambda_1,\dots\lambda_m}$, where
% $\lambda_j\defeq \pas{m-j+1}\cdot\pas{m-1}$ for $j=1,\dots, m$.
Then the entries in
$a\defeq\jtmat{\lambda}\of{x_1,\dots,x_\n}$ and in $b\defeq\jtmat{\sigma/\lambda}\of{x_{\n+1},\dots,x_{2\n}}$ are all distinct. Hence (by Observation~\ref{obs:genericSchurIdentity})
the identity
$$
\detof{a\cdot b} = \detof a \cdot \detof b
$$
%% Referee Punkt 5
is equivalent with the multiplicativity of the determinant function \EM{in general}.
\hfill\qedsymbol
%$$
%\jtmat{\lambda}=\pas{\jtmat{\sigma/\lambda}}^T=\begin{pmatrix}
%\csymf_{(m-1)m} & \csymf_{(m-2)m} & \cdots & \csymf_{m} & \csymf_{0} \\
%\csymf_{(m-1)m+1} & \csymf_{(m-2)m+1} & \cdots & \csymf_{m+1} & \csymf_{1} \\
%\vdots   & \vdots      & \vdots       & \vdots & \vdots   \\
% 
%\csymf_{m\cdot m-1} & \csymf_{(m-1)m-1} & \cdots & \csymf_{2m-1} & \csymf_{m-1}
%\end{pmatrix}.
%$$
%
%$(i,j)$\bindestrich entry in $\jtmat{\sigma/\lambda}$ is
%$\pas{\sigma_j-\pas{m-i+1}\cdot {m} - j + i}$,
%$\brk{\pas{m-j+1}\cdot {m}-j+1,\pas{m-j+1}\cdot {m}-\mu_i-j+m}$, so choosing
%$\lambda_j\defeq \pas{m-j+1}\cdot {m}$ yields all\bindestrich distinct entries in $\jtmat{\lambda}$:

% $\alljt^\prime\setminus\nijt^\prime\to\alljt^\prime\setminus\nijt^\prime$, see
%\figref{fig:lgt-modified}: The interchanging of initial segments of paths does not change
%the picture
%Assume paths $p$ with marked point $s_i$ and $p^\prime$  with marked point $s_j$ are
%involved in the interchanging operation $\invol$ As in the above proof of the Jacobi\bindestrich Trudi identity, the
%\lgt\bindestrich operation $\invol$ shows that the only weights \EM{surviving} in
%\eqref{eq:detmultgf} are the ones associated to objects in $\nijt^\prime$: It is clear
%that $\invol$ can be applied without modification to quadruples of marked paths
%in $\alljt^\prime\setminus\nijt^\prime$, the only point which (maybe) needs some
%consideration is whether it really is a mapping $\nijt^\prime\to\nijt^\prime$.
%
%
%\EM{involution} (if so, it clearly is weight\bindestrich preserving
%and sign\bindestrich reversing):
%% \EM{respects} this constraint on the lattice paths:
%If we interchange initial segments of two intersecting labeled paths $\pas{k_1,p_1}$ and
%$\pas{k_2,p_2}$,
%% both passing through points of $\asvector{y}$,
%% then the resulting paths again pass through.
%then the resulting labeled paths are 
%\bit
%\item $\pas{k_1,\invol\of{p_1}}$ and $\pas{k_2,\invol\of{p_2}}$,
%	if the minimal point of intersection of $p_1$ and $p_2$ is at a lower level than
%	the points $s_k$
%\item or $\pas{k_1,\invol\of{p_2}}$ and $\pas{k_2,\invol\of{p_1}}$,
%	if the minimal point of intersection of $p_1$ and $p_2$ is at a higher or equal level than
%	the points $s_k$,
%\eit
%
%Hence the product of the two generating functions $\detof{a}$ and $\detof{b}$ may be
%viewed as the generating function of quadruples $P$ of nonintersecting lattice paths connecting
%\bit
%\item the lower points $\asvector{r}$
%\item with the ending points $\asvector{t}$,
%\item under the
%additional \EM{constraint} that each lattice path in $P$ must pass through one of the
%points of $\asvector{s}$.
%\eit
%
%It is clear that the \lgt\bindestrich involution \EM{respects} this constraint on the lattice
%paths: If we interchange initial segments of two intersecting paths,
%both passing through points of $\asvector{y}$, then the resulting paths have the same
%property.


%So, again by the \lgt\bindestrich interpretation, $\detof{a}\cdot\detof{b}$ appears as the
%determinant of another
%matrix $c$, where entry $c_{i,j}$ equals the generating function of lattice paths connecting
%$r_j$ to $t_i$ \EM{under the constraint} that
%the paths must pass through one of the points of $\asvector{s}$. Clearly, the generating
%function of these \EM{constrained} paths is
%$$
%\sum_{k=1}^4\gfpaths{r_i}{s_k}\cdot\gfpaths{s_k}{t_j},
%$$
%which is precisely the $\pas{i,j}$\bindestrich entry of the matrix $a\cdot b$, whence we
%proved $\detof{a}\cdot\detof{b}=\detof{a\cdot b}$. At first, this proof applies to
%the special choice of $a$ and $b$ only, but it demonstrates \EM{ad oculos}
%also the validity of the general statement. 

\subsubsection{The Cauchy\bindestrich Binet formula: Another proof ``by example''.} Now it is ``almost
immediate'' to \EM{see} the Cauchy\bindestrich Binet formula as an obvious generalization of the multiplicativity of the determinant.
%Once we got accustomed
%to ``viewing determinants as nonintersecting lattice paths'', the following well\bindestrich known
%theorem appears as a simple generalization of the multiplicativity of the determinant.
%For convenience, we introduce some notation: $\brk{n}\defeq \setof{1,2,\dots,n}$, and
%for matrix $a=\pas{a_{i,j}}_{\pas{i,j}=\pas{1,1}}^{\pas{m,n}}$, we denote by
%$\minor{a}{R}{C}$ the \EM{minor}
%of $a$ which consists of the rows with indices in $R\subseteq\brk{m}$ and the columns with indices
%in $C\subseteq\brk{n}$ (with this notation, the original matrix $a$ is $\minor{a}{\brk{m}}{\brk{n}}$).

\begin{thm}[Cauchy\bindestrich Binet formula]
Let $a$ be an $m\times n$\bindestrich matrix and $b$ be an $n \times m$\bindestrich matrix. Then we have
\begin{equation}
\label{eq:CauchyBinet}
\detof{a\cdot b} = \sum_{S\subseteq\brk{n},\cardof{S}=m}
	\detof{\minor{a}{\brk{m}}{S}}\cdot\detof{\minor{b}{S}{\brk{m}}}.
\end{equation}
\end{thm}
Note that this formula holds trivially if $m>n$ (since the determinant
$\detof{a\cdot b}$ is zero in this case, as is the empty sum in \eqref{eq:CauchyBinet}), and
amounts to the multiplicativity of the determinant if $m=n$.

We illustrate this formula by a special case and consider the matrices
$$
a\defeq\pas{\csymf_{\lambda_j-j+i}\of{x_1,\dots x_7}}_{\pas{i,j}\in\pas{\brk{3}\times\brk{4}}}\text{ and }
b\defeq\pas{\csymf_{\sigma_j-\lambda_i-j+i}\of{x_8,\dots x_{13}}}_{\pas{i,j}\in\pas{\brk{4}\times\brk{3}}},
$$
where $\lambda=\pas{9,7,5,4}$ and $\sigma=\pas{19,17,14}$.

%% Referee Punkt 4:
Again, we want to employ the \lgti.
To this end, we consider the vectors of points $\asvector{r}=\pas{r_1,r_2,r_3}$ and
$\asvector{t}=\pas{t_1,t_2,t_3}$, and the vector of arcs,
$\asvector{s}=\pas{s_1,s_2,s_3,s_4}$, where
\begin{align*}
\asvector{r} &= \pas{{%\scriptstyle
	\pas{-1,1},\;\pas{-2,1},\;\pas{-3,1}}},\\
\asvector{s} &= \pas{{%\scriptstyle
	\pas{\pas{8,7},\pas{8,8}},\;\pas{\pas{5,7},\pas{5,8}},\;
	\pas{\pas{2,7},\pas{2,8}},\;\pas{\pas{0,7},\pas{0,8}}}},\\
\asvector{t} &= \pas{{%\scriptstyle
	\pas{18,13},\;\pas{15,13},\;\pas{11,13}}},
\end{align*}
(See \figref{fig:Cauchy-Binet} for an illustration.)

\begin{figure}
\caption{Illustration: The Cauchy\bindestrich Binet formula. Triples of nonintersecting
lattice paths connecting
points $\asvector{r}=\pas{r_1,r_2,r_3}$ and $\asvector{t}=\pas{t_1,t_2,t_3}$, where
each single path must pass through an arc in $\asvector{s}=\pas{s_1,s_2,s_3,s_4}$,
%(marked points are indicated by black circles in the picture),
% the
%additional constraint that each paths must pass through some fixed $3$\bindestrich element
%subvector $\asvector{s}^\prime$ of $\asvector{s}=\pas{s_1,s_2,s_3,s_4}$,
%% one of the points $\asvector{s}=\pas{s_1,s_2,s_3,s_4}$
can be ``cut in two halves'' along $\asvector{s}$. Note that the arcs used by the paths
constitute a $3$\bindestrich element subvector $\asvector{s}^\prime$, and the
halves appear as tripels of nonintersecting
lattice paths $\nijt\of{\asvector{r}\to\asvector{s}^\prime}$ and
$\nijt\of{\asvector{s}^\prime\to\asvector{t}}$,
respectively. In the picture, we have $\asvector{s}^\prime=
\pas{s_1,s_2,s_4}$.
% Disregarding the marked points, the same triple of paths
%also occurs for $\asvector{s}^{\prime\prime}=
%\pas{s_1,s_2,s_4}$.
}
\label{fig:Cauchy-Binet}
\begin{center}
\input graphics/Cauchy-Binet
\end{center}
\end{figure}

As before,
observe that the
%$\pas{i,k}$\bindestrich entry in $a$
%and the $\pas{k,j}$\bindestrich entry in $b$ appear as the generating functions
%of lattice paths
%\begin{align*}
%\csymf_{\lambda_k-k+i}\of{x_1,\dots x_7} &=\gfpaths{r_i}{s_k},\\
%\csymf_{\sigma_j-\lambda_k-j+k}\of{x_7,\dots x_{13}} &=\gfpaths{s_k}{t_j},
%\end{align*}
%respectively, and the
$\pas{i,j}$\bindestrich entry in $a\cdot b$
%is
%$$
%\sum_{k=1}^4\gfpaths{r_i}{s_k}\cdot\gfpaths{s_k}{t_j},
%$$
%which
may be viewed as the generating function $\gfpaths{r_i}{t_j}^\prime$ of
\EM{constrained} paths which must pass through one arc in $\asvector{s}$. By the \lgt\bindestrich involution, the determinant $\detof{a\cdot b}$
appears as the generating function of triples of nonintersecting labeled paths
$\nijt^\prime\of{\asvector{r}\to\asvector{t}}$, which (as in \secref{sec:multiplicativity} \gedankenstrich\ just \EM{look} at \figref{fig:Cauchy-Binet}!) appears as
$$
\nijt^\prime\of{\asvector{r}\to\asvector{t}} = \bigcup_{\asvector{s}^\prime}
	\pas{
		\nijt\of{\asvector{r}\to\asvector{s}^\prime}
		\times
		\nijt\of{\asvector{s}^\prime\to\asvector{t}},
	}
$$
where the union runs over all $3$\bindestrich element subvectors $\asvector{s}^\prime$ of
$\asvector{s}$. By the same reasoning as in \secref{sec:multiplicativity}, this shows \eqref{eq:CauchyBinet}.
\hfill\qedsymbol
% 
%length $3$
%\EM{labeled} constrained lattice paths, i.e., as % $\gfpaths{r_i}{t_j}^\prime$
%the sum of the weights of objects in the union
%$$
%\bigcup_{k=1}^4 \setof{\pas{k,p}:\text{ $p$ is lattice paths connecting $r_i$ to $t_j$ via $s_k$}},
%$$
%where the weight of the pair $\pas{k,p}$ is $\weight\of{\pas{k,p}}\defeq\weight\of p$.
%(Note that path $p$ appears more than once in this union if $p$ passes through more
%than one of the points of $\asvector{s}$.)
%%
%%generating function of \EM{constrained} lattice paths connecting
%%$r_i$ to $t_j$, where the additional \EM{constraint} is that the paths must pass through one
%%of the points in $\asvector{s}$.
%
%Clearly, the \lgt\bindestrich involution respects this constraint, whence \eqref{eq:CauchyBinet}
%follows immediately, at first only for our concrete example, but then clearly in general.
%(See \figref{fig:Cauchy-Binet} for an illustration.)
%

