The notation $\detof{x}$ has three different meanings in our presentation, depending on the
%% Referee: "respective"
type of object $x$ (the %respective
meaning should always be clear from the context):
\bit
\item if $x$ is a \EM{set}, then $\cardof{x}$ denotes the \EM{cardinality} of $x$,
\item if $x$ is a \EM{matrix}, then $\detof{x}$ denotes the \EM{determinant} of $x$,
\item if $x$ is a \EM{partition} or \EM{shape}, then $\partweightof{x}$ denotes the
	\EM{sum of parts} of $x$ (to be explained below).
\eit


In the following, we shall briefly recall basic concepts and facts needed for our presentation. (More information can be found, e.g., in \cite{macdonald:1995}.)

% The ring of symmetric functions.
\subsection{The ring of symmetric functions}
Consider the ring $\Z\brk{x_1,x_2,\dots,x_\n}$ of polynomials in $\n$ independent (commuting)
variables
$\asvector{x}\defeq\pas{x_1,x_2,\dots,x_\n}$ with integer coefficients. % (see \cite[I.2]{macdonald:1995}).
The \EM{degree} of a monomial $x_1^{k_1}\cdot x_2^{k_2}\cdots x_\n^{k_\n}$ is
the sum $k_1+k_2+\dots+k_\n$, and a polynomial $p$ is called \EM{homogeneous} of degree $k$
if all monomials of $p$ have the same degree $k$.

The symmetric group $\symm_\n$ acts on this ring by permuting the variables, and a polynomial
is \EM{symmetric} if it is invariant under this action. The set of all symmetric polynomials
forms a subring $\symfs_n\subseteq\Z\brk{x_1,x_2,\dots,x_\n}$ which is \EM{graded}, i.e.,
\begin{equation*}
\symfs_\n=\bigoplus_{k\geq 0}\symfs_\n^k,
\end{equation*}
where $\symfs_\n^k$ consists of the homogeneous symmetric polynomials of degree $k$, together
with the zero polynomial.

% Complete homogeneous symmetric functions.
%\subsection{Complete homogeneous symmetric functions}
For each $r\in\Z$, the \EM{complete homogeneous symmetric function} $\csymf_r\of{\asvector{x}}$ is the sum of
all monomials of degree $r$. % in the variables $x_1, x_2,\dots,\x_\n$.
In particular,
$\csymf_0\of{\asvector{x}}=1$ and, by convention, $\csymf_r\of{\asvector{x}}=0$ for $r<0$. % (see \cite[I.2]{macdonald:1995}).

For example, if $\n=3$, then $\asvector{x}=\pas{x_1,x_2,x_3}$ and
$$
\csymf_2\of{x_1,x_2,x_3}= x_1^2 + x_2^2 + x_3^2 + x_1 x_2 + x_1 x_2 + x_2 x_3.
$$

The following fact is well\bindestrich known (see, e.g., \cite[(2.8)]{macdonald:1995}):
%% Referee Punkt 1:
%% Achtung: Nur fr die ersten N h_i's richtig!!!!
\begin{pro}
\label{pro:hsym-alg-ind}
The set of complete homogeneous symmetric functions
$$\setof{\csymf_0\of{\asvector{x}},\csymf_1\of{\asvector{x}},\dots,
\csymf_{\n}\of{\asvector{x}}}$$
is algebraically independent, i.e., $p\equiv0$
is the \EM{only} polynomial such that 
$$p\of{\csymf_0\of{\asvector{x}},\csymf_1\of{\asvector{x}},\dots,
\csymf_{\n}\of{\asvector{x}}} \equiv 0.$$

Moreover, %for $0<k<\n$, 
% for $k=\lfloor\frac\n2\rfloor$,
the set
$$ \setof{\csymf_{i}\of{x_1,\dots,x_{\n}}:\;i=0,1,2,\dots,\n}
\cup
\setof{\csymf_{i}\of{x_{\n+1},\dots,x_{2\n}}:\;i=1,2,\dots,\n}
$$
is also algebraically independent.
\hfill\qedsymbol
\end{pro}

% Partitions and Ferrers diagrams
% \subsection{Partitions and Ferrers diagrams}
\subsection{Schur functions}
Recall the following standard definitions:
An infinite weakly decreasing series of nonnegative integers $\lambda=\pas{\lambda_i}_{i=1}^\infty$,
where only finitely many elements are positive, is called a \EM{partition}. The largest
index $i$ for which $\lambda_i>0$ is called the \EM{length} of the partition $\lambda$ and
is denoted by $\length{\lambda}$. The sum of the non\bindestrich zero parts
$\lambda_1+\lambda_2+\cdots+\lambda_{\length{\lambda}}$ of $\lambda$ is denoted by
$\partweightof\lambda$. In most cases we shall omit the trailing
zeroes, i.e.,
for $\length{\lambda}=r$ we simply write $\lambda=\pas{\lambda_1,\lambda_2,\dots,\lambda_r}$,
where $\lambda_1\geq\lambda_2\geq\dots\geq\lambda_r> 0$.

For example, $\lambda=\pas{7, 4, 4, 3, 1, 1, 1}$ is a partition of length $\length{\lambda}=7$
with $\cardof\lambda=21$.

The {\em Ferrers diagram\/} $F_\lambda$
of $\lambda$
is an array of cells with $\length{\lambda}$ left-justified rows and $\lambda_i$
cells in row $i$. For an illustration, see \figref{fig:FerrersDiagram}.

\begin{figure}
\caption{Illustration: Ferrers diagram $F_\lambda$ of the partition
$\lambda=\pas{7, 4, 4, 3, 1, 1, 1}$.}
\label{fig:FerrersDiagram}
\begin{center}
\input graphics/FerrersDiagram-slide
\end{center}
\end{figure}

For our
purposes, it is convenient to generalize this definition: By a \EM{semipartition} we understand
an infinite weakly decreasing series of integers $\pas{\lambda_i}_{i=1}^\infty$,
where
$$\limes\lambda\defeq\lim_{n\to\infty}\lambda_n>-\infty.$$
The length of semipartition $\lambda$ is the largest integer $m$ with
$\lambda_m>\lambda_\infty$, which we denote again by $\length\lambda$:
Note that every partition $\mu$ is a semipartition with $\limes\mu=0$.

Clearly, the set of semipartitions is closed under component\bindestrich wise addition
$$ \lambda+\mu=\pas{\lambda_i}_{i=1}^\infty + \pas{\mu_i}_{i=1}^\infty\defeq
	\pas{\lambda_i+\mu_i}_{i=1}^\infty.$$
For $z\in\Z$, denote by $\allconstpart{z}$ the
semipartition
$$
\allconstpart{z}\defeq\pas{z}_{i=1}^\infty.
$$
For $m,z\in\N$, %$m>0$ and $z>0$,
denote by $\constpart{z}{m}$ the
partition
$$
\constpart{z}{m}\defeq\pas{\underbrace{z,z,\dots,z}_{m\text{ times}},0,0,\dots}.
$$
%(Note that $\limes{\constpart{z}{m}}=0$.)
%with the special case $\constpart{z}{\infty}=\pas{z}_{i=1}^\infty$ (i.e., the constant
%semipartition). 
If two semipartitions
$\lambda$, $\mu$ satisfy
\bit
\item $\mu_i\leq\lambda_i$ for all $i=1,2,\dots$,
\item and $\limes{\mu} = \limes{\lambda}$,
\eit
then we denote this by $\mu\subpart\lambda$ and introduce the symbol $\lambda/\mu$,
which we call a \EM{shape}. The length of the shape $\lambda/\mu$ is defined by
$\length{\lambda/\mu}\defeq\length\lambda$,
and the (terminating!) sum 
$\sum_{i=1}^\infty\pas{\lambda_i-\mu_i}$ is denoted by
$\partweightof{\lambda/\mu}$. Note that we may view \EM{partition} $\lambda$ as the
\EM{shape} $\lambda/\!\allconstpart{0}$

The {\em Ferrers diagram\/} $F_{\lambda/\mu}$ of shape $\lambda/\mu$
is an array of cells with $\length{\lambda/\mu}$ left\bindestrich justified rows and $\pas{\lambda_i-\mu_i}$
cells in row $i$, where the first $\mu_i$ cells in row $i$ are missing, see \figref{fig:skewFerrersDiagram} for an illustration.
%An {\em $\n$\bindestrich semistandard skew Young tableau\/} of shape $\lambda/\mu$ is a filling of the cells of $F_{\lambda/\mu}$ with integers from the set
%$\{1,2,\dots,\n\}$,
%such that the numbers filled into the cells weakly increase in rows and
%strictly increase in columns
%(see the left picture in \figref{fig:skew-Young-tableau} for an illustration).

Note that for arbitrary $z\in\Z$, the Ferrers diagram $F_{\lambda/\mu}$ with $\length{\lambda/\mu}=m$ is congruent to
\begin{align}
F_{\lambda/\mu} &\simeq F_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}
	\text{ for arbitrary } z\in\Z,\notag\\
F_{\lambda/\mu} &\simeq F_{\lambda+\constpart{z}{m}/\mu+\constpart{z}{m}}
	\text{ for } z>0\in\Z.\label{eq:FerrersShiftInvariance}
\end{align}
%can be written as
%%is congruent to the Ferrers diagram
%%$F_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}$ and to $F_{\lambda+\constpart{z}{m}/\mu+\constpart{z}{m}}$ for $m=\length{\lambda/\mu}$ and \EM{arbitrary} $z\in\Z$:
%\begin{equation}
%\label{eq:FerrersShiftInvariance}
%F_{\lambda/\mu} = F_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}
% = F_{\lambda+\constpart{z}{m}/\mu+\constpart{z}{m}}.
%\end{equation}
In particular, for some
partition $\lambda$ we have $F_\lambda\simeq
%is congruent to
% may be viewed as the Ferrers diagram 
F_{\lambda+\allconstpart{z}/\allconstpart{z}}$ for \EM{arbitrary} $z\in\Z$. 
%is a semipartition, which we may
%identitify with the shape $\lambda/\constpart{0}{\infty}$.

% $\limes\lambda\defeq\lim_{n\to\infty}\lambda_n$: 

\begin{figure}
\caption{Illustration: Ferrers diagram of skew shape.
% $F_{\lambda/\mu}$ of the shape
%$\lambda/\mu=\pas{7, 4, 4, 3, 1, 1, 1}/\pas{3, 2, 2, 1}$. Note that the
%\EM{same} Ferrers diagram would arise for the shapes $\pas{\lambda+\allconstpart{z}}/\pas{\mu+\allconstpart{z}}$ for arbitrary $z\in\Z$, and for shapes $\pas{\lambda+\constpart{z}{7}}/\pas{\mu+\constpart{z}{7}}$, for $z\in\N$.
}
\label{fig:skewFerrersDiagram}
Ferrers diagram $F_{\lambda/\mu}$ of the shape
$\lambda/\mu=\pas{7, 4, 4, 3, 1, 1, 1}/\pas{3, 2, 2, 1}$. Note that the
\EM{same} Ferrers diagram would arise for the shapes $\pas{\lambda+\allconstpart{z}}/\pas{\mu+\allconstpart{z}}$ for arbitrary $z\in\Z$, and for shapes $\pas{\lambda+\constpart{z}{7}}/\pas{\mu+\constpart{z}{7}}$, for $z\in\N$.
\begin{center}
\input graphics/skewFerrersDiagram
\end{center}
\end{figure}

% Schur functions
%\subsection{Schur functions}
\EM{Schur functions}, which are irreducible general linear characters, can be defined
% as quotient of alternants
 as follows \cite{schur:1901}. Let $\lambda$ be a partition and let $\setof{x_1, \dots, x_\n}$ be a set of independent variables.
Then the Schur function $\schurf_\lambda\of{x_1,\dots, x_\n}$ indexed by $\lambda$ is defined
as the quotient of determinants  (see \cite[(3.1)]{macdonald:1995})
\begin{equation}
\label{eq:schur-determinant}
\schurf_\lambda\of{x_1,\dots, x_\n}\defeq
\frac{
	\detof{x_i^{\lambda_j+\n-j}}_{i,j=1}^\n
}{
	\detof{x_i^{\n-j}}_{i,j=1}^\n
}.
\end{equation}
It is easy to see that $\schurf_\lambda$ is a \EM{symmetric function}, which is
homogeneous of degree $\cardof\lambda$. The Jacobi\bindestrich Trudi identity
(first obtained by Jacobi \cite{jacobi:1841} and simplified by Trudi \cite{trudi:1864},
see \cite[(3.4)]{macdonald:1995}) states that the
Schur function $\schurf_\lambda$ equals the following determinant of complete homogeneous symmetric functions:
\begin{equation}
\label{eq:JacobiTrudi}
\schurf_\lambda=
\detof{\csymf_{\lambda_j-j+i}}_{i,j=1}^{m},
\end{equation}
where $m=\length{\lambda}$. Here, we introduced in passing the shorthand notations $\schurf_\lambda$ and $\csymf_r$ 
for $\schurf_\lambda\of{x_1,\dots,x_\n}$ and $\csymf_r\of{x_1,\dots,x_\n}$,
respectively.

An {\em $\n$\bindestrich semistandard Young tableau\/} of shape $\lambda$ is a filling of the cells
of the Ferrers diagram $F_\lambda$ with integers from the set $\setof{1,2,\dots,\n}$, such
that the numbers filled into the cells \EM{weakly increase in rows} and \EM{strictly increase in columns}.

Let ${\tableau}$ be a semistandard Young tableau and define $\nofentries{\tableau}{k}$ to be
the number of entries $k$ in ${\tableau}$. The \EM{weight} $\weight\of{\tableau}$ of ${\tableau}$
is defined as follows:
\begin{equation}
\label{eq:dfn-weight-tableau}
\weight\of{\tableau} = \prod_{k = 1}^{\n} x_k^{\nofentries{\tableau}{k}}.
\end{equation}
See \figref{fig:SSYT} for an illustration.

\begin{figure}
\caption{Illustration: An $8$\bindestrich semistandard Young tableau $T$ of shape
$\lambda=\pas{7, 4, 4, 3, 1, 1, 1}$ and its weight $\weight\of T$.}
\label{fig:SSYT}
\begin{center}
\input graphics/SSYT
\end{center}
\end{figure}

Then the \EM{Schur function} $\schurf_\lambda$ can equivalently be written as the following
\EM{generating function} (formal sum of weights)
\begin{equation*}
\schurf_\lambda%\of{x_1, x_2, x_3, \dots, x_\n}
= \sum_{{\tableau}}\weight\of{\tableau},
\end{equation*}
where the sum is over all $\n$\bindestrich semistandard Young tableaux ${\tableau}$ of
shape $\lambda$ (see \cite[Definition~4.4.1]{sagan:2000}). 

%Let $\lambda$, $\mu$ be two partitions. If $\mu_i\leq\lambda_i$ for all $i\geq 1$,
%then we denote this by $\mu\subpart\lambda$. Note that the Ferrers diagram of $\mu$
%is contained in the Ferrers diagram of $\lambda$ if $\mu\subpart\lambda$.

% Now consider two semipartitions $\mu\subpart\lambda$.
%The {\em skew Ferrers
%diagram\/} $F_{\lambda/\mu}$ of $\lambda/\mu$
%is an array of cells with $r$ left-justified rows and $\lambda_i-\mu_i$
%cells in row $i$, where the first $\mu_i$ cells in row $i$ are missing.
An {\em $\n$\bindestrich semistandard skew Young tableau\/} of shape $\lambda/\mu$ is a filling of the cells of $F_{\lambda/\mu}$ with integers from the set
$\{1,2,\dots,\n\}$,
such that the numbers filled into the cells weakly increase in rows and
strictly increase in columns.
See the left picture in \figref{fig:skew-Young-tableau} (or \figref{fig:SSYT} again)
for an illustration.

Then we define the \EM{skew Schur function} $\schurf_{\lambda/\mu}$
as the following generating function:
\begin{equation}
\label{eq:skewSchur}
\schurf_{\lambda/\mu}%\of{x_1, x_2, x_3, \dots, x_\n}
\defeq \sum_{{\tableau}}\weight\of{\tableau},
\end{equation}
where the sum is over all $\n$\bindestrich semistandard skew Young tableaux ${\tableau}$ of
shape $\lambda/\mu$, where the weight $\weight\of{\tableau}$ of ${\tableau}$
is defined as in \eqref{eq:dfn-weight-tableau}. 
%% Referee Punkt 2:
The function
$\schurf_{\lambda/\mu}$ also is
a symmetric function (see \cite[proof of Proposition 4.4.2]{sagan:2000}), which is
homogeneous of degree $\partweightof{\shape\lambda\mu}$.
%
%\cite[Proposition 4.9.1]{sagan:2000} or \cite[(5.9)]{macdonald:1995}
%
Let $m\defeq\length{\lambda/\mu}$, then by \eqref{eq:FerrersShiftInvariance}, we clearly have
\begin{align}
\schurf_{\lambda/\mu} &= \schurf_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}
	\text{ for arbitrary } z\in\Z,\notag\\
\schurf_{\lambda/\mu} &= \schurf_{\lambda+\constpart{z}{m}/\mu+\constpart{z}{m}}
	\text{ for } z>0\in\Z.\label{eq:SchurShiftInvariance}
\end{align}
%can be written as
%%is congruent to the Ferrers diagram
%%$F_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}$ and to $F_{\lambda+\constpart{z}{m}/\mu+\constpart{z}{m}}$ for $m=\length{\lambda/\mu}$ and \EM{arbitrary} $z\in\Z$:
%\begin{equation}
%\label{eq:FerrersShiftInvariance}
%F_{\lambda/\mu} = F_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}
% = F_{\lambda+\constpart{z}{m}/\mu+\constpart{z}{m}}.
%\end{equation}
In particular,
%$F_\lambda$ of partition $\lambda\simeq
%%is congruent to
%% may be viewed as the Ferrers diagram 
%F_{\lambda+\allconstpart{z}/\allconstpart{z}}$ for \EM{arbitrary} $z\in\Z$. 
%
%
%By \eqref{eq:FerrersShiftInvariance}, we clearly have for arbitrary $z\in\Z$:
%\begin{equation}
%\label{eq:SchurShiftInvariance}
%\schurf_{\lambda/\mu} = \schurf_{\lambda+\allconstpart{z}/\mu+\allconstpart{z}}
% = \schurf_{\lambda+\constpart{z}{\length{\lambda/\mu}}/\mu+\constpart{z}{\length{\lambda/\mu}}}.
%\end{equation}
%%
%%Note that
%%$\schurf_{\lambda/\mu} = \schurf_{\lambda+\constpart{z}{\infty}/\mu+\constpart{z}{\infty}}$;
%In particular,
$\schurf_\lambda=\schurf_{\lambda+\allconstpart{z}/\allconstpart{z}}$
for arbitrary $z\in\Z$, and $\schurf_\lambda=\schurf_{\lambda+\constpart{z}{m}/\constpart{z}{m}}$
for $z>0\in\Z$.
%for the zero partition $\pas{0}\defeq\pas{0,0,\dots}$, the skew Schur
%function $\schurf_{\lambda/\pas{0}}$
%is identical to the ``ordinary'' Schur function $\schurf_\lambda$.

Skew Schur functions, too, have an expansion in terms of complete homogeneous symmetric functions (see \cite[(5.4)]{macdonald:1995}) which generalizes \eqref{eq:JacobiTrudi}:
\begin{equation}
\label{eq:JacobiTrudiSkew}
\schurf_{\lambda/\mu}=
\detof{\csymf_{\lambda_j-\mu_i-j+i}}_{i,j=1}^{m}.
\end{equation}

\subsection{Connection between determinantal relations and skew Schur function identities}
We want to illustrate the connection between determinants and skew Schur functions,
using Dodgson's condensation formula \cite{dodgson:1866} (see also
%\cite{jacobi:1841a} or \cite[volume 1, page 253]{muir:1906})
\cite[Vol. III, page 17]{muir:1906})
as an example.

\begin{rem}
%% Referee Punkt 3:
Dodgson's condensation formula is also known as the Desnanot-Jacobi adjoint matrix theorem: According to Bressoud \cite[remarks following Theorem 3.12]{bressoud:1999}, Lagrange \cite{lagrange:1773}
discovered this theorem for $n = 3$ in 1773 (see also \cite[vol. I, p.~37ff]{muir:1906}),
Desnanot % (see also \cite[vol. I, p.~142ff]{muir:1906})
proved it for $n \leq 6$ in 1819  (see \cite[vol. I, p.~142]{muir:1906}), and Jacobi  \cite{jacobi:1841a}
published the general theorem in 1841 (see also \cite[vol. I, p.~264]{muir:1906}).
\end{rem}

%We fix some notation:
Consider some $m\times n$\bindestrich matrix
$$
\mata=\pas{\mata_{i,j}}_{\pas{i,j}=\pas{1,1}}^{\pas{m,n}}
$$
and define $\range{k}\defeq \setof{1,2,\dots,k}$ for $k\in\N$. Then we may write
%$m\times n$\bindestrich matrix $a=\pas{a_{i,j}}_{\pas{i,j}=\pas{1,1}}^{\pas{m,n}}$ as
$$
\mata=\minor{\mata}{\range{m}}{\range{n}}\defeq\pas{\mata_{i,j}}_{\pas{i,j}\in\range{m}\times\range{n}}.
$$
More generally, for subsets $R\subseteq\brk{m}$ and $C\subseteq\brk{n}$, we denote
by $\minor{\mata}{R}{C}$ the \EM{submatrix} of $\mata$ which consists of the rows $R$ and the columns $C$;
\EM{in the same order as in $\mata$.} (All sets considered in this paper are
\EM{ordered}, and all subsets inherit the order from their supersets.)

If we want to describe the same submatrix by deleting rows and columns in $a$, then we
write
$$
\minor{\mata}{R}{C} =
\deleterowcols{\mata}{X}{Y},
$$
where $X=\complement{R}\defeq\range{m}\setminus R$ and
$Y=\complement{C}\defeq\range{n}\setminus C$.

%Let $a=\pas{a_{i,j}}_{\pas{i,j}=\pas{1,1}}^{\pas{m,n}}$ be an arbitary $m\times n$\bindestrich matrix. % For positive integers $k < m$, $0\leq l< n$.
%For some subsets $\setof{i_1,\dots,i_k}\subseteq\setof{1,2,\dots,m}$ and
%$\setof{j_1,\dots,i_l}\subseteq\setof{1,2,\dots,n}$,
%denote by
%$$\deleterowcols{a}{i_1,\dots,i_k}{j_1,\dots,j_l}$$
%the result after deleting rows ${i_1,\dots,i_k}$ and columns ${j_1,\dots,i_l}$ in $a$.

\begin{pro}[Dodgson's Condensation] Let $\mata=\pas{\mata_{i,j}}_{i,j=1}^m$ be an arbitrary
$m\times m$\bindestrich matrix, $m\geq2$. Then there holds the following identity:
\begin{equation}
\label{eq:Dodgson}
\mdetof{\mata}\cdot\mdetof{\deleterowcols{\mata}{\setof{1,m}}{\setof{1,m}}} =
	\mdetof{\deleterowcols{\mata}{\setof1}{\setof1}}\cdot\mdetof{\deleterowcols{\mata}{\setof m}{\setof m}} - 
\mdetof{\deleterowcols{\mata}{\setof1}{\setof m}}\cdot\mdetof{\deleterowcols{\mata}{\setof m}{\setof1}}.
\end{equation}
We adopt the convention that the empty determinant equals $1$,
i.e., $\mdetof{\minor{\mata}{\emptyset}{\emptyset}}\equiv1$: Then for $m=2$, 
\eqref{eq:Dodgson} simply amounts to the formula for $2\times2$\bindestrich determinants.\hfill\qedsymbol
\end{pro}
%We shall provide a proof for this well\bindestrich known identity later. Here, we point out that
%by the above reasoning this immediately translates to a Schur function identity.

%In order to formulate the Schur function identity corresponding to this determinantal
%identity,
%we introduce the following notation: Given some partition $\mu=\pas{\mu_1,\mu_2,\dots,\mu_m}$ of length $\length\mu=m$, for $n\geq-\mu_m$ we denote by $\addtoparts{\mu}{n}$ the
%partition
%$$\addtoparts{\mu}{n}\defeq\pas{\mu_1+n,\mu_2+n,\dots,\mu_m+n}.$$
%Observe that by \eqref{eq:JacobiTrudiSkew}, we have
%\begin{equation}
%\label{eq:skewshifts}
%\schurf_{\addtoparts{\lambda}{n}/\addtoparts{\mu}{n}}=\schurf_{\lambda/\mu}.
%\end{equation}
%
%For some $n>0$, we denote
%by $\constpart{n}{m}$ the partition of length $m$
%$$\constpart{n}{m}\defeq\pas{n,n,\dots,n}.$$
%Note that \eqref{eq:skewshifts} implies
%\begin{equation}
%\label{eq:skewblocks}
%\schurf_{\addtoparts{\lambda}{n}/\constpart{n}{m}}=\schurf_\lambda.
%\end{equation}
%In particular, for $k<n$ we have $\schurf_{\constpart{n}{m}/\constpart{k}{m}}=\schurf_{\constpart{\pas{n-k}}{m}}$.

We shall ``translate'' this determinantal identity to a Schur function identity. To this end,
we introduce the operation ``delete part
$\lambda_k$
%$\lambda_k>\lambda_\infty$
%(i.e., $1\leq k\leq\length{\lambda}$)
in semipartition $\lambda$'', which we denote by $\deletepart{\lambda}{k}$:
\begin{equation}
\label{eq:deletepart}
\deletepart{\lambda}{k}\defeq\pas{\lambda_1,\dots,\lambda_{k-1},\lambda_{k+1}-1,\lambda_{k+2}-1,\dots}.
%,\lambda_{\length\lambda},\lambda_\infty,\lambda_\infty,\dots}.
\end{equation}
Note that
$\limes{\deletepart{\lambda}{k}}=\limes\lambda-1$
and
$\length{\deletepart{\lambda}{k}}=\max\of{\length\lambda,k}-1$.

For some subset $\setof{k_1<k_2<\dots<k_l}\subseteq
\setof{1,2,\dots,m}$, we define inductively:
$$
\deletepart{\lambda}{k_1,\dots,k_l}\defeq
\deletepart{\pas{\deletepart{\lambda}{k_2,\dots,k_{l}}}}{k_1}.
$$

% \alert{fix i<->j!}
Now assume some skew shape $\lambda/\mu$, $m=\length{\lambda/\mu}$. According
to \eqref{eq:JacobiTrudiSkew}, the skew Schur function $\schurf_{\lambda/\mu}$ equals the
determinant of the matrix
$$\jtmat{\lambda/\mu}\defeq\pas{\csymf_{\lambda_j-\mu_i-j+i}}_{\pas{i,j}={1,1}}^{\pas{m,m}}.$$
%$$
%A_{\lambda/\mu}=
%\begin{pmatrix}
%\lambda_1-\mu_1 & \lambda_1-\mu_2+1 &\dots&\lambda_1-\mu_n-1+n\\
%\lambda_2-\mu_1 - 1 & \lambda_2-\mu_2 &\dots&\lambda_2-\mu_n-2+n\\
%\vdots &\vdots & & \vdots\\
%\lambda_m-\mu_1 - m + 1 & \lambda_m-\mu_2 - m +2 & \dots&\lambda_m-\mu_n - m + n
%\end{pmatrix}.
%$$
Observe that %after deleting the row with index $k$, in the resulting
the $\pas{i,j}$\bindestrich entries in matrix $\deletecol{\jtmat{\lambda/\mu}}{k}$ are
\bit
\item $\csymf_{\lambda_{j}-\mu_i-j+i}$ for $j<k$,
\item $\csymf_{(\lambda_{j+1}-1)-\mu_i-j+i}$ for $k\leq j\leq m-1$,
\eit
i.e., deleting column $k$ in $\jtmat{\lambda/\mu}$ corresponds to deleting $\lambda_k$ in $\lambda$.
Likewise,
the $\pas{i,j}$\bindestrich entries in matrix $\deleterow{\jtmat{\lambda/\mu}}{l}$ are
\bit
\item $\csymf_{\lambda_{j}-\mu_i-j+i}$ for $i<l$,
\item $\csymf_{\lambda_{j}-\pas{\mu_{i+1}-1}-j+i}$ for $l\leq i\leq m-1$,
\eit
i.e., deleting row $l$ in $\jtmat{\lambda/\mu}$ corresponds to deleting $\mu_l$ in $\mu$.
%in $A_{\lambda/\mu}$, while the entry in row $i$, $k\leq i\leq m-1$ and column $j$ now is
%$$(\lambda_{i+1}-1)-\mu_j-i+j.$$
%So obviously, deleting row $k$ corresponds to an operation on the partition $\lambda=\pas{\lambda_1,\dots,\lambda_n}$:
%$$
%\deletepart{\lambda}{k}\defeq\pas{\lambda_1,\dots,\lambda_{k-1},\lambda_{k+1}-1,\dots,\lambda_m-1}.
%$$
These observations generalize to the following relation:
%$$
%\deletecol{\jtmat{\lambda/\mu}}{k} = \jtmat{\deletepart{\lambda}{k}/\mu},\;
%\deleterow{\jtmat{\lambda/\mu}}{l} = \jtmat{\lambda/\deletepart{\mu}{l}}\;
%\text{ and }
%\deleterowcols{\jtmat{\lambda/\mu}}{\setof k}{\setof l} = \jtmat{\deletepart{\lambda}{l}/\deletepart{\mu}{k}}.
%$$
\begin{equation}
\label{eq:minorspartitions}
\deleterowcols{\jtmat{\lambda/\mu}}{\setof{i_1,\dots,i_k}}{\setof{j_1,\dots,j_l}} = \jtmat{\deletepart{\lambda}{i_1,\dots,i_k}/\deletepart{\mu}{j_1,\dots,j_l}}.
\end{equation}
%Likewise, after deleting the column with index $l$ in $A_{\lambda/\mu}$, in the resulting
%matrix $\deletecol{A_{\lambda/\mu}}{l}$, the entries for all columns $j<k$ are the same as 
%in $A_{\lambda/\mu}$, while the entry in column $j$, $l\leq j\leq n-1$ and row $i$ now is
%$$\lambda_{i}-\pas{\mu_{j+1}-1}-i+j.$$
%This amounts to the following:
%$$
%\deletecol{A_{\lambda/\mu}}{l} = A_{\lambda/\deletepart{\mu}{l}}.
%$$
%
So from \eqref{eq:Dodgson} we immediately deduce the following Schur function identity
\eqref{eq:DodgsonSchur}.
To provide a \EM{combinatorial} proof for \eqref{eq:DodgsonSchur} (in fact, a special case
% of this identity
which was considered in \cite{kirillov:1984}) was the
starting point for the work in \cite{fulmek:2001}.
\begin{cor}
\label{cor:Dodgson}
Let $\lambda=\pas{\lambda_1,\lambda_2,\dots,\lambda_m}$ be a partition of length $\length{\lambda}=m\geq2$.
Then we have the following Schur function identiy:
\begin{multline}
\label{eq:DodgsonSchur}
\schurf_\lambda
	\cdot
\schurf_{\pas{\lambda_2,\lambda_3,\dots,\lambda_{m-1}}} = \\
\schurf_{\pas{\lambda_2,\lambda_3,\dots,\lambda_{m}}}
	\cdot
\schurf_{\pas{\lambda_1,\lambda_2,\dots,\lambda_{m-1}}} -
\schurf_{\pas{\lambda_2-1,\lambda_3-1,\dots,\lambda_{m}-1}}
	\cdot
\schurf_{\pas{\lambda_1+1,\lambda_2+1,\dots,\lambda_{m-1}+1}}
\end{multline}
\end{cor}
\begin{proof}
Consider the matrix %skew Schur function
$\mata=\jtmat{\pas{\lambda+\constpart{2}{m}}/{\constpart{2}{m}}}$ (which is equal to
$\jtmat{\lambda}$). Then
% and observe that
%$\schurf_\lambda=\schurf_{\pas{\lambda+1}/\constpart{1}{m}}$ by \eqref{eq:JacobiTrudiSkew}.
%Then
\eqref{eq:Dodgson} translates to
\begin{multline*}
\pas{\schurf_{\pas{\lambda+\constpart{2}{m}}/\constpart{2}{m}}}
	\cdot
\pas{\schurf_{\deletepart{\addtoparts{\lambda}{2}}{1,m}/\deletepart{\constpart{2}{m}}{1,m}}}
	=\\
\pas{\schurf_{\deletepart{\addtoparts{\lambda}{2}}{1}/\deletepart{\constpart{2}{m}}{1}}}
	\cdot
\pas{\schurf_{\deletepart{\addtoparts{\lambda}{2}}{m}/\deletepart{\constpart{2}{m}}{m}}}
	-\\
\pas{\schurf_{\deletepart{\addtoparts{\lambda}{2}}{1}/\deletepart{\constpart{2}{m}}{m}}}
	\cdot
\pas{\schurf_{\deletepart{\addtoparts{\lambda}{2}}{m}/\deletepart{\constpart{2}{m}}{1}}},
\end{multline*}
by \eqref{eq:JacobiTrudiSkew} and \eqref{eq:minorspartitions}. By
\eqref{eq:SchurShiftInvariance}
%and the fact that $\deletepart{\constpart{1}{m}}{1}=\allconstpart{0}$,
this simplifies to
%$$
%\pas{\schurf_\lambda}
%	\cdot
%\pas{\schurf_{\deletepart{\pas{\lambda+\constpart{1}{m}}}{1,m}}}
%	=
%\pas{\schurf_{\deletepart{\pas{\lambda+\constpart{1}{m}}}{1}}}
%	\cdot
%\pas{\schurf_{\deletepart{\pas{\lambda}}{m}}}
%	-
%\pas{\schurf_{\deletepart{\pas{\lambda}}{1}}}
%	\cdot
%\pas{\schurf_{\deletepart{\pas{\lambda+\constpart{1}{m}}}{m}}},
%$$
%which is equivalent to 
\eqref{eq:DodgsonSchur}.
\end {proof}

So it was easy to prove the \EM{Schur function identity} \eqref{eq:DodgsonSchur} by
recognizing it
as a special case of the general \EM{determinantal identity} \eqref{eq:Dodgson}. But this
also works the other way round:

\begin{obs}[Schur function identities imply equivalent determinantal identities]
\label{obs:genericSchurIdentity}
% Consider some fixed set of independent variables $\setof{x_1,x_2,\dots,x_\n}$
Assume that we have an identity $\identity{S}$ involving %$m$ 
skew Schur functions.
% $\schurf_{\lambda^{{1}}/\mu^{1}}, \schurf_{\lambda^{2}/\mu^{2}}, \dots$.

Then by \eqref{eq:JacobiTrudiSkew} and Proposition~\ref{pro:hsym-alg-ind},
%the skew Schur function identity
$\identity{S}$ translates to a % general
determinantal identity $\identity{D}$ which is
\EM{equivalent} to $\identity{S}$ %(i.e.: $\identity{S}\Leftrightarrow \identity{D}$)
as follows:
\bit
\item Rewrite each skew Schur function in $\identity{S}$ as a determinant of complete homogeneous
	symmetric functions, according to \eqref{eq:JacobiTrudiSkew},
\item and then replace each entry $\csymf_{r}$ by variable 
$y_{r}$, where $\pas{y_r}_{r=0}^\infty$ is a set of independent variables,
according to Proposition~\ref{pro:hsym-alg-ind}.
\eit
\end{obs}

% Assume we had a direct proof for \eqref{eq:DodgsonSchur}:
We call (skew) Schur function identities which are valid for \EM{arbitrary} shapes
$\lambda/\mu$ \EM{generic} (skew) Schur function identities: Note that
\eqref{eq:DodgsonSchur} is a generic identity in this sense. All identities we
shall consider in the rest of this paper are generic.

In particular, we may apply \eqref{eq:DodgsonSchur} to $\lambda=\pas{\lambda_1,\dots,\lambda_m}$,
where $\lambda_j=m\cdot\pas{m-j+1}$. It is easy to see that for this choice of $\lambda$,
all the entries
in $\pas{\csymf_{\lambda_j-j+i}}_{i,j=1}^{m}$ are \EM{distinct}, whence we may
replace them by \EM{independent} variables (by Proposition~\ref{pro:hsym-alg-ind}).
This means that \eqref{eq:DodgsonSchur} implies the \EM{general} determinantal
identity \eqref{eq:Dodgson}:
The identities are, in fact, \EM{equivalent}.

It is clear, that this phenomenon will apply also to other
determinantal identities: In this paper, we shall present \EM{generic}
Schur function identities which are equivalent to classical determinantal identities,
and give combinatorial proofs for these Schur function identities.

%, and in this sense
%we might formulate: ``Schur functions have a definition as certain determinants.
%Conversely, generic determinants (in a certain sense) \EM{are} Schur functions.''
%
%%Fr diese Konstruktionen knnen die Skew Shapes in einem gewissen Rahmen beliebig gewhlt
%%werden: Wenn umgekehrt generische Schur Identitten gezeigt werden knnen, fr die man 
%%die Shapes so whlen kann, da§ die Eintrge in jeder Determinante verschieden sind, dann
%%ergibt sich daraus wieder eine generische Determinantenidentitt!
%Clearly, a determinantal identity like
%$
%\detof{\begin{matrix}1 & 1\\2 & 1\end{matrix}}=-1
%$
%is hard to interpret as a Schur function identity.
%If some determinantal identity does not rely on any special relation between its
%entries, we call it a \EM{generic} identity. A generic determinantal identity always
%leads to a Schur function identity, for appropriate choices of the indexing partitions
%$\lambda$. For example,
%consider the entry $\pas{i,j}$ in $\jtmat{\lambda/\mu}$ together with its neighbours to
%the right and below:
%\begin{equation*}
%\begin{matrix}
%\csymf_{\lambda_j-\mu_i-j+i} & \csymf_{\lambda_{j+1}-\mu_{i}-j+i-1} \\
%\csymf_{\lambda_{j}-\mu_{i+1}-j+i+1} & \dots
%\end{matrix}
%\end{equation*}
%Since $\lambda_{j+1}-\mu_i-j+i-1 < \lambda_j-\mu_i-j+i$, it becomes clear: If the determinantal identity $I$ relies on the \EM{equality} of entry $i,j$ and entry
%$i,j+1$ in some matrix involved, then $I$ cannot be interpreted as a Schur function identity,
%while it is possible to have equality of the $i,j$\bindestrich entry and the $i+1,j$\bindestrich entry if $\mu_{i+1}=\mu_i-1$.
%
%
%Is the converse true, i.e., is there a Schur function identity for every determinantal identity?
%% This cannot be true in general, since there are determinantal relations 
%With appropriate constraints, the answer is yes: If the determinants involved in $J$ are
%minors of some fixed matrix, where all entries are distinct, independent variables, then
%there is a corresponding Schur function identity $J$!
%We shall illustrate this phenomenon by the following well\bindestrich known identity (see \dots).
%
%
%\begin{obs}
%\label{obs:determinantsareschurfunctions}
%Consider some (skew) Schur function identity $\identity{S}$. Writing each (skew) Schur
%function as a determinant (according to \eqref{eq:JacobiTrudiSkew}) and replacing each
%entry $\csymf_k$ in the corresponding determinants by the (independent) variable $y_k$
%clearly yields a determinantal identity $\identity{D}$. In particular, if for all
%determinants $D$ involved in $\identity{S}$ all entries of $D$ are different, then
%the resulting identity $\identity{D}$ is a \EM{generic} determinant identity.
%%
%% by 
%%Every (skew) Schur function identity $\identity{S}$ translates to a determinantal
%%identity $\identity{D}$, by simply replacing each entry $\csymf_k$ in (the determinants
%%involved in) $\identity{S}$ by skew Schur functions.
%
%On the other hand,
%consider some generic determinant identity $\identity{D}$: If for all
%determinants $d$ involved in $\identity{D}$, it is possible to replace the independent
%variable entries $d_{i,j}$ by complete homogeneous symmetric functions $\csymf_{\lambda_j-\mu_i-i+j}$ (for appropriate
%choices of partitions $\lambda$ and $\mu$) specializes all the determinants involved in
%$\identity{D}$ to (skew) Schur functions,
%then the result of such ``translation'' clearly is a (skew) Schur function identity
%$\identity{S}$.
%\end{obs}
