In \cite{fulmek:2001}, a combinatorial proof was given for two Schur function
identities, which were presented in \cite{kirillov:1984} and in \cite{kleber:2001}.
This combinatorial proof was shown to apply to a \EM{class} of Schur function identities
\cite[Lemma~16]{fulmek:2001}, and was used to prove bijectively Dodgson's condensation
formula and the Pl\"ucker relations %  for examples,
but was not paid further attention.
Recently, members of this class of Schur function identities received some interest
\cite{gurevichPyatovSaponov:2009}. The close connection of the result \cite[(3.3)]{gurevichPyatovSaponov:2009} to the work in \cite{fulmek:2001}
was already explained \EM{ad hoc} in \cite{fulmek:2009},
but  we take this opportunity to make obvious the much wider range of applicability of
this idea, which amounts to the following statement:

{\it
%\begin{obs}
$m\times m$\bindestrich determinants may be \EM{viewed} as (generating functions of)
$m$\bindestrich tuples of \nilp s in $\Z^2$ with starting points on some fixed horizontal
line $y=\alpha$ and ending points on some fixed horizontal line $y=\omega>\alpha$,
where
\bit
\item the \EM{rows} of the determinants (in the usual order: top to bottom)
	correspond to the \EM{starting points} of the lattice paths (ordered from
	right to left),
\item and the \EM{columns} of the determinants (in the usual order: left to right)
	correspond to the ending points of the lattice paths (ordered from
	right to left).
\eit
%\end{obs}
}

This statement, of course, has to be explained properly, and we shall give several
concrete examples to illustrate its meaning. The combinatorial constructions
are best conceived by pictures, so we give a lot of illustrations.

This paper is organized as follows: 

In Section~\ref{sec:background}, we present basic background information regarding
symmetric functions, partitions, Young tableaux and (skew) Schur functions.

In Section~\ref{sec:main}, we present the \lgti,
%\bindestrich interpretation of semi standard
%Young tableaux as \nilp s,
which (in some sense) exposes determinants as generating functions of
\nilp s. We
illustrate this view by giving a ``graphical proof''
of the Cauchy\bindestrich Binet formula. (I am grateful to the
anonymous referee for drawing my attention to the paper \cite{Aigner:2001},
which gives a similar proof in an easily comprehensible exposition.)

In Section~\ref{sec:overlays}, we present the central bijective construction
(recolouring of bicoloured trails in the overlays of families
of \nilp s, which correspond to some product of skew
Schur functions) and indicate how this construction gives rise to Schur function
identities.

In Section~\ref{sec:apps}, we present several examples: We prove 
a generalization of Dodgson's condensation rule which might be new (Theorem~\ref{thm:maybenew})
and give
``graphical proofs'' for the Pl\"ucker relations (and its generalization
\cite[(3.3)]{gurevichPyatovSaponov:2009}), for (a generalization of)
Laplace's expansion, and for Turnbull's identity.

 


