** Abstract: **We
consider
a flagged
form of the
Cauchy determinant,
for which
we provide
a combinatorial
interpretation
in terms
of nonintersecting
lattice paths.
In combination
with the
standard
determinant
for the enumeration
of nonintersecting
lattice paths,
we are able
to give a
new proof
of the Cauchy
identity
for Schur
functions.
Moreover,
by choosing
different
starting
and end points
for the lattice
paths, we are
led to a
lattice path
proof of
an identity
of Gessel
which expresses
a Cauchy-like
sum of Schur
functions
in terms
of the complete
symmetric
functions.
**AMS Classification: **05E05, 05A15.
** Keywords: ** Divided
difference,
Cauchy
theorem,
flagged
Cauchy
determinant,
multi-Schur
function,
lattice
paths.
** Download: **ps |