Summary: STAIRCASE SKEW SCHUR FUNCTIONS
ARE SCHUR P-POSITIVE
FEDERICO ARDILA AND LUIS G. SERRANO
Abstract. We prove Stanley's conjecture that, if n is the staircase shape,
then the skew Schur functions sn/µ are non-negative sums of Schur P-functions.
We prove that the coefficients in this sum count certain fillings of shifted
shapes. In particular, for the skew Schur function sn/n-2
, we discuss con-
nections with Eulerian numbers and alternating permutations.
The Schur functions s, indexed by partitions , form a basis for the ring of
symmetric functions. These are very important objects in algebraic combinatorics.
They play a fundamental role in the study of the representations of the symmetric
group and the general linear group, and the cohomology ring of the Grassman-
nian . The Schur P-functions P, indexed by strict partitions, form a basis
for an important subring of . They are crucial in the study of the projective
representations of the symmetric group, and the cohomology ring of the isotropic
Grassmannian  .
The goal of this paper is to prove the following conjecture of Richard Stanley
: If n is the staircase shape and µ n, then the staircase skew Schur function