Positivity of cylindric skew Schur functions

Abstract

Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the well-known problem of finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannians. In this paper, we prove cylindric Schur positivity of cylindric skew Schur functions, as conjectured by McNamara. We also show that all the coefficients appearing in the expansion are the same as the 3-point Gromov-Witten invariants. We start by discussing the properties of affine Stanley symmetric functions for general affine permutations and 321-avoiding affine permutations, and we explain how these functions are related to cylindric skew Schur functions. In addition, we provide an effective algorithm to compute the expansion of cylindric skew Schur functions in terms of cylindric Schur functions, as well as the expansion of affine Stanley symmetric functions in terms of affine Schur functions.