# Positivity of Cylindric skew Schur functions

## TL;DR

This paper proves the positivity of cylindric skew Schur functions, linking their coefficients to Gromov-Witten invariants, and provides algorithms for their expansion in terms of simpler functions.

## Contribution

It establishes cylindric Schur positivity for cylindric skew Schur functions and connects coefficients to Gromov-Witten invariants, also offering computational algorithms.

## Key findings

- Proved cylindric Schur positivity of cylindric skew Schur functions.
- Coefficients match 3-point Gromov-Witten invariants.
- Developed algorithms for function expansions.

## Abstract

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