# Crystal analysis of type $C$ Stanley symmetric functions

## TL;DR

This paper provides a crystal theoretic explanation for the nonnegative integer expansion of type C Stanley symmetric functions in terms of Schur functions, offering explicit combinatorial descriptions of the coefficients.

## Contribution

It introduces a crystal theoretic framework to explain and compute the Schur expansion coefficients of type C Stanley symmetric functions.

## Key findings

- Crystal theory explains nonnegative Schur expansion coefficients.
- Explicit combinatorial description of coefficients in terms of highest weight crystal elements.
- Bridges the gap between algebraic combinatorics and crystal bases for type C functions.

## Abstract

Combining results of T.K. Lam and J. Stembridge, the type $C$ Stanley symmetric function $F_w^C(\mathbf{x})$, indexed by an element $w$ in the type $C$ Coxeter group, has a nonnegative integer expansion in terms of Schur functions. We provide a crystal theoretic explanation of this fact and give an explicit combinatorial description of the coefficients in the Schur expansion in terms of highest weight crystal elements.