Crystal Structures for Double Stanley Symmetric Functions

TL;DR

This paper introduces a new combinatorial framework connecting type A and C Stanley symmetric functions through a double symmetric function, establishing a bicrystal structure and exploring algebraic relationships with conjectures for generalization.

Contribution

It defines a double Stanley symmetric function linking type A and C functions, and induces a bicrystal structure on the combinatorial objects involved.

Findings

Established a combinatorial link between type A and C Stanley symmetric functions.

Induced a bicrystal structure on the underlying combinatorial objects.

Proposed conjectures for extending the framework to type C.

Abstract

We relate the combinatorial definitions of the type $A_n$ and type $C_n$ Stanley symmetric functions, via a combinatorially defined "double Stanley symmetric function," which gives the type $A$ case at $(\mathbf{x},\mathbf{0})$ and gives the type $C$ case at $(\mathbf{x},\mathbf{x})$. We induce a type $A$ bicrystal structure on the underlying combinatorial objects of this function which has previously been done in the type $A$ and type $C$ cases. Next we prove a few statements about the algebraic relationship of these three Stanley symmetric functions. We conclude with some conjectures about what happens when we generalize our constructions to type $C$.