Affine transitions for involution Stanley symmetric functions

TL;DR

This paper introduces affine involution Stanley symmetric functions, proves a transition formula for them, and constructs an affine involution analogue of the Lascoux-Schützenberger tree, advancing combinatorial understanding of affine symmetric functions.

Contribution

It establishes a transition formula for affine involution Stanley symmetric functions and develops an affine involution analogue of the Lascoux-Schützenberger tree, extending prior combinatorial frameworks.

Findings

Proved a transition formula for affine involution Stanley symmetric functions.

Constructed an affine involution analogue of the Lascoux-Schützenberger tree.

Utilized properties of the strong Bruhat order on affine permutations.

Abstract

We study a family of symmetric functions $\hat F_z$ indexed by involutions $z$ in the affine symmetric group. These power series are analogues of Lam's affine Stanley symmetric functions and generalizations of the involution Stanley symmetric functions introduced by Hamaker, Pawlowski, and the first author. Our main result is to prove a transition formula for $\hat F_z$ which can be used to define an affine involution analogue of the Lascoux-Sch\"utzenberger tree. Our proof of this formula relies on Lam and Shimozono's transition formula for affine Stanley symmetric functions and some new technical properties of the strong Bruhat order on affine permutations.