Quasiparabolic sets and Stanley symmetric functions for affine fixed-point-free involutions

TL;DR

This paper develops affine analogues of fixed-point-free involution Stanley symmetric functions using quasiparabolic sets, providing new formulas and conjectures in the context of affine symmetric groups.

Contribution

It introduces affine fixed-point-free involution Stanley symmetric functions and proves they form a quasiparabolic set, leading to a transition formula analogous to classical Schubert polynomial results.

Findings

Affine FPF involution Stanley symmetric functions are introduced.

The set of FPF-involutions forms a quasiparabolic set.

A transition formula for these functions is proved.

Abstract

We introduce and study affine analogues of the fixed-point-free (FPF) involution Stanley symmetric functions of Hamaker, Marberg, and Pawlowski. Our methods use the theory of quasiparabolic sets introduced by Rains and Vazirani, and we prove that the subset of FPF-involutions is a quasiparabolic set for the affine symmetric group under conjugation. Using properties of quasiparabolic sets, we prove a transition formula for the affine FPF involution Stanley symmetric functions, analogous to Lascoux and Sch\"utzenberger's transition formula for Schubert polynomials. Our results suggest several conjectures and open problems.