Stanley symmetric functions for signed involutions

TL;DR

This paper extends the combinatorial framework of Stanley symmetric functions to signed involutions in Coxeter groups, establishing bijections with tableaux and reduced words, thereby generalizing known results from symmetric groups.

Contribution

It introduces a novel adaptation of Stanley symmetric functions for signed involutions, connecting involution words with tableaux in the context of Coxeter groups of type C.

Findings

Involution words for the longest element in type C are bijective with those in type A.

These involution words correspond to standard tableaux of a specific shape.

The approach generalizes symmetric function techniques to signed permutations.

Abstract

An involution in a Coxeter group has an associated set of involution words, a variation on reduced words. These words are saturated chains in a partial order first considered by Richardson and Springer in their study of symmetric varieties. In the symmetric group, involution words can be enumerated in terms of tableaux using appropriate analogues of the symmetric functions introduced by Stanley to accomplish the same task for reduced words. We adapt this approach to the group of signed permutations. We show that involution words for the longest element in the Coxeter group $C_n$ are in bijection with reduced words for the longest element in $A_n = S_{n+1}$, which are known to be in bijection with standard tableaux of shape $(n, n-1, \ldots, 2, 1)$.