Extending a word property for twisted Coxeter systems

TL;DR

This paper extends the word property for twisted Coxeter systems, exploring new variants of reduced words with commutativity and length conditions, with connections to symmetric functions.

Contribution

It introduces two novel extensions of the word property for twisted involutions in Coxeter groups, broadening the understanding of their combinatorial structure.

Findings

Extended the set of words with commutativity properties

Relaxed minimal length conditions for certain words

Connected these extensions to Schur Q-functions and K-theoretic Schur P-functions

Abstract

We prove two extensions of Hansson and Hultman's word property for certain analogues of reduced words associated to twisted involutions in Coxeter groups. Our first extension concerns the superset of such words in which terms with a natural commutativity property may be optionally primed. Our other extension involves variants of these words in which a defining minimal length condition is relaxed. In type A the sets considered are closely related to generating functions for Schur Q-functions and K-theoretic Schur P-functions.