Word posets, with applications to Coxeter groups

TL;DR

This paper explores the structure of word posets related to Coxeter groups, demonstrating their computational complexity and providing new bounds on related combinatorial objects.

Contribution

It introduces the theory of word posets for Coxeter groups and proves the #P-completeness of counting reduced words and commutation classes, offering new bounds and formulas.

Findings

Counting words in commutation classes is #P-complete.

Enumerating reduced words in Coxeter groups is #P-complete.

New recursive formulas and bounds for commutation classes and sorting networks.

Abstract

We discuss the theory of certain partially ordered sets that capture the structure of commutation classes of words in monoids. As a first application, it follows readily that counting words in commutation classes is #P-complete. We then apply the partially ordered sets to Coxeter groups. Some results are a proof that enumerating the reduced words of elements of Coxeter groups is #P-complete, a recursive formula for computing the number of commutation classes of reduced words, as well as stronger bounds on the maximum number of commutation classes than were previously known. This also allows us to improve the known bounds on the number of primitive sorting networks.