Structural properties of acyclic heaps of pieces with Kazhdan--Lusztig theory

TL;DR

This paper studies the structural properties of heaps of pieces related to Coxeter groups, proving key equivalences and establishing Property W for star reducible Coxeter groups, correcting previous errors.

Contribution

It introduces new notions in heap theory and proves that boundary vertices are linearly equivalent to effective boundary vertices in certain Coxeter group heaps, establishing Property W.

Findings

Boundary vertices are linearly equivalent to effective boundary vertices.

Property W is established for star reducible Coxeter groups.

Corrects a previous mistake in the literature.

Abstract

We introduce the notions of boundary vertex, linear equivalence and effective boundary vertex in the context of Viennot's heaps of pieces. We prove that in the heap of a fully commutative element in a star reducible Coxeter group, every boundary vertex is linearly equivalent to an effective boundary vertex. Using this result, we establish Property W (in the sense of math.QA/0509363) for star reducible Coxeter groups; this corrects a mistake in the latter paper.