Toric heaps, cyclic reducibility, and conjugacy in Coxeter groups

Shih-Wei Chao, Matthew Macauley

Published 2019-12-18Version 1

As a visualization of Cartier and Foata's "partially commutative monoid" theory, G.X. Viennot introduced "heaps of pieces" in 1986. These are essentially labeled posets satisfying a few additional properties. They naturally arise as models of reduced words in Coxeter groups. In this paper, we introduce a cyclic version, motivated by the idea of taking a heap and wrapping it into a cylinder. We call this object a "toric heap", as we formalize it as a labeled toric poset, which is a cyclic version of an ordinary poset. To define the concept of a toric extension, we develop a morphism in the category of toric heaps. We study toric heaps in Coxeter theory, in view of the fact that a cyclic shift of a reduced word is simply a conjugate by an initial or terminal generator. This allows us to formalize and study a framework of "cyclic reducibility" in Coxeter theory, and apply it to model conjugacy. We introduce the notion of "torically reduced", which is stronger than being cyclically reduced for group elements. This gives rise to a new class of elements called "torically fully commutative" (TFC), which are those that have a unique cyclic commutativity class, and comprise a strictly bigger class than the "cyclically fully commutative" (CFC) elements. We prove several cyclic analogues of results on fully commutative (FC) elements due to Stembridge. We conclude with how this framework fits into recent work in Coxeter groups, and we correct a minor flaw in a few recently published theorems.