On the Cyclically Fully Commutative Elements of Coxeter Groups

TL;DR

This paper studies cyclically fully commutative elements in Coxeter groups, providing combinatorial characterizations, enumeration, and conditions for their powers to remain fully commutative, with implications for affine and simply-laced groups.

Contribution

It introduces a combinatorial framework for CFC elements, characterizes when their powers stay fully commutative, and enumerates these elements across all Coxeter groups.

Findings

CFC elements can be described by acyclic directed graphs.

Conditions for CFC elements to be logarithmic are identified.

Enumeration formulas for CFC elements in all Coxeter groups.

Abstract

Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is "cyclically fully commutative" (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink conversions. In this paper, we explore the combinatorics of the CFC elements and enumerate them in all Coxeter groups. Additionally, we characterize precisely which CFC elements have the property that powers of them remain fully commutative, via the presence of a simple combinatorial feature called a "band." This allows us to give necessary and sufficient conditions for a CFC element w to be "logarithmic," that is, l(w^k) = k l(w) for all k > 0, for a large class of Coxeter groups that includes all affine Weyl groups and simply-laced Coxeter groups. Finally, we give a simple non-CFC element that fails to be logarithmic under these conditions.