On the length of fully commutative elements

TL;DR

This paper studies the enumeration of fully commutative elements in Coxeter groups, showing their counts follow linear recurrences and classifying groups with ultimately periodic sequences, with implications for Temperley--Lieb algebras.

Contribution

It proves that the sequence counting fully commutative elements satisfies a linear recurrence and classifies groups with ultimately periodic counts, extending previous results.

Findings

Sequences satisfy linear recurrence with constant coefficients.

Classification of Coxeter groups with ultimately periodic sequences.

Applications to growth of Temperley--Lieb algebras.

Abstract

In a Coxeter group $W$, an element is fully commutative if any two of its reduced expressions can be linked by a series of commutation of adjacent letters. These elements have particularly nice combinatorial properties, and also index a basis of the generalized Temperley--Lieb algebra attached to $W$. We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that it always satisfies a linear recurrence with constant coefficients, by showing that reduced expressions of fully commutative elements form a regular language. Then we classify those groups $W$ for which the sequence is ultimately periodic, extending a result of Stembridge. These results are applied to the growth of generalized Temperley--Lieb algebras.