On the length of fully commutative elements

Abstract

In a Coxeter group W W , an element is fully commutative if any two of its reduced expressions can be linked by a series of commutations of adjacent letters. These elements have particularly nice combinatorial properties, and index a basis of the generalized Temperley–Lieb algebra attached to W W . We give two results about the sequence counting these elements with respect to their Coxeter length. First we prove that this sequence 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 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.