Characterization of cyclically fully commutative elements in finite and affine Coxeter groups

An element of a Coxeter group is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. An element of a Coxeter group is cyclically fully commutative if any of its cyclic shifts remains fully commutative. These elements were studied by Boothby et al. In particular the authors precisely identified the Coxeter groups having a finite number of cyclically fully commutative elements and enumerated them. In this work we characterize and enumerate those elements according to their Coxeter length in all finite and all affine Coxeter groups by using an operation on heaps, the cylindric closure. In finite types, this refines the work of Boothby et al., by adding a new parameter. In affine type, all the results are new. In particular, we prove that there is a finite number of cyclically fully commutative logarithmic elements in all affine Coxeter groups. We also study the cyclically fully commutative involutions and prove that their number is finite in all Coxeter groups. Introduction Let W be a Coxeter group. An element w ∈ W is said to be fully commutative (FC) if two reduced words representing w can be transformed into each other only using commutation relations, that is relations of the form st = ts. These elements were introduced and studied independently by Fan in [5], Graham in [7] and Stembridge in [13, 14, 15]. In particular, Stembridge classified the Coxeter groups with a finite number of fully commutative elements and enumerated them in each case. Fully commutative elements appear naturally in the context of (generalized) Temperley–Lieb algebras, as they index a linear basis of those objects. Recently, in [1], Biagioli, Jouhet and Nadeau refined Stembridge’s enumeration by counting fully commutative elements according to their Coxeter length in any finite or affine Coxeter group. In this paper, we focus on a certain subset of fully commutative elements, the cyclically fully commutative (CFC) elements. These are elements w for which every cyclic shift of any reduced expression of w is a reduced expression of some FC element (not necessarily the same as w). They were introduced by Boothby et al. in [3], where the authors classified the Coxeter groups with a finite number of CFC elements (they showed that they are exactly the groups with a finite number of FC elements) and enumerated them. The main goal of [3] was to establish necessary and sufficient conditions for a CFC element w ∈ W to be logarithmic, that is to satisfy `(w) = k`(w) for any positive integer k. This is the first step towards