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. (2012). 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 cylindrical closure. In finite types, this refines the work of Boothby et al. (2012), 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.