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An element of a Coxeter group W is fully commutative if any two of itsreduced decompositions are related by a series of transpositions of adjacentcommuting generators. An element of a Coxeter group W is cyclically fullycommutative if any of its cyclic shifts remains fully commutative. Theseelements were studied in Boothby et al.. In particular the authors enumeratedcyclically fully commutative elements in all Coxeter groups having a finitenumber of them. In this work we characterize and enumerate cyclically fullycommutative elements according to their Coxeter length in all finite or affineCoxeter groups by using a new operation on heaps, the cylindric transformation.In finite types, this refines the work of Boothby et al., by adding a newparameter. In affine type, all the results are new. In particular, we provethat there is a finite number of cyclically fully commutative logarithmicelements in all affine Coxeter groups. We study afterwards the cyclically fullycommutative involutions and prove that their number is finite in all Coxetergroups.

Characterization of Cyclically Fully commutative elements in finite and affine Coxeter Groups