Fully commutative elements and lattice walks

Abstract

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. These elements were extensively studied by Stembridge in the finite case. In this work we deal with any finite or affine Coxeter group $W$, and we enumerate fully commutative elements according to their Coxeter length. Our approach consists in encoding these elements by various classes of lattice walks, and we then use recursive decompositions of these walks in order to obtain the desired generating functions. In type $A$, this reproves a theorem of Barcucci et al.; in type $\tilde{A}$, it simplifies and refines results of Hanusa and Jones. For all other finite and affine groups, our results are new.

Keywords

• fully commutative elements
• coxeter groups
• generating functions
• lattice walks
• heaps.
• [info.info-dm]computer science [cs]/discrete mathematics [cs.dm]