An element of a Coxeter group @math 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 particular in the finite case. They index naturally a basis of the generalized Temperley--Lieb algebra. In this work we deal with any finite or affine Coxeter group @math , and we give explicit descriptions of fully commutative elements. Using our characterizations we then enumerate these elements according to their Coxeter length, and find in particular that the corrresponding growth sequence is ultimately periodic in each type. When the sequence is infinite, this implies that the associated Temperley--Lieb algebra has linear growth.
The second article @cite_24 of Hagiwara deals with . In general, minuscule heaps are a strict subset of FC heaps, but here they coincide, as the author shows in his Theorem 5.1. His characterization goes by embedding posets in a family of slanted lattices @math , and Hagiwara proves that FC heaps are precisely the finite convex subsets occurring in a lattice @math . It can be easily seen that this is a corollary of our work; the gradient @math defined in [p.17] HagiwaraAtilde can be seen in the path @math as the sum of the number of up steps and the number of horizontal @math steps.
@cite_24
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A minuscule heap is a partially ordered set, together with a labeling of its ele- ments by the nodes of a Dynkin diagram, satisfying certain conditions derived by J. Stembridge. This paper classifies the minuscule heaps over the Dynkin diagram of type ˜ A.