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.
In the recent work @cite_35 by Al Harbat, the author classifies FC elements by indicating a normal form for each of them. That is, the main theorem exhibits a family of reduced FC expressions where each FC element is represented exactly once. We will not detail this here, but these normal forms correspond to a particular linear extension of FC heaps which is fairly easy to describe.
@cite_35
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We classify fully commutative elements in the affine Coxeter group of type @math . We give a normal form for such elements, then we propose an application of this normal form: we lift these fully commutative elements to the affine braid group of type @math and we get a form for "fully commutative braids".