The enumeration of fully commutative elements of Coxeter groups
An element \(w\) of a Coxeter group is said to be fully commutative, if any reduced word for \(w\) can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree \(n\), the fully commutative elements are the \(321\)-avoiding permutations. There are several different motivations for studying fully commutative elements of Coxeter groups. The author's motivation comes from his study of symmetric functions associated with the Weyl groups of type \(B\) and \(D\) [Trans. Am. Math. Soc. 349, No. 4, 1285-1332 (1997)]. A different motivation comes from the fact that fully commutative elements index a basis in a natural quotient of the Iwahori--Hecke algebra associated to a Coxeter group, as shown by \textit{J. Graham} [``Modular representations of Hecke algebras and related algebras'', Ph.D. Thesis, University of Sidney (1995)] (and \textit{C. K. Fan} [``A Hecke algebra quotient and properties of commutative elements of a Weyl group'', Ph.D. Thesis, M.I.T., 1995] in the simply-laced case). J. Graham, and independently the author [J. Algebr. Comb. 5, No. 4, 353-385 (1996; Zbl 0864.20025)] (and C. K. Fan in the simply-laced case) classified all Coxeter groups for which there is a finite number of fully commutative elements. In this paper, these numbers are explicitly determined, by either exhibiting a closed formula or a generating function. All the generating functions appearing are algebraic. (C. K. Fan did these enumerations independently for the simply-laced case [J. Algebr. Comb. 5, No. 3, 175-189 (1996; Zbl 0853.20028)], using representation-theoretic methods.) Again motivated by work of C. K. Fan, the author also determines the number of fully commutative involutions in each case.
scientific article; zbMATH DE number 1154682
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