Centralizers of Iwahori-Hecke algebras (Q2719045)

In a previous work [J. Algebra 221, No. 1, 1-28 (1999; Zbl 0940.20011)], the author established the existence of a ``minimal'' basis for the centre of an Iwahori-Hecke algebra of a finite Weyl group. A minimal basis element specialises, when \(q=1\), to a sum of conjugacy class elements; furthermore, apart from elements of this conjugacy class, the support of a minimal basis element in the Iwahori-Hecke algebra contains no shortest elements of any conjugacy class.NEWLINENEWLINENEWLINEThe original definition of the minimal basis uses character theoretical results and the isomorphism over \(\mathbb{Q}(q^{1/2})\) between the Iwahori-Hecke algebra and the group algebra of the corresponding Weyl group. The contribution of this paper is an elementary construction of the minimal basis using only combinatorics and linear algebra. As well as being more elementary, the methods of the current paper are considerably easier to generalize.NEWLINENEWLINENEWLINETheorem 2.6 is a generalization of a result of \textit{M. Geck} and \textit{G. Pfeiffer} [Adv. Math. 102, No. 1, 79-94 (1993; Zbl 0816.20034)] on conjugacy classes in Coxeter groups. This leads to the main theorem, Theorem 4.6, which establishes the existence of and characterizes the minimal basis for a finite Coxeter group. The same result can also be used to construct a minimal basis for the centralizer of a parabolic subalgebra in a Hecke algebra of type \(A\) or \(B\), and the author has now shown that this approach generalizes to arbitrary finite Coxeter groups.