Hecke algebras with independent parameters.

Huang, Jia

Journal of Algebraic Combinatorics, Tome 43 (2016) no. 3, pp. 521-551.

Résumé

We study the Hecke algebra $\mathcal H(\mathbf q)$ over an arbitrary field $\mathbb F$ of a Coxeter system $(W,S)$ with independent parameters $\mathbf q=(q_s\in\mathbb F:s\in S)$ for all generators. This algebra always has a spanning set indexed by the Coxeter group $W$, which is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receives the same parameter. In general, the dimension of $\mathcal H(\mathbf q)$ could be as small as 1. We construct a basis for $\mathcal H(\mathbf q)$ when $(W,S)$ is simply laced. We also characterize when $\mathcal H(\mathbf q)$ is commutative, which happens only if the Coxeter diagram of $(W,S)$ is simply laced and bipartite. In particular, for type $A$, we obtain a tower of semisimple commutative algebras whose dimensions are the Fibonacci numbers. We show that the representation theory of these algebras has some features in analogy/connection with the representation theory of the symmetric groups and the 0-Hecke algebras.

Détail

Classification : 20C08, 20F55, 05E10
Keywords: Hecke algebras, independent parameters, Coxeter systems, simply laced Coxeter diagrams, Fibonacci numbers, independent sets, Grothendieck groups