Hecke algebras with independent parameters

Abstract

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.