Hecke algebras with independent parameters

TL;DR

This paper investigates Hecke algebras with independent parameters over arbitrary fields, characterizing their bases, dimensions, and commutativity, and exploring their representation theory with connections to symmetric groups and Fibonacci numbers.

Contribution

It provides a basis construction for simply laced Coxeter systems, characterizes when the algebra is commutative, and links the representation theory to known algebraic structures.

Findings

Basis constructed for simply laced Coxeter systems

Characterization of when the algebra is commutative

Type A algebras form a Fibonacci-dimensional tower

Abstract

We study the Hecke algebra $\H(\bq)$ over an arbitrary field $\FF$ of a Coxeter system $(W,S)$ with independent parameters $\bq=(q_s\in\FF:s\in S)$ for all generators. This algebra is always linearly spanned by elements indexed by the Coxeter group $W$. This spanning set is indeed a basis if and only if every pair of generators joined by an odd edge in the Coxeter diagram receive the same parameter. In general, the dimension of $\H(\bq)$ could be as small as $1$. We construct a basis for $\H(\bq)$ when $(W,S)$ is simply laced. We also characterize when $\H(\bq)$ 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.