Algebras associated with Pseudo Reflection Groups: A Generalization of Brauer Algebras

TL;DR

This paper introduces a new algebra associated with pseudo reflection groups, generalizing Brauer algebras, and explores its structure, semisimplicity, and representations, including connections to Cherednik and Lawrence-Krammer theories.

Contribution

It generalizes Brauer algebras to all pseudo reflection groups and establishes their cellular structure, semisimplicity, and connections to complex braid group representations.

Findings

B_G(Υ) is isomorphic to generalized Brauer algebra for simply-laced Coxeter groups

B_G(Υ) has a cellular structure and is semisimple for rank 2 Coxeter groups with generic parameters

Introduces Cherednik type connection and Lawrence-Krammer generalization for complex braid groups

Abstract

We present a way to associate an algebra $B_G (\Upsilon) $ with every pseudo reflection group $G$. When $G$ is a Coxeter group of simply-laced type we show $B_G (\Upsilon)$ is isomorphic to the generalized Brauer algebra of simply-laced type introduced by Cohen,Gijsbers and Wales[10]. We prove $B_G (\Upsilon)$ has a cellular structure and be semisimple for generic parameters when $G$ is a rank 2 Coxeter group. In the process of construction we introduce a Cherednik type connection for BMW algebras and a generalization of Lawrence-Krammer representation to complex braid groups associated with all pseudo reflection groups.