Cellular structure of $q$-Brauer algebras

TL;DR

This paper establishes that $q$-Brauer algebras are cellular and describes their structure as iterated inflations of Hecke algebras, also analyzing their quasi-heredity and irreducible representations over fields.

Contribution

It constructs a new basis for $q$-Brauer algebras, proves their cellularity, and characterizes their quasi-hereditary properties over various fields.

Findings

$q$-Brauer algebras are cellular with a new basis.

They are iterated inflations of Hecke algebras of type A.

Conditions for quasi-heredity over fields are determined.

Abstract

In this paper we consider the $q$-Brauer algebra over $R$ a commutative noetherian domain. We first construct a new basis for $q$-Brauer algebras, and we then prove that it is a cell basis, and thus these algebras are cellular in the sense of Graham and Lehrer. In particular, they are shown to be an iterated inflation of Hecke algebras of type $A_{n-1}.$ Moreover, when $R$ is a field of arbitrary characteristic, we determine for which parameters the $q$-Brauer algebras are quasi-heredity. So the general theory of cellular algebras and quasi-hereditary algebras applies to $q$-Brauer algebras. As a consequence, we can determine all irreducible representations of $q$-Brauer algebras by linear algebra methods.