A cellular basis of the $q$-Brauer algebra related with Murphy bases of the Hecke algebras

TL;DR

This paper introduces a cellular basis for the $q$-Brauer algebra, connecting it with Murphy bases of Hecke algebras, and analyzes its module structure and semisimplicity conditions.

Contribution

It constructs a new cellular basis for the $q$-Brauer algebra that lifts Murphy bases, and characterizes simple modules and semisimplicity criteria.

Findings

Simple $q$-Brauer modules are indexed by $e(q^2)$-restricted partitions.

Provides a semisimplicity criterion for low-dimensional cases.

Shows the $q$-Brauer algebra is generally not isomorphic to the BMW-algebra.

Abstract

A new basis of the $q$-Brauer algebra is introduced, which is a lift of Murphy bases of Hecke algebras of symmetric groups. This basis is a cellular basis in the sense of Graham and Lehrer. Subsequently, using combinatorial language we prove that the non-isomorphic simple $q$-Brauer modules are indexed by the $e(q^2)$-restricted partitions of $n-2k$ where $k$ is an integer, $0 \le k \le [n/2]$. When the $q$-Brauer algebra has low-dimension a criterion of semisimplicity is given, which is used to show that the $q$-Brauer algebra is in general not isomorphic to the BMW-algebra.}