The Representations of the Brauer-Chen Algebra

TL;DR

This paper analyzes the structure and representation theory of the Brauer-Chen algebra associated with complex reflection groups, establishing semisimplicity, constructing simple modules, and computing dimensions for various groups.

Contribution

It provides a comprehensive study of the Brauer-Chen algebra, including semisimplicity proof, uniform simple module construction, and explicit dimension formulas for all complex reflection groups.

Findings

Proves the Brauer-Chen algebra is semisimple.

Constructs simple modules uniformly for all complex reflection groups.

Computes explicit dimensions for infinite series and exceptional groups.

Abstract

In this paper, we determine the structure and representation theory of the Brauer algebra associated to a complex reflection group (here called the Brauer-Chen algebra), defined by Chen in 2011. We prove that it is semisimple and provide a construction for its simple modules for generic values of the parameters, in a uniform way for all complex reflection groups. We then apply these results to the cases of all irreducible complex reflection groups: for the groups in the infinite series, we obtain a numerical formula for the dimension of the corresponding Brauer algebra, and for all exceptional complex reflection groups, we compute the dimension of the corresponding Brauer algebra explicitly, using computational methods. We also obtain a uniformly defined basis for the Brauer algebra of any complex reflection group, defined over a field. Finally, we determine for which complex reflection groups the corresponding Brauer algebra is a free module over its ring of definition.