Cellularity of Wreath Product Algebras and $A$--Brauer algebras

TL;DR

This paper proves that the property of being cyclic cellular is preserved under wreath products with symmetric groups and introduces $A$--Brauer algebras, showing they are also cyclic cellular if $A$ is.

Contribution

It establishes that wreath products and $A$--Brauer algebras inherit cyclic cellularity from the algebra $A$, expanding the class of known cyclic cellular algebras.

Findings

Wreath product of a cyclic cellular algebra with symmetric groups is cyclic cellular.

$A$--Brauer algebras are cyclic cellular if $A$ is cyclic cellular.

Includes non-abelian group $G$--Brauer algebras as special cases.

Abstract

A cellular algebra is called cyclic cellular if all cell modules are cyclic. Most important examples of cellular algebras appearing in representation theory are in fact cyclic cellular. We prove that if $A$ is a cyclic cellular algebra, then the wreath product of $A$ with the symmetric group on $n$ letters is also cyclic cellular. We also introduce $A$--Brauer algebras, for algebras $A$ with an involution and trace. This class of algebras includes, in particular, $G$--Brauer algebras for non-abelian groups $G$. We prove that if $A$ is cyclic cellular then the $A$--Brauer algebras $D_n(A)$ are also cyclic cellular.