On representation theory of partition algebras for complex reflection groups

Abstract

This paper defines the partition algebra for complex reflection group $G(r,p,n)$ acting on $k$-fold tensor product $({\mathbb{C}}^n{)}^{\otimes k}$, where ${\mathbb{C}}^n$ is the reflection representation of $G(r,p,n)$. A basis of the centralizer algebra of this action of $G(r,p,n)$ was given by Tanabe and for $p=1$, the corresponding partition algebra was studied by Orellana. We also establish a subalgebra as partition algebra of a subgroup of $G(r,p,n)$ acting on $({\mathbb{C}}^n{)}^{\otimes k}$. We call these algebras as Tanabe algebras. The aim of this paper is to study representation theory of Tanabe algebras: parametrization of their irreducible modules, and construction of Bratteli diagram for the tower of Tanabe algebras. We conclude the paper by giving Jucys-Murphy elements of Tanabe algebras and their actions on the Gelfand-Tsetlin basis, determined by this multiplicity free tower, of irreducible modules.

51 pages