# On representation theory of partition algebras for complex reflection groups

## TL;DR

This paper introduces Tanabe algebras, a class of partition algebras associated with complex reflection groups, and studies their representation theory, including irreducible modules, Bratteli diagrams, and Jucys-Murphy elements.

## Contribution

It defines Tanabe algebras for complex reflection groups and analyzes their irreducible modules, Bratteli diagrams, and Jucys-Murphy elements, extending previous work on partition algebras.

## Key findings

- Parametrization of irreducible modules of Tanabe algebras
- Construction of Bratteli diagrams for Tanabe algebra towers
- Explicit Jucys-Murphy elements and their actions

## 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.