On the representation theory of partial Brauer algebras

TL;DR

This paper investigates the structure of partial Brauer algebras over complex numbers, establishing their semisimplicity, constructing Specht modules, and analyzing their restriction rules and decomposition matrices.

Contribution

It provides a detailed representation theory framework for partial Brauer algebras, including semisimplicity conditions, Specht module construction, and restriction rules.

Findings

Partial Brauer algebras are generically semisimple.

Constructed Specht modules for these algebras.

Determined restriction rules and decomposition matrices.

Abstract

In this paper we study the partial Brauer $\mathbb{C}$-algebras $\mathfrak{R}_n(\delta,\delta')$, where $n \in \mathbb{N}$ and $\delta,\delta'\in\mathbb{C}$. We show that these algebras are generically semisimple, construct the Specht modules and determine the Specht module restriction rules for the restriction $\mathfrak{R}_{n-1} \hookrightarrow \mathfrak{R}_n$. We also determine the corresponding decomposition matrix, and the Cartan decomposition matrix.