Semisimplicity criterion for 2-tonal partition algebras

TL;DR

This paper establishes a criterion for when even partition algebras are semisimple over the complex numbers, showing they are semisimple precisely when the parameter is not a non-negative integer.

Contribution

It provides a complete characterization of semisimplicity for 2-tonal partition algebras over the complex field.

Findings

Semisimplicity holds iff δ is not a non-negative integer

The criterion applies to all n in the algebra family

The result clarifies the algebra's structure depending on δ

Abstract

We determine the semisimplicity criterion for even partition algebras over the complex field. Specifically we prove that the even/2-tonal partition algebras $P_n^2(\delta)$ over $\mathbb{C}$ are semisimple for all $n$ if and only if parameter $\delta \not\in \mathbb{N}_0$ .