Algebras in Higher Dimensional Statistical Mechanics - the Exceptional Partition (MEAN Field) Algebras

TL;DR

This paper analyzes the structure of partition algebras related to Potts models in high dimensions, identifying conditions for semi-simplicity and calculating key irreducible representation dimensions.

Contribution

It provides a detailed characterization of partition algebras for specific complex parameters, especially in the mean field limit of Potts models, including semi-simplicity criteria.

Findings

Partition algebra $P_n(Q)$ is non-semi-simple iff $Q$ is a non-negative integer less than $n$.

Dimensions of key irreducible representations are explicitly determined for all specializations.

The algebraic structure is linked to the partition function of high-dimensional Potts models.

Abstract

We determine the structure of the partition algebra $P_n(Q)$ (a generalized Temperley-Lieb algebra) for specific values of $Q \in \C$, focusing on the quotient which gives rise to the partition function of $n$ site $Q$-state Potts models (in the continuous $Q$ formulation) in arbitrarily high lattice dimensions (the mean field case). The algebra is non-semi-simple iff $Q$ is a non-negative integer less than $n$. We determine the dimension of the key irreducible representation in every specialization.