The partition algebra and the Potts model transfer matrix spectrum in high dimensions

Abstract

We construct generalizations Pmn(Q) of the partition algebra Pn(Q) (Martin P P 1996 J. Algebra 183 319), facilitating a representation theoretic approach to the n-site transfer matrix spectrum of a high-dimensional Q-state Potts model with magnetic field and source terms (and to corresponding dichromatic polynomials). For each Q ∊ ℂ we describe the irreducible representation theory of the sequence of algebras P*(Q) = {Pn(Q) ⊂ Pn1(Q) ⊂ Pn + 1(Q)| n = 0,1,2,...} approaching the large-n limit. For each positive integer Q we extend the Potts model representation ρn of Pn(Q) to a representation of P1n(Q). We show how these Potts representations embed in the representation theory of the partition algebras. These results together provide a tool with which to examine the nature of physical correlation functions.