Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model

TL;DR

This paper explores birational transformations from chiral Potts models, classifies models for prime or prime-square state counts, and demonstrates complexity reduction and integrability in associated transformations.

Contribution

It provides a complete classification of chiral q-state Potts models for prime or prime-square q and links these models to integrable birational transformations with matrix representations.

Findings

Classified all chiral q-state models for prime or prime-square q

Proved absence of certain stable patterns with more than four states

Identified a family of integrable transformations with matrix form

Abstract

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models to \emph{stable patterns} and \emph{signed-patterns}, we give general results which allow us to find \emph{all} chiral $q$-state spin-edge Potts models when the number of states $q$ is a prime or the square of a prime, as well as several $q$-dependent family of models. We also prove the absence of monocolor stable signed-pattern with more than four states. This demonstrates a conjecture about cyclic Hadamard matrices in a particular case. The birational transformations associated to these lattice spin-edge models show complexity reduction. In particular we recover a one-parameter family of integrable transformations, for which we give a matrix representation