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arXiv · 0802.1329

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

E. Preissmann·J. -Ch. Anglès d'Auriac·J. -M. Maillard
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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

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E. Preissmann, J. -Ch. Anglès d'Auriac, J. -M. Maillard. 2008-09-01. Birational Mappings and Matrix Sub-algebra from the Chiral Potts Model. https://doi.org/10.1063/1.3032564

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