Birational mappings and matrix subalgebra from the chiral Potts model

Abstract

We study birational transformations of the projective space originating from lattice statistical mechanics, specifically from various chiral Potts models. Associating these models with stable patterns and signed patterns, we give general results which allow us to find 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 with these lattice spinedge models show complexity reduction. In particular, we recover a one-parameter family of integrable transformations, for which we give a matrix representation when the parameter has a suitable value.

Degree complexity of birational maps related to matrix inversion: symmetric case

“…The map K is of interest since it represents a basic symmetry in certain problems of lattice statistical mechanics, and has been studied in [1][2][3][4][5][6][7][8]12].…”

“…The first were considered are S q (the space of symmetric matrices), C q the cyclic (also called circulant) matrices, and SC q = S q ∩ C q (see [12] for more K -invariant subspaces of P(M q )). In view of complex dynamics, as well as physical meaning, the map K as well as the restrictions of K to invariant spaces are of interest.…”

Degree complexity of birational maps related to matrix inversion: Symmetric case

“…Then the mapping K : P(M q ) → P(M q ) is defined as follows: K = I • J, where J(x) = (x −1 i,j ) takes the reciprocal of each entry of the matrix x = (x i,j ), and I(x) = (x i,j ) −1 is the matrix inverse. The map K is of interest since it represents a basic symmetry in certain problems of lattice statistical mechanics, and has been studied in [1], [2], [3], [4], [5], [6], [7], [8], and [12].…”

“…There are many K-invariant subspaces T ⊂ P(M q ). The first were considered are S q (the space of symmetric matrices), C q the cyclic (also called circulant) matrices, and SC q = S q ∩ C q (see [12] for more K-invariant subspaces of P(M q )). In view of complex dynamics, as well as physical meaning, the map K as well as the restrictions of K to invariant spaces are of interest.…”