A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain

Abstract

We study the connection between the three-color model and the polynomials $q_n(z)$ of Bazhanov and Mangazeev, which appear in the eigenvectors of the Hamiltonian of the XYZ spin chain. By specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end, we find an explicit combinatorial expression for $q_n(z)$ in terms of the partition function of the three-color model with the same boundary conditions. Bazhanov and Mangazeev conjectured that $q_n(z)$ has positive integer coefficients. We prove the weaker statement that $q_n(z+1)$ and $(z+1{)}^{n(n+1)}{q}_n(1/(z+1))$ have positive integer coefficients. Furthermore, for the three-color model, we find some results on the number of states with a given number of faces of each color, and we compute strict bounds for the possible number of faces of each color.

References in corpus (11)

• Another proof of the alternating sign matrix conjecture
• Spin chains and combinatorics
• Determinant formula for the six-vertex model with reflecting end
• Six-Vertex, Loop and Tiling models: Integrability and Combinatorics
• Eight-vertex model and non-stationary Lame equation
• The eight-vertex model and Painleve VI
• A possible combinatorial point for XYZ-spin chain
• Elliptic dynamical reflection algebra and partition function of SOS model with reflecting end
• Special polynomials related to the supersymmetric eight-vertex model: A summary
• Special polynomials related to the supersymmetric eight-vertex model. I. Behaviour at cusps
• Special polynomials related to the supersymmetric eight-vertex model. III. Painlevé VI equation

Cited by in corpus (3)

• Exact results for the six-vertex model with domain wall boundary conditions and a partially reflecting end
• Sum rules for the supersymmetric eight-vertex model
• A Combinatorial Description of Certain Polynomials Related to the XYZ Spin Chain. II. The Polynomials $p_n$