A combinatorial description of certain polynomials related to the XYZ spin chain

Abstract: The aim of this thesis is to study the connection between the three-color model and the polynomials q_n(z) of Bazhanov and Mangazeev. To give some background, we describe some exactly solvable, quantum integrable lattice models and their connections to each other and to other models. By specializing the parameters in the partition function of the six-vertex model with domain wall boundary conditions (DWBC), Kuperberg was able to enumerate the alternating sign matrices (ASMs). Razumov and Stroganov found connections between the number of ASMs and the ground state eigenvector components of the Hamiltonian for the supersymmetric XXZ spin chain of odd length. Similar problems can be studied for the eight-vertex solid-on-solid (8VSOS) model. By specializing the parameters in the partition function of the 8VSOS model with DWBC, Rosengren connected it to the three-color model with DWBC, and furthermore showed that the partition function of the three-color model can be expressed in terms of certain polynomials. Bazhanov and Mangazeev introduced other special polynomials, among them q_n(z) , that can be used to express certain ground state eigenvector components of the Hamiltonian for the supersymmetric XYZ spin chain of odd length. Many of these polynomials seem to have positive integer coefficients, which suggests that there should be a combinatorial interpretation. In Paper I, we find an explicit combinatorial expression for q_n(z) in terms of the partition function of the three-color model with DWBC and reflecting end, by specializing the parameters in the partition function of the 8VSOS model with DWBC and reflecting end. 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. For the three-color model, we find 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.