Representation Theory and Baxter's Tq Equation for the Six-vertex Model. a Pedagogical Overview

Recent years have seen renewed interest in the six and eight-vertex model at rational coupling values, that is when the crossing parameter is evaluated at roots of unity. Deguchi, Fabricius and McCoy pointed out that the extra degeneracies in the spectrum of the six-vertex transfer matrix can be understood in terms of an affine symmetry algebra in certain commensurate spin-sectors [1]. This raised the question of how the representation theory of this algebra manifests itself in the Bethe ansatz and determines the level of degeneracy [1, 2, 3]. The analogous investigation for the eight-vertex model [4, 5, 6, 7] has led to new developments and a better understanding of Baxter’s celebrated TQ equation [8, 9, 10]. Here T denotes the transfer matrix and Q stands for the auxiliary matrix. In the trigonometric limit some of the zeroes of the auxiliary matrix correspond to the solutions of the six-vertex Bethe ansatz equations first derived in [11, 12]. The significance of a better understanding of the six as well as the eight-vertex model at roots of unity has been further highlighted by results when the order of the root of unity is three. Then Baxter’s TQ equation can be explicitly solved for the groundstate eigenvalue provided the square lattice has an odd number of columns; see [13, 14] for the six and [6, 15] for the eight-vertex case. One finds two linearly independent solutions. These arise because the TQ equation is a second order difference equation, see the discussion in [16] and [17]. However, both linearly independent solutions do not always exist at roots of unity. An additional aspect which motivated further investigation is the connection with combinatorial aspects such as the enumeration of alternating-sign matrices or plane partitions, see e.g. [18, 19, 20].