Factorized Domain Wall Partition Functions in Trigonometric Vertex Models

We obtain factorized domain wall partition functions for two sets of trigonometric vertex models: 1. The N-state Deguchi-Akutsu models, for N ∈ {2, 3, 4} (and conjecture the result for all N ≥ 5), and 2. The sl(r+1|s+1) Perk-Schultz models, for {r, s ∈ N}, where (given the symmetries of these models) the result is independent of {r, s}. 0. Introduction Domain wall partition functions (DWPF’s) were first proposed and evaluated in determinant form, for the spin12 vertex model 1 on a finite square lattice, in [1, 2]. At the free fermion point of the spin1 2 model, this determinant is in Cauchy form and therefore factorizes. More recently, determinant expressions for the DWPF’s of spin2 models and also of level-1 affine so(N) models (for certain discrete values of the crossing parameter) were obtained in [3] and in [4], respectively. State variable conjugation. We are interested in models with state variables {σ}. Each state variable takes discrete integral values, σ ∈ {1, · · · , N}. We define ‘state variable conjugation’ as replacing each state variable σ by (N − σ + 1). The models mentioned above are invariant under this conjugation. The N-state Deguchi-Akutsu (N–DA) models, N ≥ 2 [5], are models with vertex weights, that depend on two sets of parameters: 1. Vertical and horizontal rapidities, and 2. Vertical and horizontal external field variables. They reduce in the limit of no external fields to the spin2 models at their respective free fermion points. Factorized DWPF. In [6], a determinant expression for the DWPF of the 2– DA model was obtained using the arguments of [1,2], but only for zero values of the rapidities. This determinant is in Cauchy form and therefore factorizes. For general values of all parameters, no determinant expression was found, and it was argued on general grounds that no such expression exists. However, using the F–basis of [7], a factorized expression for the 2–DA DWPF was obtained. In this work, we extend the above result to the N -DA models, N ∈ {2, 3, 4}. Our results are restricted to N ∈ {2, 3, 4} because our proofs require the explicit expressions of the weights, while the number of vertices grows ∼ O(N). However, our results are quite simple and have a uniform dependence on N , which allows us to conjecture that our expression extends to all N ≥ 2. Non-invariance under state variable conjugation. Our proofs rely on the non-invariance of the N -DA models under state variable conjugation (for nonvanishing external fields). This leads us to look for other models that are similarly non-invariant. 2000 Mathematics Subject Classification. Primary 82B20, 82B23.