The domain wall partition function for the Izergin-Korepin 19-vertex model at a root of unity

Abstract

We study the domain wall partition function $Z_N$ for the $U_q(A_2^{(2)})$ (Izergin-Korepin) integrable $19$-vertex model on a square lattice of size $N$. $Z_N$ is a symmetric function of two sets of parameters: horizontal $\zeta_1,..,\zeta_N$ and vertical $z_1,..,z_N$ rapidities. For generic values of the parameter $q$ we derive the recurrence relation for the domain wall partition function relating $Z_{N+1}$ to $P_N Z_N$, where $P_N$ is the proportionality factor in the recurrence, which is a polynomial symmetric in two sets of variables $\zeta_1,..,\zeta_N$ and $z_1,..,z_N$. After setting $q^3=-1$ the recurrence relation simplifies and we solve it in terms of a Jacobi-Trudi-like determinant of polynomials generated by $P_N$.