Summary: We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map \(g \mapsto R(g)\) from \(\text{GL}(2,\mathbb{C}) \times \text{GL}(1,\mathbb{C})\) to
\(\text{End}(V\otimes V)\), where \(V\) is a two-dimensional vector space such that if \(g, h\in G\) then \[ R_{12}(g) R_{13}(gh) R_{23}(h) = R_{23}(h) R_{13}(gh) R_{12}(g). \] Here \(R_{ij}\) denotes \(R\) applied to the \(i,j\) components of \(V \otimes V \otimes V\). The image of this map consists of matrices whose nonzero coefficients \(a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}\) are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy \(a_{1} a_{2} + b_{1} b_{2} - c_{1} c_{2} = 0\). This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of C. Fan and F.-Y. Wu [Ising model with next-neighbor interactions. I: Some exact results and an approximate solution. Phys. Rev. 179, 560–570 (1969); General lattice model of phase transitions. Phys. Rev. B 2, No. 3, 723–733 (1970)].
As an application, we show that with boundary conditions corresponding to integer partitions \(\lambda \), the six-vertex model is exactly solvable and equal to a Schur polynomial \(s_{\lambda }\) times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of T. Tokuyama [A generating function of strict Gelfand patterns and some formulas on characters of general linear groups. J. Math. Soc. Japan 40, No. 4, 671–685 (1988; Zbl 0639.20022)] and A. M. Hamel and R. C. King [U-turn alternating sign matrices, symplectic shifted tableaux and their weighted enumeration. J. Algebr. Comb. 21, No. 4, 395–421 (2005; Zbl 1066.05012); Bijective proofs of shifted tableau and alternating sign matrix identities. J. Algebr. Comb. 25, No. 4, 417–458 (2007; Zbl 1122.05094)].