Schur Polynomials and the Yang-Baxter Equation

We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map g 7→ R(g) from GL(2,C)×GL(1,C) to End(V ⊗V ) where V is a two-dimensional vector space such that if g, h ∈ G then R12(g)R13(gh)R23(h) = R23(h)R13(gh)R12(g). Here Rij denotes R applied to the i, j components of V ⊗V ⊗V . The image of this map consists of matrices whose nonzero coefficients a1, a2, b1, b2, c1, c2 are the Boltzmann weights for the non-field-free six-vertex model, constrained to satisfy a1a2 +b1b2−c1c2 = 0. This is the exact center of the disordered regime, and is contained within the free fermionic eight-vertex models of Fan and Wu. As an application, we show that with boundary conditions corresponding to integer partitions λ, the six-vertex model is exactly solvable and equal to a Schur polynomial sλ times a deformation of the Weyl denominator. This generalizes and gives a new proof of results of Tokuyama and Hamel and King. Baxter’s method of solving lattice models in statistical mechanics is based on the star-triangle relation, which is the identity R12S13T23 = T23S13R12, (1) where R, S, T are endomorphisms of V ⊗V for some vector space V . Here Rij is the endomorphism of V ⊗ V ⊗ V in which R is applied to the i-th and j-th copies of V 1 and the identity map to the k-th component, where i, j, k are 1, 2, 3 in some order. If the endomorphisms R, S, T are all equal, this is the Yang-Baxter equation (cf. [15], [25]). A related construction is the parametrized Yang-Baxter equation R12(g)R13(g · h)R23(h) = R23(h)R13(g · h)R12(g) (2) where the endomorphism R now depends on a parameter g in a group G and g, h ∈ G in (2). There are many such examples in the literature in which the group G is an abelian group such as R or R×. In this paper we present an example of (2) having a non-abelian parameter group. The example arises from a two-dimensional lattice model—the six-vertex model. We now briefly review the connection between lattice models and instances of (1) and (2). In statistical mechanics, one attempts to understand global behavior of a system from local interactions. To this end, one defines the partition function of a model to be the sum of certain locally determined Boltzmann weights over all admissible states of the system. Baxter (see [1] and [2], Chapter 9) recognized that instances of the star-triangle relation allowed one to explicitly determine the partition function of a lattice model. The six-vertex, or ‘ice-type,’ model is one such example that is much studied in the literature, and we revisit it in detail in the next section. For the moment, we offer a few general remarks needed to describe our results. In our presentation of the six-vertex model, each state is represented by a labeling of the edges of a finite rectangular lattice by ± signs, called spins . If the Boltzmann weights are invariant under sign reversal the system is called field-free, corresponding to the physical assumption of the absence of an external field. For field-free weights, the six-vertex model was solved by Lieb [23] and Sutherland [32], meaning that the partition function can be exactly computed. The papers of Lieb, Sutherland and Baxter assume periodic boundary conditions, but non-periodic boundary conditions were treated by Korepin [18] and Izergin [14]. Much of the literature assumes that the model is field-free. In this case, Baxter shows there is one such parametrized YangBaxter equation with parameter group C× for each value of a certain real invariant 4, defined below in (7) in terms of the Boltzmann weights. One may ask whether the parameter subgroup C× may be enlarged by including endomorphisms whose associated Boltzmann weights lie outside the field-free case. If4 6 = 0 the group may not be so enlarged. However we will show in Theorem 4 that if 4 = 0, then the group C× may be enlarged to GL(2,C)×GL(1,C) by expanding the set of endomorphisms to include non-field-free ones. In this expanded 4 = 0 regime, R(g) is not field-free for general g. It is contained within the set of exactly 2 solvable eight-vertex models called the free fermionic model by Fan and Wu [8], [9]. Our calculations suggest that it is not possible to enlarge the group G to the entire free fermionic domain in the eight vertex model. As an application of these results, we study the partition function for ice-type models having boundary conditions determined by an integer partition λ. More precisely, we give an explicit evaluation of the partition function for any set of Boltzmann weights chosen so that 4 = 0. This leads to an alternate proof of a deformation of the Weyl character formula for GLn found by Hamel and King [13], [12]. That result was a substantial generalization of an earlier generating function identity found by Tokuyama [33], expressed in the language of Gelfand-Tsetlin patterns. Our boundary conditions depend on the choice of a partition λ. Once this choice is made, the states of the model are in bijection with strict Gelfand-Tsetlin patterns having a fixed top row. These are triangular arrays of integers with strictly decreasing rows that interleave (Section 3). In its original form, Tokuyama’s formula expresses the partition function of certain ice models as a sum over strict Gelfand-Tsetlin patterns. This connection between states of the ice model and strict Gelfand-Tsetlin patterns has one historical origin in the literature for alternating sign matrices. (An independent historical origin is in the Bethe Ansatz. See Baxter [2] Chapter 8 and Kirillov and Reshetikhin [17].) The bijection between the set of alternating sign matrices and strict Gelfand-Tsetlin patterns having smallest possible top row is in Mills, Robbins and Rumsey [27], while the connection with what are recognizably states of the six-vertex model is in Robbins and Rumsey [29]. This connection was used by Kuperberg [19] who gave a second proof (after Zeilberger) of the alternating sign matrix conjecture of Mills, Robbins and Rumsey. It was observed by Okada [28] and Stroganov [31] that the number of n×n alternating sign matrices, that is, the value of Kuperberg’s ice (with particular Boltzmann weights involving cube roots of unity) is a special value of the particular Schur function in 2n variables with λ = (n, n, n − 1, n − 1, · · · , 1, 1) divided by a power of 3. Moreover Stroganov gave a proof using the Yang-Baxter equation. This occurrence of Schur polynomials in the six-vertex model is different from the one we discuss, since Baxter’s parameter 4 is nonzero for those investigations. There are other works relating symmetric function theory to vertex models or spin chains. Lascoux [21], [20] gave six-vertex model representations of Schubert and Grothendieck polynomials of Lascoux and Schützenberger [22] and related these to the Yang-Baxter equation. Fomin and Kirillov [10], [11] also gave theories of the Schubert and Grothendieck polynomials based on the Yang-Baxter equation. Tsilevich [34] gives an interpretation of Schur polynomials and Hall-Littlewood poly-