We describe a parametrized Yang-Baxter equation with nonabelian parameter group. That is, we show that there is an injective map g → 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 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 ⊗ V ⊗ 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 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 s λ 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 identitywhere R, S, T are endomorphisms of V ⊗ V for some vector space V . Here R ij is the endomorphism of V ⊗ V ⊗ V in which R is applied to the i-th and j-th copies of V 1

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We show that the Whittaker coefficients of Borel Eisenstein series on the metaplectic covers of GLr+1 can be described as multiple Dirichlet series in r complex variables, whose coefficients are computed by attaching a number-theoretic quantity (a product of Gauss sums) to each vertex in a crystal graph. These Gauss sums depend on "string data" previously introduced in work of Lusztig, Berenstein and Zelevinsky, and Littelmann. These data are the lengths of segments in a path from the given vertex to the vertex of lowest weight, depending on a factorization of the long Weyl group element into simple reflections. The coefficients may also be described as sums over strict Gelfand-Tsetlin patterns. The description is uniform in the degree of the metaplectic cover.

Weyl group multiple Dirichlet series were associated with a root system Φ and a number field F containing the n-th roots of unity by Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] provided n is sufficiently large; their coefficients involve n-th order Gauss sums. The case where n is small is harder, and is addressed in this paper when Φ = A r . "Twisted" Dirichet series are considered, which contain the series of [4] as a special case. These series are not Euler products, but due to the twisted multiplicativity of their coefficients, they are determined by their p-parts. The p-part is given as a sum of products of Gauss sums, parametrized by strict Gelfand-Tsetlin patterns. It is conjectured that these multiple Dirichlet series are Whittaker coefficients of Eisenstein series on the n-fold metaplectic cover of GL r+1 , and this is proved if r = 2 or n = 1. The equivalence of our definition with that of Chinta [11] when n = 2 and r 5 is also established.Let F be a totally complex algebraic number field containing the group μ 2n of 2n-th roots of unity. Thus −1 is an n-th power in F . Let Φ ⊂ R r be a reduced root system. It has been shown in Brubaker, Bump, Chinta, Friedberg and Hoffstein [3] and Brubaker, Bump and Friedberg [4] how one can associate a multiple Dirichlet series with Φ; its coefficients involve n-th order Gauss sums. A condition of stability is imposed in this definition, which amounts to n being sufficiently large, depending on Φ. In this paper we will propose a description of the Weyl group multiple Dirichlet series in the unstable case when Φ has Cartan type A r , and present the evidence in support of this description. We will refer to this as the Gelfand-Tsetlin description whose striking characteristic is that it gives a single formula valid for all n for these coefficients, that reduces to the stable description when n is sufficiently large.We conjecture that this Weyl group multiple Dirichlet series coincides with the Whittaker coefficient of an Eisenstein series. The Eisenstein series E(g; s 1 , · · · , s r ) is of minimal parabolic type, on an n-fold metaplectic cover of an algebraic group defined over F whose root system is the dual root system 294 B. BRUBAKER, D. BUMP, S. FRIEDBERG, AND J. HOFFSTEIN of Φ. We refer to this identification of the series with a Whittaker coefficient of E as the Eisenstein conjecture.We will present some evidence for the Eisenstein conjecture by proving it when Φ is of type A 2 (for all n) or when Φ is of type A r and n = 1. We will also present evidence for the Gelfand-Tsetlin description (but not the Eisenstein conjecture) for general n.There is a good reason not to use the Eisenstein series as a primary foundational tool in the study of the Weyl group multiple Dirichlet series. This is the relative complexity of the Matsumoto cocycle describing the metaplectic group. The approach taken in [3] and [4] had its origin in Bump, Friedberg and Hoffstein [9], where it was proposed that multiple Dirichlet...

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