Abstract. In the mid-1960s Borevič and Faddeev initiated the study of the Galois module structure of groups of pth-power classes of cyclic extensions K/F of pth-power degree. They determined the structure of these modules in the case when F is a local field. In this paper we determine these Galois modules for all base fields F .In 1947Šafarevič initiated the study of Galois groups of maximal pextensions of fields with the case of local fields [12], and this study has grown into what is both an elegant theory as well as an efficient tool in the arithmetic of fields. From the very beginning it became clear that the groups of pth-power classes of the various field extensions of a base field encode basic information about the structure of the Galois groups of maximal p-extensions. (See [7] and [13].) Such groups of pth-power classes arise naturally in studies in arithmetic algebraic geometry, for example in the study of elliptic curves.In 1960 Faddeev began to study the Galois module structure of pthpower classes of cyclic p-extensions, again in the case of local fields, and during the mid-1960s he and Borevič established the structure of these Galois modules using basic arithmetic invariants attached to Galois extensions. (See [6] and [4].) In 2003 two of the authors ascertained the Galois module structure of pth-power classes in the case of cyclic extensions of degree p over all base fields F containing a primitive pth root of unity [9]. Very recently, this work paved the way for the determination of the entire Galois cohomology as a Galois module in

We use statistical mechanics-variants of the six-vertex model in the plane studied by means of the Yang-Baxter equation-to give new deformations of Weyl's character formula for classical groups of Cartan type B, C, and D, and a character formula of Proctor for type BC. In each case, the corresponding Boltzmann weights are associated with the free fermion point of the six-vertex model. These deformations add to the earlier known examples in types A and C by Tokuyama and Hamel-King, respectively. A special case for classical types recovers deformations of the Weyl denominator formula due to Okada.

No abstract

Abstract. In this paper, we explain a connection between a family of free-fermionic sixvertex models and a discrete time evolution operator on one-dimensional Fermionic Fock space. The family of ice models generalize those with domain wall boundary, and we focus on two sets of Boltzmann weights whose partition functions were previously shown to generalize a generating function identity of Tokuyama. We produce associated Hamiltonians that recover these Boltzmann weights, and furthermore calculate the partition functions using commutation relations and elementary combinatorics. We give an expression for these partition functions as determinants, akin to the Jacobi-Trudi identity for Schur polynomials. OverviewHamiltonians arising from Fock representations of Clifford algebras were explored by the Kyoto school, for example in [5,6,12]. The Boson-Fermion correspondence gives an explicit isomorphism between this Fermionic Fock representation and a polynomial algebra, the Bosonic Fock space. The image of elements under this correspondence are commonly called "τ -functions." The Kyoto school papers show that τ -functions are solutions to integrable hierarchies of nonlinear differential equations. Moreover, the Bosonic Fock space may be identified with the ring of symmetric functions over a field. In particular if the Clifford algebra is gl(∞), then there exists a simple family of τ -functions equal to Schur polynomials.Thinking of each application of the Hamiltonian operator as a step in discrete time, the evolution of a one-dimensional model gives rise to a two-dimensional lattice model. In the case above of the Hamiltonian for gl(∞), the resulting two-dimensional model is the fivevertex model; a nice exposition of this fact may be found in Zinn-Justin [21]. Our first result is a generalization of this fact, using a deformation of the above Hamiltonian operator for gl(∞) whose evolution produces the six-vertex model studied in [3]. These models have boundary conditions generalizing the more familiar domain wall boundary conditions. As a consequence, the partition functions for these six-vertex models may be studied using methods for evaluating τ -functions, e.g., the time evolution of fermionic fields (given in Proposition 4) and Wick's theorem. This theme is also present in [21], and we borrow many of the same techniques for analyzing our more complicated six-vertex model. The partition function of our family of six-vertex models was computed (upon using a combinatorial bijection with Gelfand-Tsetlin patterns) by Tokuyama [20] and subsequently reproved using the Yang-Baxter equation in [3]. Here we give an alternate proof of Tokuyama's result using only commutation relations for Hamiltonians and some elementary combinatorial facts. Moreover, we provide new explicit expressions for partition functions as a determinant, akin to the Jacobi-Trudi identity for ordinary Schur polynomials.Date: June 2, 2016.

1 The interested reader should consult [6] and [3] for some initial background on Hilbert's Theorem 90 and [13, p. 30] for its cohomological generalization. To observe the use of Hilbert 90-type theorems in the partially published work of Markus Rost and Voevodsky on the Bloch-Kato conjecture, see [11] and [12]. For further original sources on Hilbert 90 and its cohomological generalization see Ernst Kummer's early discovery of a special case [4], followed by Speiser's result [9] and Noether's application [7]. toral, and habilitation degrees at the University of Warsaw, and then for many years he taught at the same institution. He has been a member of the Polish Academy of Sciences, as well as scientific director of its Institute of Mathematics and the Banach Center. Once transplanted halfway around the world, he met Ján Mináč when Ján, absorbed in thought, quite literally crashed into him.JÁN MINÁČ spent years blissfully unaware of the fact that the Fields Medal clock had already begun ticking from the moment of his birth. He happily wasted his time daydreaming, playing soccer, reading all kinds of literature, and toying with prime numbers. Although now past forty, he is as excited as ever to keep playing with mathematics, blessed with great collaborators and the support of his wonderful wife Leslie. Summing these up, he reaches but one result, his own sort of Fields Medal.