On Hamiltonians for six-vertex models

Abstract

We show that a deformation of Schur polynomials (matching the Shintani–Casselman–Shalika formula for the p-adic spherical Whittaker function) is obtained from a Hamiltonian operator on Fermionic Fock space. The discrete time evolution of this operator gives rise to states of a free-fermionic six-vertex model with boundary conditions generalizing the “domain wall boundary conditions,” which are not field-free. This is analogous to results of the Kyoto school in which ordinary Schur functions arise in the Boson–Fermion correspondence, and the Hamiltonian operator produces states of the five-vertex model. Our Hamiltonian arises naturally from super Clifford algebras studied by Kac and van de Leur. As an application, we give a new proof of a formula of Tokuyama [25] and Jacobi–Trudi type identities for the deformation of Schur polynomials. Variants leading to deformations of characters for other classical groups and their finite covers are also presented.