On Hamiltonians for six-vertex models

Citation data:

Journal of Combinatorial Theory, Series A, ISSN: 0097-3165, Vol: 155, Page: 100-121

Publication Year:
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Ben Brubaker; Andrew Schultz
Elsevier BV
Mathematics; Computer Science
article description
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.