On Hamiltonians for six-vertex models

Ben Brubaker, Andrew Schultz

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)100-121
Number of pages22
JournalJournal of Combinatorial Theory. Series A
Volume155
DOIs
StatePublished - Apr 2018

Bibliographical note

Funding Information:
We thank an anonymous referee of this paper for careful comments leading to improved exposition, and for pointing out a similarity between some of our results and those sketched in a talk by Zinn-Justin [26] . This work was supported by NSF grant DMS-1406238 .

Publisher Copyright:
© 2017 Elsevier Inc.

Copyright:
Copyright 2017 Elsevier B.V., All rights reserved.

Keywords

  • Discrete time evolution
  • Fock space
  • Partition function
  • Six-vertex model
  • τ-functions

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