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Lattice Models, Hamiltonian Operators, and Symmetric Functions

Andrew Hardt

Abstract

We give general conditions for the existence of a Hamiltonian operator whose discrete time evolution matches the partition function of certain solvable lattice models. In particular, we examine two classes of lattice models: the classical six-vertex model and a generalized family of $(2n+4)$-vertex models for each positive integer $n$. These models depend on a statistic called charge, and are associated to the quantum group $U_q(\widehat{\mathfrak{gl}}(1|n))$. Our results show a close and unexpected connection between Hamiltonian operators and the Yang-Baxter equation. The six-vertex model can be associated with Hamiltonians from classical Fock space, and we show that such a correspondence exists precisely when the Boltzmann weights are free fermionic. This allows us to prove that the free fermionic partition function is always a (skew) supersymmetric Schur function and then use the Berele-Regev formula to correct a result from of Brubaker, Bump, and Friedberg on the free fermionic domain-wall partition function. In this context, the supersymmetric function involution takes us between two lattice models that generalize the vicious walker and osculating walker models. Then, we prove a sharp solvability criterion for the six-vertex model with charge that provides the proper analogue of the free fermion condition. Building on results by Brubaker, Buciumas, Bump, and Gustafsson, we show that this criterion exactly dictates when a charged model has a Hamiltonian operator acting on a Drinfeld twist of $q$-Fock space. The resulting partition function is then a (skew) supersymmetric LLT polynomial, and almost all supersymmetric LLT polynomials appear as partition functions of our lattice models. We also prove a Cauchy identity for skew supersymmetric LLT polynomials. (see 2024 author's note below)

Lattice Models, Hamiltonian Operators, and Symmetric Functions

Abstract

We give general conditions for the existence of a Hamiltonian operator whose discrete time evolution matches the partition function of certain solvable lattice models. In particular, we examine two classes of lattice models: the classical six-vertex model and a generalized family of -vertex models for each positive integer . These models depend on a statistic called charge, and are associated to the quantum group . Our results show a close and unexpected connection between Hamiltonian operators and the Yang-Baxter equation. The six-vertex model can be associated with Hamiltonians from classical Fock space, and we show that such a correspondence exists precisely when the Boltzmann weights are free fermionic. This allows us to prove that the free fermionic partition function is always a (skew) supersymmetric Schur function and then use the Berele-Regev formula to correct a result from of Brubaker, Bump, and Friedberg on the free fermionic domain-wall partition function. In this context, the supersymmetric function involution takes us between two lattice models that generalize the vicious walker and osculating walker models. Then, we prove a sharp solvability criterion for the six-vertex model with charge that provides the proper analogue of the free fermion condition. Building on results by Brubaker, Buciumas, Bump, and Gustafsson, we show that this criterion exactly dictates when a charged model has a Hamiltonian operator acting on a Drinfeld twist of -Fock space. The resulting partition function is then a (skew) supersymmetric LLT polynomial, and almost all supersymmetric LLT polynomials appear as partition functions of our lattice models. We also prove a Cauchy identity for skew supersymmetric LLT polynomials. (see 2024 author's note below)

Paper Structure

This paper contains 28 sections, 57 theorems, 246 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fock Space and Hamiltonian Operators
  3. Partitions
  4. Fock Space
  5. Hamiltonians and symmetric functions
  6. Cauchy, Pieri, and branching rules for Hamiltonians
  7. The six-vertex model
  8. Relating $\mathfrak{S}$ and $\mathfrak{S}^*$
  9. Six-vertex models and Hamiltonians
  10. The $\mathfrak{S}$ lattice model
  11. The $\mathfrak{S}^*$ lattice model
  12. The free fermionic partition function
  13. Supersymmetric Schur functions
  14. Evaluating the partition function
  15. Identities: Pieri rule, Cauchy identity, and branching rule
  16. ...and 13 more sections

Key Result

Theorem \oldthetheorem

Figures (7)

  • Figure 1: The Boltzmann weights for $\mathfrak{S}$. Here, $x_i, y_i, A_i$, and $B_i$ are parameters associated to each row.
  • Figure 2: The Boltzmann weights for $\mathfrak{S}^*$. Here, $x_i, y_i, A_i$, and $B_i$ are parameters associated to each row.
  • Figure 3: A state of the lattice model $\mathfrak{S}_{\lambda/\mu}$, where $\lambda = (5,3,1)$ and $\mu = (3,1,0)$.
  • Figure 4: The Boltzmann weights for $\mathfrak{S}$ with charge. Here, $x_i, y_i, A_i$, and $B_i$ are parameters associated to each row, while $f(a)$ and $h(a)$ depend only on the change $a$.
  • Figure 5: The Boltzmann weights for $\mathfrak{S}^*$ with charge. Here, $z_i, w_i, A_i$, and $B_i$ are parameters associated to each row, while $f(a)$ and $h(a)$ depend only on the change $a$.
  • ...and 2 more figures

Theorems & Definitions (94)

  • Theorem \oldthetheorem: Theorems \ref{['Delta-lattice-Hamiltonian']} and \ref{['Gamma-lattice-Hamiltonian']}, and Corollary \ref{['Partition-Function-is-Supersymmetric-Schur']}
  • Theorem \oldthetheorem: Theorem \ref{['partition-function-all-boundaries']}, Corollary \ref{['domain-wall-corollary']}
  • Theorem \oldthetheorem: Theorem \ref{['charge-solvability-theorem']}
  • Theorem \oldthetheorem: Theorems \ref{['charged-Delta-lattice-Hamiltonian']}, \ref{['charged-Gamma-lattice-Hamiltonian']}, and \ref{['Partition-Function-is-Supersymmetric-LLT']}
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem: Wick's Theorem
  • Proposition \oldthetheorem: Boson-Fermion Correspondence
  • Lemma \oldthetheorem: Duality
  • proof
  • ...and 84 more