Lattice Models, Hamiltonian Operators, and Symmetric Functions

TL;DR

This paper establishes a connection between Hamiltonian operators, lattice models, and symmetric functions, particularly focusing on the six-vertex model, charge-dependent models, and supersymmetric LLT polynomials, revealing new solvability criteria and identities.

Contribution

It introduces a sharp solvability criterion for charge-dependent six-vertex models and links their partition functions to supersymmetric LLT polynomials, expanding understanding of integrable lattice models.

Findings

Hamiltonian operators correspond to free fermionic weights in the six-vertex model.

Partition functions of certain lattice models are supersymmetric LLT polynomials.

A Cauchy identity for skew supersymmetric LLT polynomials is proved.

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)