Lattice Models, Hamiltonian Operators, and Symmetric Functions

Abstract

We explore the connections between two types of integrability phenomena arising from quantum groups: solvable lattice models and Hamiltonian operators from Heisenberg algebras. The fundamental question this paper explores is when there exists a Hamiltonian operator whose discrete time evolution matches the partition function of a solvable lattice model.We deal with two types of lattice models: the classical six-vertex model and a modified six-vertex model involving charge. 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. In this context, the supersymmetric function involution takes us between two lattice models that are generalizations of the vicious walker and osculating walker models. We also observe that the supersymmetric Jacobi-Trudi formula, through Wick's Theorem, can be seen as a six-vertex analogue of the Lindström-Gessel-Viennot Lemma.The six-vertex model with charge has been used to study metaplectic Whittaker functions. It can be associated with Hamiltonian operators acting on Drinfeld twists of q-Fock space [3], and we determine that such a correspondence exists exactly when the weights satisfy what we call a generalized free fermion condition, in which case the partition function is always a (skew) supersymmetric LLT polynomial.

Free-fermions and canonical Grothendieck polynomials

“…Let us briefly digress to discuss vertex models as there is a well-known relationship with freefermions; see, e.g., [Har21] and references therein. There is a vertex model known for Grothendieck polynomials [MS13, MS14, WZJ19, ZJ09].…”

Free Fermionic Schur Functions

“…With another, it produces Schur functions, and more generally, spherical Whittaker functions for the general linear group over a non-archimedean local field by means of the Casselman-Shalika formula [BBF11]. Additionally, using yet another set of weights, the six vertex model generates supersymmetric Schur functions [Har21]. Recently, by using more general weights, it has been demonstrated that the six vertex model can produce various generalizations of Schur functions [Mot17b,Mot17a,ABPW21].…”

Integrable systems and crystals for edge labeled tableaux

“…We note that our vertex model satisfies the free fermion condition, where the weights a 1 a 2 + b 1 b 2 = c 1 c 2 , where a 2 is (the weight of) the vertex with all boundaries 1. This does not naturally fit into the formalism defined by Hardt [Har21] since we have the column dependency. However, if we set a = α with a i = 0 for i ≫ 1, then, in the notation of [Har21, Fig.…”

“…We can also introduce an extra parameter from [Har21] to get a six-vertex model. If we take the extra parameter to be x i = ty j , which reduces to our edge labeled tableaux case when t = 0, then we expect to obtain a generalization of Tokuyama's theorem for spherical Whittaker functions (see, e.g., [BS18]).…”