Algebraic Operations via Solvable Lattice Models

Abstract

This thesis explores solvable lattice models in several contexts. Our overarching goal is to un- derstand and exploit the flexibility of lattice models in their ability to express functorial operations in algebra. In particular, we study lattice models whose partition functions are special functions in representation theory and Schubert calculus. These functions tend to have nice properties relating to their algebraic structure, and we try to connect these properties to combinatorial operations on lattice models. Chapter 2 studies the connection between solvable lattice models and discrete-time Hamiltonian operators. 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(gl(1|n)) [1]. Our results show a close and unexpected connection between Hamiltonian operators and solvability. 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 [2]. 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 in [3], 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 always a (skew) supersymmetric LLT polynomial. Chapter 3 considers the connections between lattice models and formal group laws. In particular, we exhibit a substitution corresponding to any formal group law into any solution to the Yang-Baxter equation. When applied to the R-matrix from the standard evaluation module for U_q(sl(n+1)), the resulting lattice models are related to those studied in [4], and their partition functions may have interpretations in higher cohomology of Schubert varieties. Then, Chapter 4 gives an exposition of lattice model proofs of some well-known identities for Schur polynomials. In addition, we introduce what we call a symmetrized version of the Yang-Baxter algebra and show how the Schur polynomial identities come from relations in this algebra. Running through this work is a thematic aspiration: that lattice models are “unreasonably effective” (in Ben Brubaker’s words) in expressing various phenomena throughout mathematics.