The Flexibility of the Six Vertex Lattice Model in the Study of Special Functions

Abstract

In this thesis, we examine the six-vertex lattice model and three generalizations thereof, whose partition functions give three different kinds of special functions: double biaxial (β,q)-Grothendieck polynomials, supersymmetric LLT functions, and metaplectic spherical Whittaker functions. Modelling these functions on a solvable lattice allows us to prove functional equations and identities by using the Yang-Baxter equations associated to the lattice model. Lattice models also encode an immense amount of data from the underlying structure of a space of special functions, and we will examine how different interpretations of this data visualize different properties of the functions, including fundamental connections to quantum groups.