Combinatorics of Symmetric Functions through Lattice Models

Abstract

Symmetric functions emerge in many fields of mathematics, serving as characters in representation theory, polynomial representatives in the cohomology rings of varieties in algebraic geometry, and generating series in enumerative combinatorics, among others. Beyond these connections, the ring of symmetric functions itself contains a rich combinatorial theory that propels their systematic study. This PhD thesis explores the application of exactly solvable lattice models to symmetric functions in four distinct papers. Paper 1: Vertex models of canonical Grothendieck polynomials and their duals We study exactly solvable lattice models associated to canonical Grothendieck polynomials and their..