Geometric Lattice Models And Irrational Conformal Field Theories PDF Download

Integrability theory has long been a productive area of research in mathematical physics, particularly in statistical mechanics. Originally introduced to explain the residual entropy of water ice, integrable lattice models have subsequently found numerous applications across diverse mathematical domains, including algebraic combinatorics, integrable probability, special functions, the representation theory of $p$-adic groups, and conformal field theory. This thesis delves into the combinatorics of integrable lattice models. The distinctive combinatorics arises from the integrability condition that is manifested in the Yang-Baxter equation. We employ the resulting combinatorial framework to establish applications in the theory of special functions and representation theory. The work presented here is organized into three distinct sections. The first part revolves around the six vertex model with integrable free fermionic weights. We utilize this model to introduce a novel family of Schur functions, which are dependent on two sets of variables and two sets of parameters. This newly presented family both generalizes and unifies diverse families of Schur functions from the literature, providing a consistent framework for studying their combinatorics. The second section continues our exploration with the six vertex model, investigating its integrability independently. We provide a complete solution to the parametrized Yang-Baxter equation within the context of this model. Our results unveil an unexpected algebraic structure within these solutions, forming a groupoid in relation to the operation that resolves the Yang-Baxter equation. The third and final section offers a concise overview of the application of an alternative lattice model, the bosonic lattice models, to the representation theory of $p$-adic groups. We clarify how the refined bosonic models, termed the colored bosonic lattice models, yield values of the spherical-Iwahori matrix coefficients for the general linear group over nonarchimedean local fields. We also demonstrate that these colored bosonic models satisfy the local lifting property, which allows us to establish a connection with the uncolored bosonic lattice models and provide new proofs to many known results.