On Combinatorics, Integrability and Puzzles

Abstract

In the last decade, many old and new results in combinatorics have been shown using the theory of quantum integrable systems from particle physics. The key to solving such problems is the derivation of an underlying Yang-Baxter equation. In this thesis, we explore some of the results in this area, focusing on two proofs due to Zinn-Justin in. The first is a proof of Knutson, Tao and Woodward’s puzzle rule which states that Littlewood-Richardson coefficients count the number of tilings of an equilateral triangle with three different types of tiles. The second result concerns Knutson and Tao's product rule for two factorial Schur functions. We present an extension of Zinn-Justin's constructions to Grothendieck polynomials and close with an overview of integrable vertex models. The purpose of this thesis is to make "combinatorics and integrability" more accessible to the general mathematician and illustrate the power and elegance of these ideas.