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Theses

Alternating Sign Matrices, completely Packed Loops and Plane Partitions

Abstract : This thesis is devoted to the study of identities which one observes at the interface between integrable models in statistical physics and combinatorics. The story begins with Mills, Robbins and Rumsey studying the Alternating Sign Matrices (ASM). In 1982, they came out with a compact enumeration formula. When looking for a proof of such formula they discovered the existence of other objects counted by the same formula: Totally Symmetric Self-Complementary Plane Partitions (TSSCPP). It was only some years later that Zeilberger was able to prove this equality, proving that both objects are counted by the same formula. At the same year, Kuperberg, using quantum integrability (a concept coming from statistical physics), gave a simpler and more elegant proof. In 2001, Razumov and Stroganov conjectured one intriguing relation between ASM and the ground state of the XXZ spin model (with Delta=-1/2), also integrable. This conjecture was proved by Cantini and Sportiello in 2010. The principal goal of this manuscript is to understand the role of integrability in this story, notably, the role played by the quantum Knizhnik-Zamolodchikov equation. Using this equation, we prove some combinatorial conjectures. We prove a refined version of the equality between the number of ASM and TSSCPP conjectured in 1986 by Mills, Robbins and Rumsey. We prove some conjectured properties of the components of the XXZ groundstate. Finally we present new conjectures concerning the groundstate.
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Contributor : Tiago Fonseca <>
Submitted on : Wednesday, November 17, 2010 - 10:18:37 PM
Last modification on : Thursday, December 10, 2020 - 10:54:50 AM
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Tiago Fonseca. Alternating Sign Matrices, completely Packed Loops and Plane Partitions. Mathematical Physics [math-ph]. Université Pierre et Marie Curie - Paris VI, 2010. English. ⟨tel-00521884v2⟩

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