LIC

"Loop models, integrability and combinatorics"

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 Obiettivo del progetto (Objective)

'The purpose of this proposal is to investigate new connections which have emerged in the recent years between problems from statistical mechanics, namely two-dimensional exactly solvable models, and a variety of combinatorial problems, among which: the enumeration of plane partitions, alternating sign matrices and related objects; combinatorial properties of certain algebro-geometric objects such as orbital varieties or the Brauer loop scheme; or finally certain problems in free probability. One of the key methods that emerged in recent years is the use of quantum integrability and more precisely the quantum Knizhnik--Zamolodchikov equation, which itself is related to many deep results in representation theory. The fruitful interaction between all these ideas has led to many advances in the last few years, including proofs of some old conjectures but also completely new results. More specifically, loop models are a class of statistical models where the PI has made significant progress, in particular in relation to the so-called Razumov--Stroganov conjecture (now Cantini--Sportiello theorem).

New directions that should be pursued include: further applications to enumerative combinatorics such as proofs of various open conjectures relating Alternating Sign Matrices, Plane Partitions and their symmetry classes; a full understanding of the quantum integrability of the Fully Packed Loop model, a specific loop model at the heart of the Razumov--Stroganov correspondence; a complete description of the Brauer loop scheme, including its defining equations, and of the underlying poset; the extension of the work on Di Francesco and Zinn-Justin on the loop model/6-vertex vertex relation to the case of the 8-vertex model (corresponding to elliptic solutions of the Yang--Baxter equation); the study of solvable tilings models, in relation to generalizations of the Littlewood--Richardson rule, and the determination of their limiting shapes.'