INTEGRABLE COMBINATORICS

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

We consider combinatorial objects such as: plane partitions, alternating sign matrices, tilings, multi-degree of matrix varieties. We approach their enumeration in the light of their common underlying integrable structure. A sign of integrability is the possibility of enumeration via models of weighted non-intersecting paths on graphs. Another is the existence of some underlying quantum group symmetry which allows to perform computations via representation theory. We also address cluster algebras, introduced by Fomin and Zelevinsky, with a wide range of applications in mathematics (Teichm\"uller theory, total positivity, somos-like sequences, quantum groups, triangulated categories) as well as in physics (Donaldson-Thomas invariants, wall-crossing, dimer models). We will concentrate on applications to discrete integrable evolution equations, and try to prove the Laurent positivity conjecture of cluster algebras in this context, by use of graph statistical models. This proposal aims at developing a new field: Integrable Combinatorics. The general idea is to solve combinatorial problems by use of concepts and techniques borrowed from the statistical physics of integrable systems. We have learnt from physics that symmetries determine much of the structure of matter or particles. Analogously, hidden symmetries underlying simple yet hard combinatorial problems can shed a new light and sometimes lead to exact solutions. A large class of combinatorial problems is connected to models of statistical physics which possess rich and sophisticated algebraic structures. The combinatorial problems at hand are just the tip of a huge iceberg, which is rooted in the theory of representations of the underlying symmetry algebras, a machinery that allows to reformulate and hopefully solve the problems in terms of basic matrix theory and linear algebra.