Title: Discrete Integrable Systems and Cluster Algebras
Speaker: Philippe Di Francesco, CEA-Saclay
Date: Tuesday, April 2, 2013
Time: 1:00pm – 2:00pm
Place: Lecture Hall 102, Simons Center
View the Slides.
Abstract: We discuss discrete time non-linear evolution equations arising from the study of quantum integrable spin chains in physics, and show how they fit into the cluster algebra structure of Fomin and Zelevinsky. The latter is a sort of dynamical system on formal coordinate patches that are transformed according to rational mutation rules, guaranteeing the Laurent property that each mutated coordinate is a Laurent polynomial of the variables in any other coordinate patch. We shall focus on the open positivity conjecture of cluster algebra, stating that these Laurent polynomials have only non-negative integer coefficients, and explore non-commutative generalizations.
The combinatorial content of our evolution equations is revealed by their exact solutions, which make use of their discrete integrable structure to express them as partition functions for statistical models of paths on target graphs. Cluster algebra mutations are implemented by local transformations in various guises: local continued fraction rearrangements, matrix representation local moves à la Yang-Baxter, etc. Positivity comes as a consequence of positivity of the Boltzmann weights of the underlying statistical models.
(based on work in collaboration with R. Kedem).