Combinatorics Reading Seminar: Ice Models for Type A

Seminar | October 16 | 11:10 a.m.-12:10 p.m. | 748 Evans Hall

 Yulia Alexandr, UC Berkeley

 Department of Mathematics

This talk will focus on combinatorial objects called ice models, which arise in statistical mechanics. We will start by exploring the relationship between semi-standard Young tableaux and Gelfand-Tsetlin patterns, and see how the Shur polynomial can be defined in terms of those objects. In general, given rules for a tableaux representing a branching rule for GL(n, C), we define a bijection between the tableaux and Gelfand-Tsetlin patterns. Restricting our attention to only strict Gelfand-Tsetlin patterns and the corresponding so-called “shifted” tableaux, we can construct a certain ice model in bijection with those objects. Assigning appropriate weights to the vertices of the resulting ice model, we obtain a partition function that is equal to the product of the type A deformation formula and the character of GL(n,C), which is precisely the Shur polynomial. We will sketch a proof that the weights proposed by Brubaker, Bump and Friedberg give the desired equation using the star-triangle identity.