University of Minnesota Combinatorics Seminar: 2021-2022

Date: 10/01/2021

Speaker: Gabriel Frieden

Crystal Invariant Theory
Berenstein and Kazhdan's theory of geometric crystals gives rise to four families of rational actions on the space of $m \times n$ matrices. Two of the families consist of $SL_m$ and $SL_n$ geometric crystal operators, which we view as "crystal analogues" of the usual actions of $SL_m$ and $SL_n$ by matrix multiplication. The other two families consist of geometric R-matrices, which we view as crystal versions of the diagonal actions of $S_m$ and $S_n$. We study the fields (and rings) of invariants of all possible combinations of these four actions. Our primary tool is Noumi and Yamada's de-tropicalization of the RSK correspondence. The two rings of geometric R-matrix invariants were shown by Lam and Pylyavskyy to consist of loop symmetric functions, a generalization of symmetric functions in which the variables (and boxes of Young diagrams) come in multiple "colors." We identify algebraically independent sets of loop symmetric functions which conjecturally generate each of the remaining invariant fields. As an application, we give a new derivation of a mysterious piecewise-linear formula for cocharge due to Kirillov and Berenstein. This is joint work with Ben Brubaker, Pasha Pylyavskyy, and Travis Scrimshaw.

Crystal Invariant Theory

Berenstein and Kazhdan's theory of geometric crystals gives rise to four families of rational actions on the space of $m \times n$ matrices. Two of the families consist of $SL_m$ and $SL_n$ geometric crystal operators, which we view as "crystal analogues" of the usual actions of $SL_m$ and $SL_n$ by matrix multiplication. The other two families consist of geometric R-matrices, which we view as crystal versions of the diagonal actions of $S_m$ and $S_n$. We study the fields (and rings) of invariants of all possible combinations of these four actions. Our primary tool is Noumi and Yamada's de-tropicalization of the RSK correspondence. The two rings of geometric R-matrix invariants were shown by Lam and Pylyavskyy to consist of loop symmetric functions, a generalization of symmetric functions in which the variables (and boxes of Young diagrams) come in multiple "colors." We identify algebraically independent sets of loop symmetric functions which conjecturally generate each of the remaining invariant fields. As an application, we give a new derivation of a mysterious piecewise-linear formula for cocharge due to Kirillov and Berenstein. This is joint work with Ben Brubaker, Pasha Pylyavskyy, and Travis Scrimshaw.