University of Minnesota Combinatorics Seminar: 2021-2022

Date: 11/12/2021

Speaker: Iva Halacheva

Crystals and Cacti in Representation Theory
One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. While the original representations carry an action of the braid group, their crystals carry an action of a closely related group known as the cactus group. I will describe how we can realize this action using generalized Schützenberger involutions, as well as its properties under $GL(n) \times GL(m)$ skew Howe duality for crystals. I will also give an overview of how this action can be realized geometrically, as a monodromy action coming from a family of "shift of argument" algebras, as well as categorically through certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.

Crystals and Cacti in Representation Theory

One approach to studying the representation theory of Lie algebras and their associated quantum groups is through combinatorial shadows known as crystals. While the original representations carry an action of the braid group, their crystals carry an action of a closely related group known as the cactus group. I will describe how we can realize this action using generalized Schützenberger involutions, as well as its properties under $GL(n) \times GL(m)$ skew Howe duality for crystals. I will also give an overview of how this action can be realized geometrically, as a monodromy action coming from a family of "shift of argument" algebras, as well as categorically through certain equivalences on triangulated categories known as Rickard complexes. Parts of this talk are based on joint work with Joel Kamnitzer, Leonid Rybnikov, and Alex Weekes, as well as Tony Licata, Ivan Losev, and Oded Yacobi.