Title: Crystal invariant theory and geometric RSK
| Speaker: | Gabriel Frieden |
| Affiliation: |
Université du Québec à Montréal (UQAM) |
| Zoom: | Contact Olya Mandelshtam |
Abstract:
The original problem of classical invariant theory was to describe the invariants of SL_m acting on a polynomial ring in an m \times n matrix of variables. One way to solve this problem is to consider the polynomial ring as a GL_m \times GL_n representation, and decompose this representation into its irreducible components.
Berenstein and Kazhdan's theory of geometric crystals gives rise to two families of rational actions on the space of m \times n complex matrices, which we view as "crystallized versions" of the usual GL_m and GL_n actions. We describe the invariants of these two families of actions. Our main tool is Noumi and Yamada's geometric lifting (or de-tropicalization) of the RSK correspondence, which is analogous to the above-mentioned irreducible decomposition in the classical setting. This is joint work with Ben Brubaker, Pasha Pylyavskyy, and Travis Scrimshaw.
I will not assume prior knowledge of crystals or geometric crystals; the basic building blocks of these theories will be introduced through examples.