# GSRTS

#### UCR Graduate Student Representation Theory Seminar

This is the website for the grad-student run Graduate Student Representation Theory Seminar (GSRTS) at UC Riverside, which meets on Thursdays from 12:30-1:50pm. If you wish to attend, please contact Raymond Matson (email located at the bottom of the page) for any questions or comments concerning the seminar. GSRTS is an extension of the Lie Theory Seminar that is run on Tuesdays from 12:30-1:50pm. The website for that seminar is available at here.

#### Upcoming/Past Speakers

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# Fall 2022

• November 17, 2022:
• Title:
• Abstract:

• November 3, 2022: Anthony Muljat
• Title: Representation stability for generalized hyperoctahedral subgroups of wreath products
• Abstract: A general goal in the field of representation stability is to show that, given an ascending chain of groups $$\{G_n\}$$ and a field $$k$$, any sequence $$\{V_n\}$$ of finite-dimensional representations of $$G_n$$ over $$k$$ that satisfies a certain finite generation condition is representation stable. This result holds when $$G_n$$ are the Weyl groups for types $$B$$/$$C$$, and $$k$$ is algebraically closed and of characteristic 0. Wilson (2014) showed that this property descends to the Weyl groups for type $$D$$. In this talk, we will discuss our application of Wilson’s method to the wreath products $$A \sim S_n$$ and analogous subgroups. In doing so, we generalize the stability result from the case where $$A = \mathbb{Z}_2$$ to an arbitrary finite abelian group $$A$$.

• October 27, 2022: Joseph Wagner
• Title: Generalized Demazure Modules of the Twisted Current Algebra $$^2\tilde{A}_{2\ell-1}$$
• Abstract: In this talk, I will first give some background information on Lie algebras, including some of their properties and how they can be classified. I'll then introduce twisted Lie algebras, affine Lie algebras, and current algebras, and explain how they are constructed. Finally, I'll define certain types of Lie algebra representations called Demazure modules, as well as a certain family of generalized Demazure modules which my research focuses on. I'll end the talk with a brief look at what I'm trying to show in my research, and some of the progress I've made so far.

• October 13, 2022: Raymond Matson
• Title: Stated Skein Modules and DAHAs
• Abstract: Some knot invariants come from looking at highly noncommutative associated groups. As these groups can be incredibly difficult to work with, one can instead consider corresponding commutative algebras and representations. However, if you want to still extract knot invariants you need to quantize these algebras and skein theory provides a breathable way to understand these deformations. In 2012, Berest and Samuelson provided a geometric way to understand these invariants and in the process uncovered certain defining modules for a bigger underlying beast, double affine Hecke algebras. I will discuss these module structures and how they act in the context of a newer, more general skein theory recently established by Thang Lê. This new theory, called stated skein theory, provides significant additional algebraic structure to these algebras and modules and will hopefully lead to more insights into a nicer presentation of these DAHAs.

• October 6, 2022: Raymond Matson
• Title: Two Truths and a Lie
• Abstract: In this talk we will go over fundamental definitions and ideas in Lie theory that anyone going into Lie theory or representation theory will learn about at some point. My goal is to give a small and hopefully comfortable crash course on some topics in the Lie Algebras electives offered here at UCR and if there's time, introduce a couple of the topics that the algebraists here (in particular us graduate students) are interested in.