April 21, 2021
Speaker: Brian Pinsky
February 10, 2021
Speaker: Jason Saied
Title: Connections in type A representation theory: Schur polynomials, GL_n, and S_n
Abstract: I will give an overview of the connections between a few different topics in type A representation theory. I will introduce Schur polynomials, the representation theory of the general linear group GL_n, and the representation theory of the symmetric group S_n. We will discuss (mostly without proof) the connections between each pair of these objects: Schur polynomials and GL_n through character theory, Schur polynomials and S_n through the characteristic map, and GL_n and S_n through Schur-Weyl duality. I won't assume anything beyond first-year algebra.
In the fall, I gave a detailed talk on the characteristic map connecting symmetric groups and Schur functions. It will not be necessary in order to understand this week's talk, but if you're interested, you can find the video and slides here.
March 10, 2021
Speaker: Terence Coelho
Abstract: Begin with some set of hyperplanes in R^n. Reflecting these hyperplanes over each other and iterating will almost always cover the entire space densely. If it doesn't, we call the group generated by reflections over these hyperplanes a Reflection Group.
Reflection groups have many beautiful properties that allow us to classify them fully. From a representation theory perspective, the infinite reflection groups are precisely the Affine Weyl groups (in fact, they can be thought of as a motivation for Affine Lie algebras) and the finite reflection subgroups of infinite reflection groups are precisely the Weyl groups.
No background is needed for this talk; we will only mention Lie algebras in passing.
March 24, 2021
Speaker: Yael Davidov
Title: The Brauer Group through the Lens of Crossed Product Algebras
Abstract: In this talk I will define (and hopefully motivate a bit) the Brauer group of a field. I will also prove(ish) that it is in fact a group! Then we're going to take an abrupt turn and talk about crossed product algebras for a bit in the hopes that I can hint at the connection between Brauer groups and Galois cohomology at the end of the talk. This talk should be completely accessible and will only assume knowledge of some first year algebra.
April 7, 2021
Speaker: Tamar Lichter Blanks
Title: Schur's Lemma and Beyond
Abstract: We often think of Schur's lemma as a fact about representations... but is it? In this talk, we'll start by stating and proving a classic version of Schur's lemma, and see how it tells us something surprising about quadratic forms. Then we will broaden our perspective and see that Schur's lemma runs much deeper than just representation theory (in fact it holds in any abelian category!), and look at an example of Schur's lemma applied to sheaves. We'll also mention a nice connection to division algebras and the Brauer group.