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

Title: Flips

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.

April 28, 2021

Speaker: Brian Pinsky

Title: The Robinson Field and other Ordered Fields

Abstract: Most of you probably saw ordered fields way back in your first analysis class. You probably learned 2 nice examples; R and Q. It turns out, there's actually a lot of interesting examples. There are examples coming from algebraists, analysts, and logicians, with lots of structure which is occasionally useful. I plan to give many examples of different flavors. The Robinson field in particular preserves much of the analytic structure of the hyperreals, but also has a valuation structure. It turned out this was exactly the structure needed to solve a question in geometric group theory. I'm hoping to construct it, and give suitable hypotheses for it to be unique up to isomorphism.

May 5, 2021 **at 7 PM EST**

Speaker: Slava Naprienko (Stanford University)

Title: Tokuyama's Formula — Combinatorics of Deformed Schur Polynomials

Abstract: There is a cute combinatorial formula for the deformation of the Schur polynomial. The deformation behaves fantastically: the formula generalizes Gelfand's parametrization, Jacobi's bialternant formula, and Stanley's Formula on Hall-Littlewood polynomials. Moreover, the arising combinatorics is identical to combinatorics of alternating sign matrices and six-vertex lattice models. In the end, I will explain why everything works so well (spoiler: representations of p-adic groups!). Moreover, I will talk about open problems related to Tokuyama's Formulas for other Lie types.

The talk is based on my post on Thuses.