Student Algebra and Representation Theory Seminar (StARTS)

Autumn 2025

0. Week 0 (September 22): Justin Bloom
   • Title: What is representation theory?
   • Abstract: We will give an overview of the central topics of representation theory from a modern perspective. We will discuss character theory for finite groups and where it goes wrong, and we will briefly discuss Lie groups, group schemes, and Lie algebras.
1. Week 1 (September 29): Justin Bloom
   • Title: Group Schemes, (co)induction, restriction, and Frobenius reciprocity
   • Abstract: We will discuss properties of some of the most fundamental operations in the representation theory of groups and of group schemes, including Frobenius reciprocity, featuring connections with algebraic geometry, and homological perspectives.
2. Week 2 (October 6): Wolfgang Allred
• Title: The Lyndon-Hochschild-Serre Spectral Sequence for Group Schemes
• Abstract: An essential tool for computing group cohomology, the Lyndon-Hochschild-Serre(LHS) spectral sequence is a special case of the Grothendieck spectral sequence that relates the group cohomology of a group G to the cohomology of its normal subgruops and associated quotients. We will examine the LHS in the context of group schemes, but this will first require us to talk a little bit about the subtleties involved in the quotients of group schemes. After this, we will examine some of the applications of the LHS.
3. Week 3 (October 13): Charlie Magland
• Title: What is a Reductive Group? A World Record Attempt!
• Abstract: I will be the first to attempt the challenge posed by Justin Bloom on September 22nd, 2025. Along the way we will learn what a reductive group is, some examples, and why we care about them.
4. Week 4 (October 20): Ting Gong
   • Title: What's the problem with taking quotients?
   • Abstract: As we have noted earlier in this seminar, taking quotients can be quite painstaking. There is no guarantee that the naive functor G/H gives a group scheme, and there is even no information on whether the universal property guarantees the existence of group schemes. What do we need to make this work? In this talk, we are going to briefly include Grothendieck topologies and conditions for making quotients work nicely, so that we can put our hearts back to our chest.
5. Week 5 (October 27): Monty McGovern
   • Title: Introduction to the flag variety
   • Abstract: Following Fulton's treatment in his book Young Tableaux, I will define the flag variety F_n (for the general linear group GL_n) and give explicit defining equations for it. Using Young tableaux, I will show how these relations lead naturally to the realization of all finite-dimensional polynomial representations of this group, showing how these appear in the coordinate ring of F_n. I will also briefly indicate how the equations generalize to other classical (i.e., symplectic and orthogonal) groups.
6. Week 6 (November 3): Justin Bloom
   • Title: Finite flat group schemes and cohomology
   • Abstract: We will discuss the geometry of finite flat group schemes and some beautiful applications of moduli theory to representation theory. We'll discuss a sixty year history of finite generation results in cohomology, culminating in van der Kallen's recent work over a noetherian base ring.
7. Week 7 (November 10): Ian Martin
   • Title: Introduction to quantum groups
   • Abstract: Since its establishment in the 1980s, the theory of quantum groups has become a rich subject, with many connections to other areas of mathematics. We'll cover some of the basic results on the structure of quantum groups, the classification of their finite dimensional modules (for q not a root of unity), and their quasitriangular structure. If time permits, we'll aslo briefly discuss some of their applications to knot theory.
8. Week 8 (November 17): Andrew Aguilar
   • Title: A green way to locally source indecomposable modules
   • Abstract: The representation theory of a finite group G, over a field k of characteristic not dividing the order, is well understood and relies solely on classifying the finite list of simple modules. In the modular case simple and indecomposable no longer coincide and the list of indecomposables may not be finite. How does one even begin to classify the indecomposables? One approach is to do so locally. We will attempt to prove the Green correspondence, which gives a correspondence between the indecomposables of G and the indecomposables of its p-local subgroups. This provides a local-to-global approach to studying the representation theory of G by its p-subgroups which, often, have simpler representation theory.
9. Week 9 (November 24): TBA
10. Week 10 (December 1): TBA