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
4. Week 4 (October 20): Ting Gong
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): TBA
8. Week 8 (November 17): TBA
9. Week 9 (November 24): TBA
10. Week 10 (December 1): TBA