Summer Graduate School

Blocks, but not as you know them

How much of the structure of a finite group can be recovered from its representations over a field of positive characteristic? Modular representation theory approaches this question by studying the indecomposable pieces, called blocks, into which the group algebra decomposes.

The summer school will provide a comprehensive introduction to the structure and classification of blocks. Participants will explore the algebraic, homological, combinatorial, categorical, geometric, and computational perspectives that shape the area. Beginning with basic concepts such as defect groups, decomposition numbers, and Cartan matrices, the lectures will develop Morita and derived equivalences and higher representation-theoretic methods, before turning to finite reductive groups and geometric constructions arising from Deligne–Lusztig varieties. Hands-on computational sessions will complement this theoretical material, illustrating how character tables, decomposition matrices, and block invariants can be explored using Julia, and in particular its Chevie package for computations involving finite reductive groups.

Together, these viewpoints will equip participants with the background and tools needed to engage with current research, including major open problems in the field such as Donovan’s Finiteness Conjecture and Broué’s Abelian Defect Conjecture.

School Structure

Each day will consist of two teaching blocks, one in the morning and one in the afternoon. Each lecture will be followed by a collaboration session led by the teaching assistants, devoted to exploring the material through problem solving, discussion, or computational work in Julia. This alternation of lectures and sessions will allow participants to consolidate new concepts immediately and to connect algebraic, geometric, and combinatorial viewpoints throughout the school.

In addition, each day will begin with a 20-minute “triage” session, during which participants can ask questions and clarify material from the previous day.

Prerequisites

Students should be familiar with basic finite group theory (including normal subgroups, quotient groups, group actions, and Sylow theory); rings and modules (including ideals, quotient rings, tensor products, and finite-dimensional algebras); basic category theory (including categories, functors, and natural transformations); elementary homological algebra (including exact sequences, projective modules, and the basic notions of Ext and Tor); and the rudiments of representation theory and character theory of finite groups (including irreducible representations, Maschke’s theorem, characters, and character tables).

The following chapters of David S. Dummit and Richard M. Foote’s "Abstract Algebra", 3rd edition, provide a rough guide to the expected background and level:
    - Chapters 1-5: basic group theory;
    - Chapter 7: basic ring theory;
    - Chapters 10-11: module theory and linear algebra;
    - Chapter 17: homological algebra;
    - Chapter 18: representation theory and character theory.

We also suggest the following selections from P. Etingof et al., "Introduction to Representation Theory", as preparation for the school:
- Sections 2.1-2.5 and 2.11, together with Problems 2.3.16, 2.3.17, and 2.11.16;
- Chapter 3, in particular Problems 3.9.1 and 3.10.1;
- Sections 4.1-4.3, together with Exercise 4.3.1;
- Section 8.1;
- Sections 9.1-9.3.
Section 2.15 will be explored during a problem session. These selections presuppose the algebraic background described above and are intended to introduce or review the representation-theoretic language used during the school. Participants should be able to work through them with reasonable effort, but prior mastery of all the material and exercises is not expected.

Application Procedure

For eligibility and how to apply, see the Summer Graduate Schools homepage.