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Summer Graduate School

New Directions in Representation Theory (AMSI, Brisbane, Australia) June 29, 2020 - July 10, 2020
Parent Program: --
Location: University of Queensland, Brisbane, Australia
Organizers Tim Brown (Australian Mathematical Sciences Institute), Joseph Grotowski (University of Queensland), Chloe Pearse (Australian Mathematical Sciences Institute), Jacqui Ramagge (University of Sydney), Ole Warnaar (University of Queensland), Geordie Williamson (University of Sydney)
Description

Representation Theory has undergone a revolution in recent years, with the development of what is now known as higher representation theory. In particular, the notion of categorification has led to the resolution of many problems previously considered to be intractable.

The school will begin by providing students with a brief but thorough introduction to what could be termed the “bread and butter of modern representation theory”, i.e., compact Lie groups and their representation theory; character theory; structure theory of algebraic groups.

We will then continue on to a number of more specialized topics. The final mix will depend on discussions with the prospective lecturers, but we envisage such topics as:

• modular representation theory of finite groups (blocks, defect groups, Broué’s conjecture);

• perverse sheaves and the geometric Satake correspondence;

• the representation theory of real Lie groups.

Suggested prerequisites

Prerequisites in algebra and representation theory is the material covered in the following texts (or equivalent):

Dummit and Foote, Abstract Algebra
In particular:

  • Part I (Group theory)
  • Part II (Ring theory)
  • Part III (Modules and vector spaces)
  • Part V (Introduction to commutative rings, algebraic geometry, and homological algebra)
  • Part VI (Introduction to the representation theory of finite groups)


Atiyah and Macdonald, Introduction to Commutative Algebra
In particular:

  • Chapter 1 (Rings and ideals)
  • Chapter 2 (Modules)
  • Chapter 3 (Rings and modules of fractions)
  • Chapter 4 (Primary decomposition)
  • Chapter 5 (Integral dependence and valuations)
  • Chapter 6 (Chain conditions)
  • Chapter 7 (Noetherian rings)


James and Liebeck, Representations and Characters of Groups
In particular: 

  • Chapter 13 (Characters)
  • Chapter 16 (Character tables and orthogonality relations)
  • Chapter 19 (Tensor products)
  • Chapter 29 (Permutations and characters)

 

Useful, but non-essential additional reading material may be found in

  • James and Kerber, The Representation Theory of the Symmetric Group
  • Mathas, Iwahori-Hecke Algebras and Schur Algebras of the Symmetric Group
  • Alperin, Local representation theory
  • Humphreys, Introduction to Lie Algebras and Representation Theory
  • Springer, Linear algebraic groups


Any material needed from these additional sources during the lectures and tutorials will be recalled, and is not assumed knowledge.

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

Due to the small number of students supported by MSRI, only one student per institution will be funded by MSRI.

Keywords and Mathematics Subject Classification (MSC)
Tags/Keywords
  • modular representation theory

  • character theory

  • homological algebra

  • symmetric groups

  • Algebraic groups

  • root systems

  • Lusztig’s conjecture

  • higher representation theory

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification