# Summer Graduate School

Parent Program: |
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Location: |
University of Queensland, Brisbane, Australia |

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

Lusztigâ€™s conjecture

higher representation theory

**Primary Mathematics Subject Classification**

**Secondary Mathematics Subject Classification**