In this series of talks, I will give an introduction to some of the ideas of “categorification” which have created a new point of view in representation theory centered around various monoidal categories of a diagrammatic nature. I will likely start by discussing classical examples such as the Temperley-Lieb and HOMFLY-PT skein categories, before focussing on the Kac-Moody 2-category of Khovanov, Lauda and Rouquier. Many of the categories appearing in classical representation theory, especially of symmetric and general linear groups, admit additional structure making them into module categories (“2-representations”) over the Kac-Moody 2-category. This has consequences both at a combinatorial level (related to crystals and labelling sets of irreducible modules) and at a categorical level (related to the construction of Morita and derived equivalences between blocks)

In these lectures we provide an introduction to Lusztig's classification
of the irreducible characters of a finite group of Lie type. This essentially relies on
structural properties of the underlying algebraic group, which will be surveyed
in the first lecture. We then go on to discuss the partition of the set of characters
into series and the Jordan decomposition of characters. Finally, we address the
problem of computing character values, in the framework of Lusztig's theory of
character sheaves."

 

This series of lectures will be centered on decomposition numbers for a special class of finite groups such as GL_n(q), SO_n(q),... E_8(q). I will first present what kind of numerical invariants decomposition numbers are, and what representation-theoretic problems they can solve. For finite reductive groups, I will explain how one can use Deligne--Lusztig theory to get basic sets of ordinary characters and to compute decomposition numbers. If time permits, I will mention a few open problems, including the case of small characteristic.

Lecture 1 - Generalities on decomposition numbers

Lecture 2 - Basic sets for finite reductive groups

Lecture 3 - Computing decomposition numbers

This series of lectures will be centered on decomposition numbers for a special class of finite groups such as GL_n(q), SO_n(q),... E_8(q). I will first present what kind of numerical invariants decomposition numbers are, and what representation-theoretic problems they can solve. For finite reductive groups, I will explain how one can use Deligne--Lusztig theory to get basic sets of ordinary characters and to compute decomposition numbers. If time permits, I will mention a few open problems, including the case of small characteristic.

Lecture 1 - Generalities on decomposition numbers

Lecture 2 - Basic sets for finite reductive groups

Lecture 3 - Computing decomposition numbers

In this series of talks, I will give an introduction to some of the ideas of “categorification” which have created a new point of view in representation theory centered around various monoidal categories of a diagrammatic nature. I will likely start by discussing classical examples such as the Temperley-Lieb and HOMFLY-PT skein categories, before focussing on the Kac-Moody 2-category of Khovanov, Lauda and Rouquier. Many of the categories appearing in classical representation theory, especially of symmetric and general linear groups, admit additional structure making them into module categories (“2-representations”) over the Kac-Moody 2-category. This has consequences both at a combinatorial level (related to crystals and labelling sets of irreducible modules) and at a categorical level (related to the construction of Morita and derived equivalences between blocks)

The lectures will provide an introduction to the the theory of p-blocks of finite groups . The emphasis will be on the interplay between p-local invariants and global structure.