# General Linear Groups

## Introduction

A general linear group is a group consisting of all invertible square matrices of a fixed size, with the matrix multiplication as the group law.

As their name says, these groups are quite general, since many of the interesting groups (finite groups, classical groups, etc…) are subgroups of general linear groups. Nevertheless, as a particular family of groups, they are quite special. The methods and ideas used to study these groups are very fundamental in algebraic, combinatorial and geometric representation theory.

The goal of this lecture is to study the structure and representation theory of general linear groups. We will focus on different constructions of finite dimensional irreducible representations of these groups:
- combinatorial: Young tableaux;
- algebraic: Schur-Weyl duality;
- geometric: Borel-Weil theory.

On the way of achieving this goal, we plan to cover the following topics: reduction of endomorphisms, nilpotent orbits, bicommutant theorem, Bruhat decomposition, Schubert varieties, etc… If time permits, we will study the tensor product decompositions (Littlewood-Richardson coefficients) or the Springer resolution of nilpotent orbits.

## Informations

• (04.02.2019) If you plan to participate in the Nachklausur, please write me an e-mail to fix the schedule. The Nachklausur will be an oral exam on the same content as in the Klausur.
• (30.01.2019) The solution to Sheet 6, Exercise 2 (3) contains mistakes: to compute a,b,c,d,e,f, we have to take \lambda+\rho_{k-1}=(3,1,1)+(3,2,1)=(6,3,2). Please replace the subscripts (5,3,2,1), (4,3,1), (5,2,1), (4,2), (5,1) and (5) all by (6,3,2). The answer is then (a,b,c,d,e,f)=(0,-2,0,0,0,1). Sorry!
• (14.01.2019) The exercise class on 17.01.2019 will be canceled. Solutions to the exercises will be posted online soon.
• (07.01.2019) Partial solutions to the first two exercise sheets is now available.
• (07.01.2019) The sixth exercise sheet is now available.
• (13.12.2018) The fifth exercise sheet is now available.
• (28.11.2018) The fourth exercise sheet is now available.
• (14.11.2018) The third exercise sheet is now available.
• (14.11.2018) More comments on Exercise 5, Sheet 2: in the part 4, one needs to show two things: the pre-image of I is non-empty, and it is ONE GL_n(C)-orbit.
• (06.11.2018) Exercise 5 in sheet 2: in the beginning of the third line, after "Recall that", "a sub-vector space" is added.
• (31.10.2018) The second exercise sheet is now available.
• (16.10.2018) The first exercise sheet is now available.
• (12.10.2018) The Nachklausur will be on 27.03.2019, 14h in Seminarraum 1.
• (12.10.2018) For those who need 9CP from the lecture, you need to hand in 10 Exercises where: 5 exercises from Exercise sheets 1 and 2; and 5 exercises from Exercise sheets 3, 4 and 5. The deadline of handing in the first 5 exercises is 06.12.2018 in the lecture, and the deadline of handing in the other 5 exercises is 10.01.2019 in the lecture. Among the 10 exercises you choose, there must be at least one long exercise (Exercise 5 in sheets 1-5). Your final grade will be: 85% final exam + 15% homework. NOTE: the deadlines are strict, late submissions will not be accepted.
• (10.10.2018) The exam will be on 31.01.2019, from 15h45 to 17h45 in Hörsaal, MI.
• (18.05.2018) There will be no exercise class for the lecture. Exercise sheets will be distributed every two weeks.
• (18.05.2018) The lectures will be held on Wednesday 14h-15h30 and Thursday 16h-17h30 in Cohen-Vossen Raum (3.13). Office hour will be on Wednesday 15h30-16h30 or by appointment.

## Exercises

Exercise Sheet 1 (16.10.2018)
Exercise Sheet 2 (31.10.2018)
Exercise Sheet 3 (14.11.2018)
Exercise Sheet 4 (28.11.2018)
Exercise Sheet 5 (13.12.2018)
Exercise Sheet 6 (07.01.2019)
Partial solutions 1 (07.01.2019)

## Contents

• Lecture 1 (10.10.2018): Introduction.
• Lecture 2 (11.10.2018): Group action, orbits, stabilizers.
• Lecture 3 (17.10.2018): Invariants, symmetric polynomials.
• Lecture 4 (18.10.2018): Flag variety, rank.
• Lecture 5 (24.10.2018): Rank, Chevalley order.
• Lecture 6 (25.10.2018): Jordan-Dunford-Chevalley decomposition.
• Lecture 7 (31.10.2018): Semi-simple orbits.
• Lecture 8 (07.11.2018): Semi-simple and nilpotent orbits.
• Lecture 9 (08.11.2018): Nilpotent orbits: orbit space.
• Lecture 10 (14.11.2018): Nilpotent orbits: orbit closure.
• Lecture 11 (15.11.2018): Nilpotent orbits: orbit closure. Conjugation orbits.
• Lecture 12 (21.11.2018): Conjugation orbits.
• Lecture 13 (22.11.2018): Conjugation orbit closures, tensor product.
• Lecture 14 (28.11.2018): Tensor product.
• Lecture 15 (29.11.2018): Construction of representations of groups.
• Lecture 16 (05.12.2018): Concrete examples of representations.
• Lecture 17 (06.12.2018): Character theory.
• Lecture 18 (12.12.2018): Character theory.
• Lecture 19 (13.12.2018): Character theory.
• Lecture 20 (19.12.2018): Representations of symmetric groups.
• Lecture 21 (20.12.2018): Exercise class.
• Lecture 22 (09.01.2019): Representations of symmetric groups.
• Lecture 23 (10.01.2019): Representations of symmetric groups.
• Lecture 24 (16.01.2019): Representations of symmetric groups.
• Lecture 25 (17.01.2019): Cancelled (I will be out of town).
• Lecture 26 (23.01.2019): Applications of characters.
• Lecture 27 (24.01.2019): Revision.
• Lecture 28 (30.01.2019): Question session.

## References

• . W. Fulton and J. Harris, Representation Theory: A First Course, Graduate Texts in Mathematics Vol. 129, 2004, Springer.
• . R. Goodman and N. Wallach, Symmetry, Representations, and Invariants. Graduate Texts in Mathematics Vol. 255, 2009, Springer.
• . R. Mneimné and F. Testard, Introduction à la théorie des groupes de Lie classiques. Hermann, 1986.
• . R. Mneimné, Réduction des endomorphismes, Calvage et Mounet, Paris, 2006.

• Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln