Representation Theory of Finite Groups, spring 2019

Organisation

Lectures: Mondays, 11:00–12:45, Snellius, room 412
4 February–20 May, except 11 March, 15 and 22 April
Lecturer: Peter Bruin, office 229, P.J.Bruin@math.leidenuniv.nl
Teaching assistant: Kevin van Helden, office 207
Office hours: Fridays, 15:00–16:00
E-mail for questions and
handing in homework:		 reptheory2019@gmail.com
Exam: Monday 3 June, 10:00–13:00, Huygens, room 106–109
Resit: Thursday 27 June, 10:00–13:00, Snellius, room 401

The final mark will be based on regular hand-in exercises (25%) and a written exam (75%). The mark for the exam must be at least 5.0 and the overall mark at least 5.5.

The exam will cover the material treated in the lectures and the exercises. You may consult books and lecture notes. The use of electronic devices is not allowed.

A practice exam together with example solutions is available.

The exam of 3 June and the retake of 27 June are available. The marked exams may be inspected (ingezien) in office 229 of the Snellius building on Thursday 11 July between 11:00 and 12:00, or by appointment.

Description

Representation theory is about understanding and exploiting symmetry using linear algebra. The central objects of study are linear actions of groups on vector spaces. This gives rise to a very structured and beautiful theory, which can be viewed as a generalisation of Fourier analysis to a non-commutative setting.

Representation theory plays a major role in mathematics and physics. For example, it provides a framework for understanding finite groups, special functions, and Lie groups and algebras. In number theory, Galois groups are studied via their representations; this is closely related to modular forms. In physics, representation theory is the mathematical basis for the theory of elementary particles.

After introducing the concept of a representation of a group, we will study decompositions of representations into irreducible constituents. A finite group only has finitely many distinct irreducible representations; these are encoded in a matrix called the character table of the group. One of the goals of this course is to use representation theory to prove Burnside's theorem on solvability of groups whose order is divisible by at most two prime numbers. Along the way, we will also meet categories, modules and tensor products.

Literature

Of the above references, the contents of this course are closest to Lenstra's notes.

Programme

Date Material Etingof Lang Lenstra Serre Exercises
4 February Introduction and motivation.
Solvable groups.
Representations, algebras and modules.		 §2.1–2.3 §II.3, §III.1, §XVIII.1 §1–4 §1.1–1.2, §6.1 Problem Sheet 1
11 February Representations as k[G]-modules.
Exact sequences.		 §2.3–2.5 §III.1–3 §4–5 §6.1 Problem Sheet 2
18 February Semi-simple modules.
Composition chains, semi-simplification.		 §3.1, §3.4, §3.7 §I.3, §III.8, §XVII.2 §7, §9 — Problem Sheet 3
25 February Semi-simplification (continued).
Categories and functors.
Universal properties of sums, products and (co)kernels.		 §7.1–7.2 §I.3, §I.11 §7, §6 — Problem Sheet 4
4 March Abelian categories.
Exactness, Hom functors.		 §7.7, §7.9 §III.3 §6 — Problem Sheet 5
18 March Tensor products.
Maschke's theorem, semi-simple rings. 		§2.11, §4.1 §XVI.1–2, §XVII.2, §XVII.4, §XVIII.1 §6, §9 §1.4–1.5 Problem Sheet 6
25 March Structure theorem for semi-simple rings.
Applications to classifying k[G]-modules.		 §3.3, §3.5, §4.1 §XVII.1, §XVII.5, §XVIII.4 §9 §2.4–2.5, §6.2–6.3 Problem Sheet 7
1 April Characters.
(Lecture by Roland van der Veen.)		 §4.2, §4.4–4.5 §XVIII.2, §XVIII.4–5 §10 §2.1, §2.3–2.5 Problem Sheet 8
8 April Characters (continued).
Interpretation of the character table.		 §4.5, §4.8 §XVIII.4–5 §10 §2.3–2.5 Problem Sheet 9
29 April Algebraic integers.
Divisibility of character values.		 §5.2–5.3 §VII.1 §11 §6.4–6.5 Problem Sheet 10
6 May Proof of Burnside's theorem.
Frobenius's theorem.		 §5.4 — §11–12 — Problem Sheet 11
13 May Proof of Frobenius's theorem.
Induced representations.		 §5.8, §5.10 §XVIII.7 §12, §14 §7.1 Problem Sheet 12
20 May Induced representations (continued). §5.9–5.10 §XVIII.7 §14 §7.2 Practice exam

Homework

The exercises listed below should be handed in on the due date at the beginning of the lecture (11:00). You may hand the homework in on paper or by e-mail to reptheory2019@gmail.com.

Due date Hand-in exercises
18 February Problem sheet 1: 5, 13
Problem sheet 2: 4, 10
4 March Problem sheet 3: 5, 11
Problem sheet 4: 5, 8
25 March Problem sheet 5: 7, 11
Problem sheet 6: 5, 12
815 April Problem sheet 7: 1, 8
Problem sheet 8: 2, 10
6 May Problem sheet 9: 1, 5
Problem sheet 10: 5, 8
20 May Problem sheet 11: 3, 6
Problem sheet 12: 7, 11