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 
Email 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 handin 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.
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 noncommutative 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.
Of the above references, the contents of this course are closest to Lenstra's notes.
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  Semisimple modules. Composition chains, semisimplification. 
§3.1, §3.4, §3.7  §I.3, §III.8, §XVII.2  §7, §9  —  Problem Sheet 3 
25 February  Semisimplification (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, semisimple 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 semisimple 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 
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 email to reptheory2019@gmail.com.
Due date  Handin 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 
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 