# Module MA3416: Group Representations

**Term taught**

**Lecturer**

E-mail: lebed (at) maths.tcd.ie

Office: 1.5, School of Mathematics

Any questions or remarks or calls for help are welcome!

**Schedule**Wed 4pm Sal1

Thu 11am Syn2

**Tutorials and homework**

Tutor: Christian Marboe

E-mail: marboec (at) maths.tcd.ie

Homework 3 is already marked. You are welcome to collect your scripts from my office!

Problem sheets: T1, T2, T3, T4.

Problem sheets with partial solutions: T1A, T2A, T3A. Corrections, suggestions and questions are welcome!

Past homework and quizzes, with solutions: HW1, HW1A, Q1, Q1A, HW2, HW2A, Q2, Q2A, HW3, HW3A.

A remark on HW3: if you want to understand where the 3-dimensional representations of A

_{5}come from, have a look here and here.

**Continuous assessment**

= 20% of the final mark

**Suggested reading**

Main references:

- J.-P. Serre, "Linear representations of finite groups"

Clear, concise and efficient. Several copies are provided by the School of
Mathematics. You can borrow them from me when necessary.

- W. Fulton, J. Harris, "Representation theory: a first course"

An example-driven approach. More detailed, and might be easier to read.

Additional references:

- P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, E. Yudovina, "Introduction to representation theory" [PDF]
- C. Curtis, I. Reiner, "Representation Theory of Finite Groups and
Associative Algebras"

- The philosophy of representation theory. First examples. Notes

- Representations of S
_{3}seen geometrically and algebraically. Notes - Building blocks: irreducible representations. Notes
- Characters as a tool for tracing representations: introduction. Notes
- Characters as a tool for tracing representations: basic results. Notes
- Character tables: an example and an algorithm. Notes
- Representations of abelian and "close to abelian" groups. Examples: cyclic groups. Notes
- Relations between irreducible representations. Schur's lemma.
- Irreducible characters form a basis of the space of class functions: proof.
- Schur's orthogonality relations. Isotypic components. Notes for lectures 8-10.
- Tensor products of vector spaces.

- Tensor products of representations. Notes for lectures 11-12 and 15.
- Representations of symmetric groups and Young diagrams: introduction.
- Representations of symmetric groups and Young diagrams: more techniques and examples.

- Representations of S
_{5}. Notes for lectures 13-15. - Representations of symmetric groups and Young diagrams: some proofs.
- An algorithm for constructing the character table of S
_{n}for general n. Notes for lectures 16-17. - More on characters of S
_{n}: Frobenius formula. Notes - Relating representations of different groups: induction and restriction.
- Induction and restriction for subgroups of index 2.

- Representations of the alternating groups A
_{n}. Notes for lectures 19-21. - Braid groups.

- Burau representations for braid groups.
- How much do we know about the representation theory of braid groups?
- Character varieties. Applications in topology: invariants of braids and knots. Notes for lectures 22-25.
- Representations of the general linear groups GL
_{n}.

- Schur-Weyl duality. Notes for lectures 26-27.