Representation Theory

Recommended Prerequisites

Familiarity with the basic concepts and objects of algebra: linear and multilinear algebra, groups and rings.

Teaching Methods

Lectures have an expository character, being up to the professor the choice of the most appropriate way to do it and the degree of participation of the students. As an integral part of the learning process, it may be recommended or required  the solution of exercises, course projects or oral presentations.

Learning Outcomes

The goal of this course is to provide a general knowledge of Representation Theory at an advanced level. It is intended that the students become familiar with the main basic techniques and results of this area of Algebra, or at least attain enough familiarity with some of  them to be able to acquire others by themselves that may later prove to be useful.

Work Placement(s)

No

Syllabus

“Very roughly speaking, representation theory studies symmetry in linear spaces.” [Etingof et al.].

Representation theory has applications to group theory, combinatorics, number theory, probability, geometry and physics.

The syllabus will vary from year to year. The following items cover the basic aspects of representation

theory:

1.Associative algebras,quivers and path algebras, irreducible and indecomposable representations, semisimple algebras, Jordan-Holder and Krull-Schmidt Theorem, finite-dimensional algebras.

2. Quivers: indecomposable representations of type A1, A2, A3, D4. Simply laced root systems, Gabriel's theorem.

3. Lie groups and algebras: Classification of semisimple Lie algebras. Quantized enveloping algebras.

4. Finite groups: Maschke's theorem, characters,Burnside's theorem, induced representations, Mackey formula,Frobenius reciprocity.

5. Symmetric and general linear group:Schur-Weyl duality, first fundamental theorem of invariant theory.

Assessment Methods

Assessment
Synthesis work: 20.0%
Exam: 40.0%
Resolution Problems: 40.0%

Bibliography

I. M. Isaacs, Character theory of finite groups. AMS Chelsea Publishing, Providence, RI, 2006.

C.W. Curtis and I. Reiner, Representation theory of finite groups and associative algebras, AMS Chelsea Publishing, Providence, RI, 2006.

C.W, Curtis and Irving Reiner, Methods of representation theory - with applications to finite groups and orders, vol 1, John Wiley & Sons, New York, 1990.

G. James and M. Liebeck, Representations and characters of groups, 2nd edition, Cambridge University Press, New York, 2001.

I. Assem, D. Simson, and A. Skowronski, Elements of representation theory of associative algebras, vol 1: Techniques of representation theory, Cambridge University Press, Cambridge, 2006.

W. Fulton and J. Harris, Representation Theory: A First Course, Graduate texts in mathematics, vol. 129, Springer-Verlag, New York, 1991.

J.P. Serre, Linear Representations of Finite Groups, Graduate texts in mathematics, vol. 42, Springer-Verlag, New York, 1977.

K. Erdmann and M.J. Wildon, Introduction to Lie algebras, Springer Undergraduate Mathematics Series, Springer-Verlag, London, 2006.

P. Etingof, O. Golberg, S. Hensel, T. Liu, A. Schwendner, D. Vaintrob, and E. Yudovina, Introduction to Representation Theory, Student Mathematical Library, vol. 59, American Mathematical Society, Providence, RI, 2011.

J.E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9,  Springer-Verlag, New York-Berlin, 1978.