Prospectus

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Representation theory of finite groups (BM)

Course 2018-2019

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.

Examination

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.

Prerequisites

Linear Algebra 1 and 2, Algebra 1.
Recommended: Algebra 2.

Recommended literature

Pavel Etingof, Introduction to Representation Theory.
American Mathematical Society, 2011.
Freely available on the author's website

Serge Lang, Algebra.
Springer-Verlag, 2002.
Chapter XVIII is about representations of finite groups.

Hendrik Lenstra, Representatietheorie.
Lecture notes (in Dutch).

Jean-Pierre Serre, Représentations lineaires des groupes finis.
Hermann, Paris, 1967.

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