- Lecturer: Konstantin Ardakov
General Prerequisites:
Rings and Modules is essential. Group Theory is recommended.
Course Term: Michaelmas
Course Lecture Information: 16 lectures
Course Weight: 1
Course Level: H
Assessment Type: Written Examination
Course Overview:
This course gives an introduction to the representation theory of finite groups. Representation theory is a fundamental tool for studying symmetry by means of linear algebra: it is studied in a way in which a given group may act on vector spaces, giving rise to the notion of a representation.
The first part of the course will deal with the structure theory of semisimple algebras and their modules (representations). We will prove that any finite-dimensional semisimple algebra is isomorphic to a product of matrix rings (Wedderburn’s Theorem).
In later parts of the course we apply the developed material to group algebras, and classify when group algebras are semisimple (Maschke’s Theorem). All of this material will be applied to the study of characters and representations of finite groups.
The first part of the course will deal with the structure theory of semisimple algebras and their modules (representations). We will prove that any finite-dimensional semisimple algebra is isomorphic to a product of matrix rings (Wedderburn’s Theorem).
In later parts of the course we apply the developed material to group algebras, and classify when group algebras are semisimple (Maschke’s Theorem). All of this material will be applied to the study of characters and representations of finite groups.
Learning Outcomes:
They will know in particular simple modules and semisimple algebras and they will be familiar with examples. They will appreciate important results in the course such as Schur’s Lemma, Wedderburn’s Theorem, the Row Orthogonality Theorem and Burnside’s \(p^\alpha q^\beta\) Theorem. They will be familiar with the classification of semisimple algebras over \({\mathbb C}\,\) and be able to apply this to representations and characters of finite groups.
Course Synopsis:
Representations of groups. Maschke’s Theorem. The group ring.
Modules and their relationship with representations. Semisimple algebras, Schur’s Lemma and the Wedderburn Theorem. Characters of complex representations. Orthogonality relations, finding character tables. Tensor product of representations. Induction and restriction of representations. Application: Burnside’s \(p^\alpha
q^\beta\) Theorem.
Modules and their relationship with representations. Semisimple algebras, Schur’s Lemma and the Wedderburn Theorem. Characters of complex representations. Orthogonality relations, finding character tables. Tensor product of representations. Induction and restriction of representations. Application: Burnside’s \(p^\alpha
q^\beta\) Theorem.