Module MA3416: Group Representations
- Credit weighting (ECTS)
- 5 credits
- Semester/term taught
- Hilary term 2014-15
- Contact Hours
- 11 weeks, 3 lectures including tutorials per week
- Prof. Vladimir Dotsenko
- Learning Outcomes
On successful completion of this module, students will be able to:
- construct complex irreducible representations for various finite groups of small orders;
- reproduce proofs of basic results that create theoretical background for dealing with group representations;
- apply orthogonality relations for characters of finite groups to find multiplicities of irreducible constituents of a representation;
- apply representation theoretic methods to simplify problems from other areas that ``admit symmetries'';
- identify group theoretic questions arising in representation theoretic problems, and use results in group theory to solve problems on group representations.
- Module Content
The purpose of this module is to give an introduction to representation theory for the case of finite groups (and demonstrate that most of those approaches work well for infinite compact groups). A really important idea in mathematics is that a proper theory for anything should possess enough symmetries, and that for studying mathematical theories it makes sense to study their symmetries. Representation theory is the main instrument for studying symmetries.
The module covers the following topics:
- Representation of a group. Examples of representations. Trivial representation. Regular representation.
- Equivalent representations. Arithmetics of representations. Irreducible representations. Schur's lemma.
- Characters and matrix elements. Orthogonality relations for matrix elements and characters.
- Applications. Representations of a product of two groups. Tensor powers of a faithful representation contain all irreducibles as constituents. Dimensions of irreducibles divide the order of the group. Burnside's $p^aq^b$-theorem.
- Set representations. Orbits, intertwining numbers etc.
- Representations and character table of $A_5$.
- Module Prerequisite
- MA1212 Linear Algebra II (compulsory), MA1214 Introduction to Group Theory (compulsory) MA2215 Fields, Rings & Modules (desirable). Students who did not take MA2215 have to obtain the lecturer's permission by showing adequate preparation through background reading.
- Assessment Detail
- This module will be examined in a 2 hour examination in Trinity term. Continuous assessment is via bi-weekly home assignments. The final mark is 80% of the exam mark plus 20% of continuous assessment. Supplemental exams if required will consist of 100% exam.