Groups and Representations

Course guarantor:

Goce Chadzitaskos

Lecturer:

Goce Chadzitaskos, Lenka Motlochová

Tutor:

Goce Chadzitaskos, Lenka Motlochová

Supervisor:

Department of Physics

Synopsis:

The aim of the lectures is to acquaint students with the basic concepts of discrete group theory and their representations. The student will be thoroughly acquainted with the methods of classification of finite groups, decomposition of groups into direct and semidirect products, and with the properties of reducible and irreducible representations.

Requirements:

None

Syllabus of lectures:

1. Symmetry in physics and its mathematical realization: overview of basic concepts of symmetry in physics and of solving differential equations.

2. Basics of group theory: discrete finite and infinite groups and their description, Abelian and cyclic groups, group generators, homomorphism and isomorphism, subgroups, order of a group.

3. Factor groups, simple groups, kernel of a homomorphism, normal subgroups, equivalence classes, isomorphism theorems.

4. Group actions on sets: permutation representations, orbit, stabilizer, normalizer, centralizer.

5. Sylow's theorem, classification of non-isomorphic non-commutative groups of a given order, classification of

non-isomorphic abelian groups.

6. Basics of representation theory: reducible and irreducible group representations, equivalent representations, uni-

tary representations.

7. Schur’s lemmas, criteria of irreducibility

8. Irreducible representations of finite groups, orthogonality, geometric interpretation, basis of representation

space.

9. Character of a representation, character tables, orthogonality, Frobenius criterion, dimensions of irreducible rep-

resentations.

10. Regular representation, classification of irreducible representations.

Syllabus of tutorials:

Semigroups, groups, group properties, application of Sylow`s theorem, classification of groups of a given order, group representations, non-isomorphic irreducible representations of symmetric groups, Young diagrams.

Study Objective:

Knowledge:

become acquainted with the methods of classification of discrete groups and their representations.

Abilities:

classify of the groups of some order

Study materials:

Key references:

[1] C.W. Curtis and I. Reiner: Representation Theory of Finite Groups and Associative Algebras, AMS Chelsea Pu- blishing, 2006.

[2] A. P. Isaev, V. A. Rubakov: Theory of Groups and Symmetries: Finite Groups, Lie Groups, and Lie Algebras, World Scientific 2018.

Recommended references:

[3] D.S. Dummit, R.M. Foote: Abstract Algebra, John Wiley and Sons, 2004.

[4] H.F. Jones: Groups, Representations and Physics, 2nd Ed., IOP, Bristol 1998.

[5] I.M. Isaacs: Character Theory of Finite Groups, Dover, NY 1976.

Note:

Time-table for winter semester 2024/2025:

	06:00–08:0008:00–10:0010:00–12:0012:00–14:0014:00–16:0016:00–18:0018:00–20:0020:00–22:0022:00–24:00
Mon
Tue
Wed	roomBR:11 Chadzitaskos G. 16:00–18:50 (lecture parallel1) Břehová 7
Thu
Fri

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Matematická fyzika (compulsory course in the program)