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Classical groups and their invariants - NMAG448

Title:	Klasické grupy a jejich invarianty
Guaranteed by:	Mathematical Institute of Charles University (32-MUUK)
Faculty:	Faculty of Mathematics and Physics
Actual:	from 2021
Semester:	summer
E-Credits:	5
Hours per week, examination:	summer s.:2/2, C+Ex [HT]
Capacity:	unlimited
Min. number of students:	unlimited
4EU+:	no
Virtual mobility / capacity:	no
State of the course:	taught
Language:	Czech, English
Teaching methods:	full-time

Guarantor:	doc. RNDr. Petr Somberg, Ph.D.
Teacher(s):	doc. RNDr. Petr Somberg, Ph.D.
Class:	M Mgr. MSTR M Mgr. MSTR > Povinně volitelné
Classification:	Mathematics > Geometry

Opinion survey results Examination dates Noticeboard

Annotation -

Invariants of classical groups will be studied, mainly those induced by actions on vector spaces. Invariants are traces of endomorphisms or determinants. Hereby we mean that these objects are invariants with respect to the adjoint action of the appropriate general linear group. In the application of the theory, we focus to the case of polynomial invariants of binary n-ics, i.e., invariant polynomials defined on the space of polynomials P^n(C^2). The discriminant of a binary quadric is an example of such an invariant for n = 2.

Last update: T_MUUK (13.05.2015)

Aim of the course -

Study of invariants for classical groups

Last update: Souček Vladimír, prof. RNDr., DrSc. (01.03.2020)

Course completion requirements - Czech

Podmínky pro udělení zápočtu - referát.

Last update: Souček Vladimír, prof. RNDr., DrSc. (05.05.2020)

Literature -

Weyl, H., Quantumtheorie und Gruppenlehre

Goodman, R., Walalch, N., Representations and Invariants of the Classical Groups,

Griffiths, Harris, Representation theory

Last update: Souček Vladimír, prof. RNDr., DrSc. (01.03.2020)

Teaching methods -

Lecture notes prepared by the lecturer.

Last update: Souček Vladimír, prof. RNDr., DrSc. (05.05.2020)

Requirements to the exam -

Examination is written, its aim is to test the knowledge of the material contained in the lecture notes prepared by the lecturer.

Last update: Souček Vladimír, prof. RNDr., DrSc. (05.05.2020)

Syllabus -

1) Linear algebraic groups, classical groups and their Lie algebras, real forms of complex algebraic groups.

2) Regular and locally regular representations.

3) Classification of irreducible representations of classical groups using highest weights, complete reducibility.

4) Fourier analysis on the group algebra of a finite group, classification of irreducible representations of the group of permutations.

5) Representations of associative algebras, double commutant theorm.

6) Invariants for GL, SO and Sp groups.

Last update: Souček Vladimír, prof. RNDr., DrSc. (01.03.2020)

Entry requirements -

Linear algebra, analysis on manifolds, basics of Lie algebras.

Last update: Šmíd Dalibor, Mgr., Ph.D. (11.06.2021)