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Course, academic year 2023/2024
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Group Representations 2 - NMAG567
Title: Reprezentace grup 2
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 6
Hours per week, examination: winter 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
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. Mgr. Pavel Příhoda, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Volitelné
Classification: Mathematics > Algebra
Incompatibility : NALG124
Interchangeability : NALG124
Is interchangeable with: NALG124
Annotation -
Last update: T_KA (14.05.2013)
The course gives a brief overview of some classical results on modular and integral representations of finite groups.
Course completion requirements - Czech
Last update: doc. Mgr. Pavel Příhoda, Ph.D. (13.10.2023)

Zápočet bude udělen buď za průběžné řešení úloh ze cvičení nebo za vyřešení sady domácích úkolů, které zadám ke konci semestru.

Literature -
Last update: T_KA (14.05.2013)

1. Charles W. Curtis, Irving Reiner: Representation theory of finite groups and associative algebras, John Wiley & Sons, New York, 1988.

2. Walter Feit: The representation theory of finite groups, North-Holland mathematical library, Amsterdam, 1982

3. Steven H. Weintraub: Representation Theory of Finite Groups: Algebra and Arithmetic (Graduate Studies in Mathematics, Vol. 59), AMS, Providence 2003.

Requirements to the exam - Czech
Last update: doc. Mgr. Pavel Příhoda, Ph.D. (13.10.2023)

Zkouška bude ústní - dvě otázky z probrané látky. K úspěšnému složení zkoušky stačí prokázat základní přehled u každé otázky.

Syllabus -
Last update: T_KA (14.05.2013)

1. Summary of some results from algebraic number theory and commutative algebra. Tensor product of algebras.

2. Further results on induced representations. Theorem of Artin and Brauer giving expressions of a character as a combination of induced characters. Any complex representation of a finite group G is defined over the exp(G)-th cyclotomic field.

3. Very basic methods from modular representation theory. Composition series, Jacobson radical, finite representation type. Brauer characters and their relation to composition series. Blocks.

4. Integral representations of finite groups. Lattices, the notion of finite representation type for integral representation. Representation type of cyclic groups. The relation between K_0(Z[C_n]) and the ideal class group of cyclotomic integers. Integral representations from the point of view of the representation theory of artin algebras, Klinger-Levy programme.

5. Local-global methods in integral representation theory. Jordan-Zassenhaus theorem, genus. Projective modules over Z[G], a theorem of Swan. Induced indecomposable representations, Green correspondence.

6. (only informatively) Some aspects of representation theory of compact groups or the theory of maximal orders.

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