Representation theory of finite groups and compact groups
This course is addressed to master students and doctoral students not specialising in representation theory. The material of the course covers the basics of representation theory with a special emphasis to characters and partially uses personal material of the lecturer. It is contained in several books (together) including
1) J.P.Serre, Linear Representations of Finite Groups
2) B.E.Sagan, The Symmetric Group
Structure of the course:
24 lectures (no home assignments or exam).
(1) Lectures 1–5 are devoted to group theory.
(2) Lectures 6–16 are devoted to the general representation theory of finite groups with Lecture 15 dealing with compact groups instead of finite.
(3) Lectures 17–24 are devoted to representations of symmetric group and symmetric functions.
1) J.P.Serre, Linear Representations of Finite Groups
2) B.E.Sagan, The Symmetric Group
Structure of the course:
24 lectures (no home assignments or exam).
(1) Lectures 1–5 are devoted to group theory.
(2) Lectures 6–16 are devoted to the general representation theory of finite groups with Lecture 15 dealing with compact groups instead of finite.
(3) Lectures 17–24 are devoted to representations of symmetric group and symmetric functions.
Lecturer
Boris Shapiro
Date
18th September ~ 13th December, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Monday,Wednesday | 09:50 - 11:25 | A3-1-101 | ZOOM 03 | 242 742 6089 | BIMSA |
Syllabus
September 18, Lecture 1. Preliminaries on group theory. Groups acting on sets.
September 20, Lecture 2. The Sylow theorems. Finitely generated abelian groups.
September 25, Lecture 3. Solvable groups. The symmetric and alternating groups
September 27, Lecture 4. Groups of low order.
October 9, Lecture 5. Linear algebra. Group representations.
October 10, Lecture 6. Characters. Properties of characters.
October 16, Lecture 7. The group algebra.
October 18, Lecture 8. Character tables.
October 23, Lecture 9. Algebraic numbers. Applications.
October 25, Lecture 10. Roots of unity. Burnside’s p^aq^b-theorem.
October 30, Lecture 11. Results on integer-values characters. Burnside’s theorem on zeros in the character table.
November 1, Lecture 12. Products of characters. Representations on exterior and symmetric powers.
November 6, Lecture 13. Squares.
November 8, Lecture 14. Induced characters. Induced representations.
November 13, Lecture 15. Representations of compact groups.
November 15, Lecture 16. Representations of compact groups. (Cont.)
November 20, Lecture 17. Representations of the symmetric group. Young subgroups, Tableaux and Tabloids. Lexicographic order.
November 22, Lecture 18. Specht Modules. The submodule theorem.
November 27, Lecture 19. Standard Tableaux and a Basis for S^alpha.
November 29, Lecture 20. Garner Elements. Young’s natural representation.
December 4, Lecture 21. The Branching Rule. Decomposition of M^mu.
December 6, Lecture 22. The semi standard Bais for Hom(S¨\lambda, M^\mu). Kostka numbers.
December 11, Lecture 23. Schur functions. The Jacobi-Trudi determinants.
December 13, Lecture 24. Other Definitions of the Schur functions. The characteristic map.
September 20, Lecture 2. The Sylow theorems. Finitely generated abelian groups.
September 25, Lecture 3. Solvable groups. The symmetric and alternating groups
September 27, Lecture 4. Groups of low order.
October 9, Lecture 5. Linear algebra. Group representations.
October 10, Lecture 6. Characters. Properties of characters.
October 16, Lecture 7. The group algebra.
October 18, Lecture 8. Character tables.
October 23, Lecture 9. Algebraic numbers. Applications.
October 25, Lecture 10. Roots of unity. Burnside’s p^aq^b-theorem.
October 30, Lecture 11. Results on integer-values characters. Burnside’s theorem on zeros in the character table.
November 1, Lecture 12. Products of characters. Representations on exterior and symmetric powers.
November 6, Lecture 13. Squares.
November 8, Lecture 14. Induced characters. Induced representations.
November 13, Lecture 15. Representations of compact groups.
November 15, Lecture 16. Representations of compact groups. (Cont.)
November 20, Lecture 17. Representations of the symmetric group. Young subgroups, Tableaux and Tabloids. Lexicographic order.
November 22, Lecture 18. Specht Modules. The submodule theorem.
November 27, Lecture 19. Standard Tableaux and a Basis for S^alpha.
November 29, Lecture 20. Garner Elements. Young’s natural representation.
December 4, Lecture 21. The Branching Rule. Decomposition of M^mu.
December 6, Lecture 22. The semi standard Bais for Hom(S¨\lambda, M^\mu). Kostka numbers.
December 11, Lecture 23. Schur functions. The Jacobi-Trudi determinants.
December 13, Lecture 24. Other Definitions of the Schur functions. The characteristic map.
Reference
The material of the course is based on several books including
1) J.P.Serre, Linear Representations of Finite Groups
2) B.E.Sagan, The Symmetric Group
1) J.P.Serre, Linear Representations of Finite Groups
2) B.E.Sagan, The Symmetric Group
Video Public
Yes
Notes Public
Yes