Mikhail Bershtein

Plan of the Course

1. Group representations.
2. Algebras, semisimple representations.
3. Semisimple algebras.
4. Characters, orthogonality.
5. Representations of S_n.
6. Irreducible representations of S_n. Restriction and induction.
7. Characters of representations of the symmetric group.
8. Hook length formula.
9. New (Okounkov–Vershik) approach to representation theory of S_n.
10. Schur-Weyl duality.
11. Schur functions and representation theory of GL_N.
12. Schur functions and representation theory of S_n.
13. Lie algebra sl₂.
14. Finite-dimensional representations of sl₂.
15. Representations of sl₂ and flag variety.
16. Group SL₂(R).
17. Unitary representations of group SL₂(R).
18. Representations of group SL₂(F_p).
19. Quantum groups.
20. Representations of U_q(sl₂).

Problems. 1, 2, 3.

Plan of the Course

1. Representation theory of finite groups. Definitions, characters, orthogonality, restriction and induction.
2. Classical representation theory. Irreducible representations of S_n. Schur polynomials. Hook length formula. Schur-Weyl duality. Finite-dimensional representations of GL_N.
3. Further topics. Quantum groups and Hecke algebras.