## Math 252—Representation Threory Fall 2012

Time and place: MWF 12-1pm, Room 6 Evans Hall

Professor: Mark Haiman
Office: 855 Evans
Office hours: MW 10:30am-12:30pm
Phone: (510) 642-4318

Outline: The core topics for this course are representation theory of finite-dimensional algebras, especially semisimple algebras, and representation theory of finite groups. To this I will add some other topics, on which I haven't precisely settled yet. I expect it will be some mix of representation and invariant theory for the classical Lie groups (or algebraic groups), plus `q-analogs' of some of the above: namely, Hecke algebras and quantum groups, with a bit of Kazhdan-Lusztig theory (for Hecke algebras) and canonical basis theory (for quantum groups).

Prerequisites: Algebra background equivalent to 250A.

Textbooks:

• Fulton and Harris, Representation Theory, A First Course, Springer, 1991-2004.
• Goodman and Wallach, Symmetry, Representations, and Invariants, Springer, 2009. (This is a revised version of their earlier and now out-of-print book Representations and Invariants of the Classical Groups.)

Homework and grading: Grading based entirely on homework. I will maintain a running list of homework problems, organized by lecture (and the LaTeX source file for your convenience if you are writing solutions on a computer). Errata: 9/11 - changed Lect. 4 #4. 11/24 - changed Lect. 19-24 #2.

Officially, homework is due the second Friday following the lecture for which it is assigned, or the next class day if that Friday is a holiday. Problems for the last two weeks of lectures are due by Friday, December 14, the last day of final exam week. To keep up with the lectures you really should try to do the problems on time. However, I don't insist on this, since in practice, in graduate courses I usually don't manage to grade and return homework in a timely fashion.

Lecture topics

1. Matrix representations and linear representations of finite groups: definitions and examples.
2. Group algebra. Representations are kG-modules.
3. A-modules in general: Jordan-Holder theorem, semisimple modules.
4. Schur's Lemma. Jacobson radical J(A). Characterization of algebras A such that every A-module is semisimple (Wedderburn theorem).
5. Proof of Wedderburn theorem. Homk(V,W) as a G-module. Maschke's theorem.
6. Proof of Maschke's theorem. Applications and examples. Reminder on tensor products.
7. Tensor products. kG as a Hopf algebra.
8. Characters: basic properties, orthogonality.
9. Almost-irreducibility of doubly-transitive permutation representations. Character table of S4. Using characters to construct central idempotents.
10. Frobenius reciprocity.
11. Examples and applications of Frobenius reciprocity.
12. Construction of irreducible representations of Sn.
13. The Frobenius ring.
14. Introduction to the algebra of symmetric functions. Bases of monomial, elementary and power-sum symmetric functions.
15. Basis of complete homogeneous symmetric functions. Relations among the bases. Involution ω. Isomorphism with the Frobenius ring.
16. Inner product, Cauchy identities.
17. Schur functions.
18. Characters of Sn, completed. Bases of the irreducible modules.
19. Algebraic groups—classical examples.
20. Algebraic geometry over C: affine varieties, coordinate ring, morphisms. Affine algebraic groups and algebraic G-modules.
21. O(G) comodules, action of O(G)*. Representations of Gm and Ga. Reductive and unipotent groups.
22. Tangent vectors and vector fields on algebraic varieties and algebraic groups.
23. O(G) as a comodule for itself. O(G)* as algebra of left invariant operators.
24. Lie(G)=T1G as Lie algebra of left invariant derivations in O(G)*.
25. Formal exponential map gln → GLn(C[[t]]). Calculation of Lie bracket in gln.
26. Lie algebras of the classical groups.
27. Determination of the irreducible SL2 modules Sd(C2).
28. Criterion for semisimplicity of comodules for a coalgebra, Peter-Weyl theorem.
29. Semisimplicity and Peter-Weyl, continued.
30. Peter-Weyl for SL2.
31. Algebraic tori, character and co-character lattices.
32. Root data for GLn, SLn, PGLn.
33. Root subgroups (P)SL2. Roots and co-roots.
34. The Weyl group.
35. Borel subgroup, positive roots. Root data and Weyl group for SO2n+1.
36. SO2n+1 continued. Langlands duality: SO2n+1 and Sp2n, Spin2n+1 and Sp2n/{±1}
37. Highest weight modules and Weyl character formula.