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
- Matrix representations and linear representations of finite
groups: definitions and examples.
- Group algebra. Representations are kG-modules.
- A-modules in general: Jordan-Holder theorem, semisimple
modules.
- Schur's Lemma. Jacobson radical J(A). Characterization of
algebras A such that every A-module is semisimple
(Wedderburn theorem).
- Proof of Wedderburn theorem. Hom_{k}(V,W) as
a G-module. Maschke's theorem.
- Proof of Maschke's theorem. Applications and examples. Reminder
on tensor products.
- Tensor products. kG as a Hopf algebra.
- Characters: basic properties, orthogonality.
- Almost-irreducibility of doubly-transitive permutation
representations. Character table of S_{4}. Using
characters to construct central idempotents.
- Frobenius reciprocity.
- Examples and applications of Frobenius reciprocity.
- Construction of irreducible representations of S_{n}.
- The Frobenius ring.
- Introduction to the algebra of symmetric functions. Bases of
monomial, elementary and power-sum symmetric functions.
- Basis of complete homogeneous symmetric functions. Relations
among the bases. Involution ω. Isomorphism with the
Frobenius ring.
- Inner product, Cauchy identities.
- Schur functions.
- Characters of S_{n}, completed. Bases of the
irreducible modules.
- Algebraic groups—classical examples.
- Algebraic geometry over C: affine varieties,
coordinate ring, morphisms. Affine algebraic groups and
algebraic G-modules.
- O(G) comodules, action of O(G)^{*}.
Representations of G_{m} and G_{a}.
Reductive and unipotent groups.
- Tangent vectors and vector fields on algebraic varieties and
algebraic groups.
- O(G) as a comodule for itself.
O(G)^{*} as algebra of left invariant operators.
- Lie(G)=T_{1}G as Lie algebra of left invariant
derivations in O(G)^{*}.
- Formal exponential
map gl_{n} → GL_{n}(C[[t]]).
Calculation of Lie bracket in gl_{n}.
- Lie algebras of the classical groups.
- Determination of the irreducible SL_{2}
modules S^{d}(C^{2}).
- Criterion for semisimplicity of comodules for a coalgebra,
Peter-Weyl theorem.
- Semisimplicity and Peter-Weyl, continued.
- Peter-Weyl for SL_{2}.
- Algebraic tori, character and co-character lattices.
- Root data
for GL_{n}, SL_{n}, PGL_{n}.
- Root subgroups (P)SL_{2}. Roots and co-roots.
- The Weyl group.
- Borel subgroup, positive roots. Root data and Weyl group
for SO_{2n+1}.
- SO_{2n+1} continued. Langlands
duality: SO_{2n+1}
and Sp_{2n}, Spin_{2n+1}
and Sp_{2n}/{±1}
- Highest weight modules and Weyl character formula.
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