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| Week # | Date | Topics covered | Homework Problems | Home Work Due | |
|---|---|---|---|---|---|
| Week 1 | Tuesday 01/17 | First Day of class. Introduction. Fourier Analysis as the representation theory of the circle group. Classification of irreps of sl_2, and of finite type quivers. | . | ||
| Thursday 01/19 | Algebras, left, right representations. Intertwiners. Schur's lemma and corollaries. Indecomposable modules and Jordan blocks. Ideals, twosided and quotients. Generators and relations. | First assignment here. | . | ||
| Week 2 | Tuesday 01/24 | Weyl algebra, q-Weyl algebra, bases. The path algebra P_Q of a quiver Q, generators and relations for P_Q. Representations of Q and of P_Q. Lie algebras, examples, derivations and automorphisms, sl_2. | . | ||
| Thursday 01/26 | Universal enveloping algebras and group algebras as examples of universal constructions and adjoint functors. Tensor algebra, symmetric algebra, super symmetric algebra. | Second assignment here. | |||
| Week 3 | Tuesday 01/31 | Tensor product of representations for a group or Lie algebra. Clebsch-Gordan coefficients. Hopf algebra (just a hint). Semi simple representations. Characterization of semi simple representations: sum of irreducibles iff direct sum of irreducibles iff all subrepresentations are direct summands. | . | ||
| Thursday 02/02 | Proof of characterization of semi simple modules | Third assignment here. | |||
| Week 4 | Tuesday 02/07 | Background on Weyl algebra, paralellism char p and q-Weyl, graded dimensions. Matrix algebras. | . | ||
| Thursday 02/09 | Filtrations, simple filtrations. Radical, semi simple algebra if Rad(A)=0. Main theorem characterizing semi simple algebras: Rad(A)=0 iff all dim A is sum of squares of dim irreps iff A sum of matrix algebras iff all fd reps are semi simple iff A is ss as left A reps. Filtrations and extensions. Characters, irreducible characters independent. For A ss characters are a basis. | Fourth assignment with correction here. | |||
| Week 5 | Tuesday 02/14 | Characters as decategorification. Examples of (de)categorification. Application of characters to the char 0 proof of Jordan-Hoelder. Finite groups. Maschke's theorem. | . | ||
| Thursday 02/16 | Converse of Maschke's theorem: if kG is ss then |G| is not zero in k. The trivial projector P_t, central, acts by scalars on irreps, picks out invariants. Group characters as class functions. Number of irreps is number on conjugacy classes. In char 0 character determines reps. Dual character. Inner product on class functions. | Fifth assignment here. | |||
| Week 6 | Tuesday 02/21 | Orthogonality of characters. Characters and Hom spaces. Characters of S_3 and tensor products. Unitary representations. | . | ||
| Thursday 02/23 | Hermitian inner product on complex representations, semi-simplicity again. Quaternion group and quaternions. | ||||
| Week 7 | Tuesday 02/28 | Representations over R and C. Real, complex and quaternionic representations over R and C. Proof of Frobenius theorem on division algebras over R. | . | ||
| Thursday 03/01 | Bilinear invariant forms on representations, and real, complex, quaternionic. Frobenius Schur indicator. | Sixth assignment here. | Due March 8th. | ||
| Week 8 | Tuesday 03/06 | Induction and restriction functors. Character of induced representation (Mackey formula). Frobenius reciprocity. | . | ||
| Thursday 03/08 | Induced representations for Lie algebra sl(2). Verma modules.Poincare-Birkhoff-Witt (without proof), BGG for sl(2). Geometry of projective space and sl(2). | ||||
| Week 9 | Tuesday 03/13 | Representations of S_n. Young diagrams, Young Tableau, subgroups P_\lambda, Q_\lambda, elements a_\lambda, b_\lambda, c_\lambda, Specht Module. Specht modules are irreducible and conversely, beginning of proof. | . | ||
| Thursday 03/15 | Finish proof of classification of irreducibles of S_n over C. Induced modules U_\lambda. Kostka numbers. Characters of U_\lambda and V_\lambda in terms of symmetric polynomials. | Seventh assignment here. | Due March 29th. | ||
| Week | Tuesday 03/20 | Spring break | |||
| Thursday 03/22 | Spring Break | ||||
| Week 10 | Tuesday 03/27 | . | |||
| Thursday 03/29 | . | . | |||
| Week 11 | Tuesday 04/03 | . | |||
| Thursday 04/05 | Eighth assignment here. | Due April 19th. | |||
| Week 12 | Tuesday 04/10 | . | |||
| Thursday 04/12 | . | . | |||
| Week 13 | Tuesday 04/17 | . | |||
| Thursday 04/19 | . | . | |||
| Week 15 | Tuesday 05/01 | Last Day of Class | . | . | |
| Thursday 05/03 | Reading Day | . | . | ||
| Monday May 7 | Final: 8:00-11:00AM, in Class. (Tentative date) |