Topics in Representation Theory
August-December 2006
Complete Course Notes
Solutions to the mid-semester examination.
Cartan matrix of a centraliser algebra.
Lecture notes for the last two weeks (updated 16-11-06, 1pm).
Solutions to assignment given by Ragahavan.
Course information sheet.
Assignments (arranged by due dates)
August: 11th,
18th,
25th.
September: 1st,
8th,
15th.
October: 4th,
18th (project),
27th.
November: 8th, 17th.
Topics Covered
Lecture 1: Basic definitions (rings, modules, homomorphisms, direct sums). invariance of rank.
Lecture 2: Submodule of a free module over a PID is free. Equivalence of matrices over a PID. Smith canonical form. Structure theorem for finitely generated modules over a PID. invariant factors,
Lecture 3: Torsion module. Primary decomposition. Structure of primary modules. Invariance theorem.
Lecture 4: Finitely generated torsion k[t]-modules and the similarity problem for matrices over k. Companion matrix. Direct sum of matrices. Similarity classes of matrices and equivalence classes of their characteristic matrices. Rational canonical form.
Lecture 5: Primary decomposition of matrices, a related canonical form (analogous to the Jordan canonical form). Centraliser ring of a matrix. Hensel's lemma.
Lecture 6: Multiplicative system of representatives (see [3], II.4). Generalised Jordan canonical form, simple, indecomposable and semi-simple modules. Schur's lemma.
Lecture 7: Semisimple modules and semisimple matrices, Cyclic modules and cyclic matrices. Jordan decomposition. Semi-simplicity preserved under polynomial maps.
Lecture 8: Artinian and Noetherian modules. Fitting's theorem. Endomorphism ring of an indecomposable Noetherian module is local. Krull-Remak-Schmidt theorem. Quivers. Representations of quivers. Morphisms between representations of quivers.
Lecture 9: Path algebra of a quiver. The classification problem. Solution of the classification problem for the linear quiver and the one loop quiver.
Lecture 10: The classification problem for the two loop quiver contains the classification problem for any quiver.
Lecture 11: Filtrations. Composition Series. The Jordan-Holder theorem. Nilpotent ideals. Sum of nilpotent ideals in nilpotent. Uniqueness of maximal nilpotent ideal.
Lecture 12: Radical of a ring. Semisimplicity and the radical.
Lectures 13, 14 and 15: Guest lectures by K. N. Raghavan.
Lecture 16: Principal indecomposable modules. Radical of an ideal. Correspondence between principal indecomposable and irreducible modules. The Cartan matrix.
Lecture 17: Partition of principal indecomposable and irreducible modules into blocks. Composition factors of an indecomposable module come from the same block.
Lecture 18: Linkage classes are blocks. Number fields. Extension of ground field. A theorem of Noether and Deuring. Absolutely irreducible modules.
Lecture 19: Frobenius algebras. Symmetric algebras.
Lecture 20: Brauer's theorem on the correspondence between irreducible representations in characteristic p and p-regular conjugacy classes.
Lecture 21: Splitting fields.
Lecture 22: Brauer-Nesbitt theorem (D'D=C).
References
1. Basic Algebra, by Nathan Jacobson.
2. Linear Algebra, by Kenneth Hoffman and Ray Kunze.
3. Local Fields, by Jean-Pierre Serre.
4. Michael Barot's introduction to representations of quivers.
5. Representation Theory of Finite Groups and Associative Algebras, by Charles W. Curtis and Irving Reiner.
6. On the modular characters of groups, by Richard Brauer and Cecil Nesbitt.