Representation Theory: Lecture notes

Lecture Notes

General results on representations of associative algebras.

Lecture 0 (January 9, 2023) Overview. Definitions: associative algebras, representations, homomorphisms, subrepresentations, irreducible and indecomposable representations. Examples.
Lecture 1 (January 11, 2023) Schur's lemma. Semisimple representations. Density theorem.
Lecture 2 (January 13, 2023) Density theorem and corollaries. Finite-dimensional, semisimple algebras.
Lecture 3 (January 18, 2023) Krull-Schmidt theorem.
Lecture 4 (January 20, 2023) (Correction: proof of Krull-Schmidt thm). Jordan-Holder series. Summary of important results.

Representations of finite groups

Lecture 5 (January 23, 2023) Group representations. Group algebra of a finite group. Maschke's theorem. Tensor product of two representations. Dual of a representation.
Lecture 6 (January 25, 2023) Decomposition of group algebra. Class functions.
Lecture 7 (January 27, 2023) Frobenius' character theory. Orthogonality relations.
Lecture 8 (January 30, 2023) Matrix coefficients. Example.
Lecture 9 (slides) (February 1, 2023) Frobenius' work on the group determinant.

Symmetric groups and symmetric functions

Lecture 10 (February 3, 2023) Representations of symmetric groups.
Lecture 11 (February 6, 2023) Frobenius character map. Inner product on the space of symmetric functions.
Lecture 12 (February 8, 2023) Orthonormality of Schur functions. Jacobi-Trudi identity.
Lecture 13 (February 10, 2023) Young tableaux. Lindstorm-Gessel-Viennot lemma. Schur polynomials vs semi-standard Young tableau.
Lecture 14 (February 13, 2023) Partition representations and complete symmetric functions. End of the proof of Frobenius' main theorem.
Lecture 15 (February 15, 2023) Induced representations and their characters. Frobenius reciprocity.
Lecture 16 (February 17, 2023) Summary. Pieri rule and branchings. Young lattice. Suggestions for further reading.

Polynomial functors and polynomial representations of the general linear group

Lecture 17 (February 20, 2023) Polynomial functors. Decomposition into homogeneous components. Linearization.
Lecture 18 (February 22, 2023) Classification theorem of homogeneous polynomial functors.
Lecture 19 (February 24, 2023) Tensor product of polynomial functors vs induction product. Trace map vs characteristic map.
Lecture 20 (February 27, 2023) Classification of finite-dimensional, polynomial representations of GL(m).
Lecture 21 (March 1, 2023) Double commutatant theorem. Another proof of classification of irr-reps of GL(m).
Lecture 22 (March 3, 2023) Weyl dimension formula. Summary (notes coming soon!).

Lie algebras and their representations

Lecture 23 (March 6, 2023) Lie algebras and their representations. Definitions and examples.
Lecture 24 (March 8, 2023) Universal enveloping algebra.
Lecture 25 (March 10, 2023) Complete reducibility theorem for sl(2).
Lecture 26 (March 20, 2023) Summary of sl(2) reps. Weyl group, Category O, and Verma modules.

Lie algebras associated to Cartan matrices.

Lecture 27 (March 22, 2023) Cartan matrices. Root system associated to a Cartan matrix.
Lecture 28 (March 24, 2023) Properties of Weyl groups. Coxeter presentation.
Lecture 29 (March 27, 2023) Lie algebra associated to a Cartan matrix. Serre relations.
Lecture 30 (March 29, 2023) Integrable representations and Weyl group symmetry.
Lecture 31 (March 31, 2023) Classification theorem of finite-dimensional irreducible representations.
Lecture 32 (April 3, 2023) Invariant forms and Casimir operator.
Lecture 33 (April 7, 2023) Proof of the existence of the invariant form. Applications of the Casimir operator.
Lecture 34 (April 10, 2023) Weyl character formula.
Lecture 35 (April 12, 2023) Dimension formulae. Complete reducibility theorem.

Further topics

Lecture 36 (April 17, 2023) Weyl-Kac character formula vs Jacobi triple product identity. Affine sl(2).