Lecture Notes

• Lecture 00 (January 8, 2024) Overview, remarks and basic definitions.

• Lecture 01 (January 10, 2024) Definitions. Maschke's theorem and Schur's lemma.
• Lecture 02 (January 12, 2024) Decomposition of (left) regular representation.
• Lecture 03 (January 17, 2024) Decomposition of group algebra. Class functions and characters. Orthogonality relation.
• Lecture 04 (January 19, 2024) Character table. Generalized averages and projections.
• Lecture 05 (January 22, 2024) Matrix coefficients and their orthogonality.
• Lecture 06 (January 24, 2024) Frobenius and the group determinant (slides)

• Lecture 07 (January 26 and 29, 2024) Basic facts about permutations and partitions. Ring of symmetric functions and Frobenius' map.
• Lecture 08 (January 31, 2024) Symmetric polynomials, Green's inner product.
• Lecture 09 (February 2, 2024) Schur polynomials. Jacobi-Trudi identity.
• Lecture 10 (February 5, 2024) Lindstrom-Gessel-Viennot lemma. Young's rule.
• Lecture 11 (February 7, 2024) Kostka numbers. Complete symmetric polynomials are characters of partition representations.
• Lecture 12 (February 9, 2024) Induced representations and their characters. Frobenius reciprocity theorem.
• Lecture 13 (February 12, 2024) Summary of Frobenius' theorem. Branching rules for representations of symmetric groups.

• Lecture 14 Polynomial representations of GL(N). Polynomial functors.
• Lecture 15 Linearization and symmetric group action. Schur's theorem on homogeneous polynomial functors.
• Lecture 16 Equivalence between homogeneous polynomial functors and representations of symmetric groups.
• Lecture 17 Polynomial representations of GL(n). Peter-Weyl type theorem.
• Lecture 18 Some remarks on representations of GL(n). Note by Abhay Chaudhary.

Associative algebras

• Lecture 19 Semisimple representations. Density theorem.
• Lecture 20 Artin-Wedderburn and Krull-Schmidt theorems.
• Lecture 21 Double commutant theorem. Another proof of Schur-Weyl duality.

Lie algebras and their representations.

• Lecture 22 Lie algebra basics.
• Lecture 23 Universal enveloping algebras.
• Lecture 24 (notes not available) Discussion of sl(2) representations, after Problem 2.15.1 of Etingof's book.
• Lecture 25 More on sl(2) representations: Weyl symmetry, Verma modules and BGG resolution.
• Lecture 27 Cartan matrices, Weyl groups and root systems.
• Lecture 28 Sign coherence of roots, Coxeter presentation.
• Lecture 29 Lie algebra associated to Cartan matrix.
• Lecture 30 Integrable representations and Weyl symmetry.
• Lecture 31 Classification of f.d. irreducible representations.
• Lecture 32 Invariant forms and Casimir element.
• Lecture 33 Applications of Casimir element. Harish-Chandra presentation of irred. reps.
• Lecture 34 Weyl character formula.
• Lecture 35 Weyl dimensional formula. Complete reducibility.