Representation Theory, Fall 2022

北京大学

Instructor: Evgeny Smirnov, esmirnov@hse.ru, https://www.hse.ru/en/staff/esmirnov

Course assistant: Wang Yihan (王一涵), 2001110016@pku.edu.cn

Time: Mon 15:10-17:00, Thu 18:40-20:30 CST (GMT+8).

Lectures

• Lecture 1, 05/09/22. What is representation theory? Main definitions. Irreducibility and indecomposability, Maschke’s theorem. Blackboard
• Lecture 2, 08/09/22. Morphisms of representations. Schur’s lemma, uniqueness of decomposition into irreducibles. Representations of abelian groups and S₃. Blackboard
• Lecture 3, 12/09/22. Multilinear maps. Tensor product of vector spaces, universality. Symmetric powers. Blackboard
• Lecture 4, 15/09/22. Exterior powers. Characters of representations, their properties. Blackboard
• Lecture 5, 19/09/22. Orthonormality of characters. Characters form a basis in class functions. Blackboard
• Lecture 6, 22/09/22. Representations of S₄. Character tables, tensor products. Blackboard
• Lecture 7, 26/09/22. Representations of A₄, S₅, and A₅. Blackboard
• Lecture 8, 29/09/22. Representations of A₅ and symmetries of dodecahedron. Irreducibility of exterior powers of the tautological representation of S_n. The group algebra. Blackboard | Pictures
• Lecture 9, 06/10/22. Induced representations. Frobenius reciprocity. Blackboard
• Lecture 10, 13/10/22. Representations of S_d: construction of Young symmetrizers. Blackboard
• Lecture 11, 17/10/22. Representations of S_d. The Frobenius character formula. Blackboard
• Lecture 12, 20/10/22. Euler’s pentagonal theorem. Gaussian binomial coefficients. Blackboard
• Lecture 13, 24/10/22. Quiver representations. Indecomposable representations. Examples. Blackboard
• Lecture 14, 27/10/22. Gabriel’s theorem. Dynkin diagrams and positive definite Cartan matrices. Blackboard
• Lecture 15, 31/10/22. Roots. Simple and positive roots. Reflection functors. Blackboard
• Lecture 16, 03/11/22. Coxeter elements. Proof of Gabriel’s theorem. Path algebra of a quiver. Blackboard
• Lecture 17, 07/11/22. Representations of associative algebras. The density theorem. Representations of direct sums of matrix algebras. Blackboard
• Lecture 18, 14/11/22. Radical. Semisimple algebras. Jordan-Hölder theorem. Blackboard
• Lecture 19, 17/11/22. Krull-Schmidt theorem. Extensions, Ext¹(W,V). Blackboard
• Lecture 20, 21/11/22. Algebraic integers. Frobenius divisibility theorem. Burnside’s theorem. Blackboard
• Lecture 21, 24/11/22. Lie groups, tangent spaces. The exponential map. Blackboard
• Lecture 22, 28/11/22. Lie algebras. Adjoint representation. Epimorphism SU₂->SO₃. Blackboard
• Lecture 23, 01/12/22. Compact Lie groups, complete reducibility. Weyl’s unitary trick. Representations of sl(2). Blackboard
• Lecture 24, 05/12/22. Universal enveloping algebra. Double centralizer theorem. Schur-Weyl duality. Blackboard

Videos are accessible here: https://disk.pku.edu.cn:443/link/EFB541F8F4E3C0D5723B713D7F4BECCE

Textbooks

1. William Fulton, Joe Harris. Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. Springer, 1991. This will be our main textbook
2. Jean-Pierre Serre. Linear Representations of Finite Groups. Graduate Texts in Mathematics. Vol. 42. Springer, 1996.
3. Pavel Etingof et al. Introduction to Representation Theory. AMS, 2011. Available at the author’s website: https://math.mit.edu/~etingof/repb.pdf
4. J.N. Bernstein, I.M. Gelfand, V.A.Ponomarev. Coxeter functors and Gabriel’s theorem. https://www.math.uni-bielefeld.de/~sek/kau/bgp.pdf
5. Ernest B. Vinberg. A Course in Algebra. Graduate Texts in Mathematics, Vol. 56. AMS, 2003.

Notes and other links

1. J.J.Jiang’s notes on semisimplicity (in Chinese)