MATH353/533 (Spring 2023): Introduction to Representation theory.

Instructor: Prof. Ivan Loseu (email: ivan.loseu@yale.edu)

Lectures: MW 2.30-3.45pm, Location: LOM 206. The first class is on 1/18, and the last class is on 4/26

Office hours: DL 416, T11-12, T4-5, W11-12.

A detailed description of the course containing some information on what is (and is not) going to be covered as well as discussions of references and prerequisites.

Homeworks:

  • Homework 1, due Feb 7.
  • Homework 2, due Feb 21.
  • Homework 3, due Mar 30.
  • Homework 4, due Apr 13.
  • Homework 5, due Apr 27.

    Index of terminology.

    Schedule:

  • Lec 1, 1/18: Introduction to representations of groups, notes: The goal of this lecture is to give a brief introduction into roughly the first half of the class. We will discuss the concept of representations, talk about irreducible representations and characters. Time permitting we will also discuss some applications to the structure theory of finite groups: the Burnside theorem. An example: going from actions to representations.
  • Lec 2, 1/20:Basics, pt 1, notes: More on going from actions to representations. Constructions with representations. Bonus (in dark blue -- not necessary for understanding the main part of the lecture): more on representations in functions: invariants, harmonic analysis, and modular forms.
  • Lec 3, 1/23:Basics, pt 2, notes: Homomorphisms of representations. Associative algebras and their representations. Bonus: Lie algebras and their universal enveloping algebras.
  • Lec 4, 1/25: Basics, pt 3, notes: Tensor products and duals of vector spaces and of group representations.
  • Lec 5, 1/30: Irreducible and completely reducible representations, pt 1, notes: Irreducible and completely reducible representations (incl. examples).
  • Lec 6, 2/1: Irreducible and completely reducible representations, pt 2, notes: Proof of Macshke's theorem. Decomposition into irreducibles and the Schur lemma. Bonus: averaging for infinite groups.
  • Bonus Lec 6.5: Application of averaging to the finite generation of algebras of invariants, notes.
  • Lec 7, 2/6: Irreducible and completely reducible representations, pt 3, notes: Decomposition into irreducibles, cont'd. Decomposition for the regular representation. Skew-fields incl. quaternions. Bonus: Schur lemma for infinite dimensional representations.
  • Lec 8, 2/8: Irreducible and completely reducible representations, pt 4/ Characters, pt 1, notes: More applications of the Schur lemma. Characters.
  • Lec 9, 2/13:Characters, pt 2, notes (updated): Orthogonality of characters.
  • Lec 10, 2/15: Characters, pt 3, notes: Orthogonality of characters: discussion and examples. More orthogonality.
  • Lec 11, 2/20: Characters, pt 4, notes: Applications of orthonormality. Representations of direct products. What's next?
  • Lec 12, 2/22: Digression: algebraic integers, notes.
  • Lec 13, 2/27: Applications of characters, notes: Integrality properties of characters. Burnside theorem.
  • Lec 14, 3/1: Induction of representations, I, notes: Motivation and definition. Frobenius reciprocity. Updated 3/5.
  • Midterm, 3/6.
  • Lec 15, 3/8: Induction of representations, II, updated (3/9) notes with Sec 2.1 rewritten and Sec 2.2 revised; notes (preliminary version): Proof of Frobenius reciprocity. The character formula for induced representations.
  • Bonus lecture A1 --why do we consider these groups?: Reflection groups and regular polytopes.
  • Bonus lecture A2 --why do we consider these groups?: Root systems.
  • Bonus lecture A3 --why do we consider these groups?: Complex reflection groups.
  • Bonus lecture A4 --why do we consider these groups?: Symplectic reflection groups.
  • Lec 16, 3/27: Representations of symmetric groups, I, notes: Classification of irreducibles.
  • Lec 17, 3/29: Representations of symmetric groups, II, notes: Classification of irreducibles, finished. Characters of irreducibles.
  • Lec 18, 4/3: Representations of symmetric groups, III, notes, Characters of irreducibles.
  • Bonus Lec B1, notes: Alternative construction of irreducible representations.
  • Bonus Lec B2, notes: Schur-Weyl duality.
  • Bonus Lec B3, notes: Schur polynomials.
  • Lec 19, 4/5: Representations of finite dimensional associative algebras, I, notes: Introduction and basics.
  • Lec 20, 4/10: Representations of finite dimensional associative algebras, II, notes: Partial proof of the three theorems from Lec 19; modules over skew-fields.
  • Lec 21, 4/12: Representations of finite dimensional associative algebras, III, notes: Tensor products over algebras, canonical decomposition into irreducibles, submodules in completely reducible modules.
  • Lec 22, 4/17: Representations of finite dimensional associative algebras, IV, notes: Submodules in completely reducible modules. The proof of Density theorem.
  • Lec 23, 4/19: Representations of finite dimensional associative algebras, V, notes (w. fixed mistake from the lecture): Criteria for semisimplicity (via the radical and the trace form), applications to representations of finite groups.
  • Lec 24, 4/24: Skew-fields, I, notes: Maximal subfields, Frobenius theorem on skew-fields over R, Tensor product of algebras and base change.
  • Lec 25 (a.k.a. the last), 4/26, Skew-fields, II, notes: Proof of main theorem from Lec 24. Finite skew-fields are commutative. Bonus: Brauer groups.

    References:

  • [E] Etingof et al, Introduction to Representation theory, available here.
  • [V] Vinberg, A course in Algebra. Graduate Studies in Mathematics, 56.