A short course on finite group representations

In Fall 2020, I taught an undergraduate course on abstract algebra. I chose to spend two weeks on the theory of complex representations of finite groups. I covered the basic concepts, leading to the classification of representations by characters. I also briefly addressed a few more advanced topics, notably induced representations and Frobenius divisibility. I'm making the lectures and the associated notes for this material publicly available.

The material here is standard, and is mainly based on Steinberg, Representation theory of finite groups, Ch 2-4, whose notation I mostly follow. I also used Serre, Linear representations of finite groups, Ch 1-3. More precisely, I'm following Steinberg, except that I'm avoiding all references to ``unitary representations''. Where this notion appears in proofs, I'm instead using arguments based on Serre's elegant proofs. (There's nothing wrong with knowing about unitary representations, but it's overkill given that I don't go very far into the material.)

Here are the lecture notes.

Here are some exercises

Here are the lectures (about 4.5 hours at normal speed), together with slides.

  1. Group representations (5:24) [Video, Slides]
  2. Examples of Group Representations (9:30) [Video, Slides]
  3. Equivalence of Representations (9:14) [Video, Slides]
  4. Invariant subspaces (6:36) [Video, Slides]
  5. Irreducible representations (6:00) [Video, Slides]
  6. Direct sum of representations (8:08) [Video, Slides]
  7. Morphisms of representations (5:33) [Video, Slides]
  8. The averaging trick (7:15) [Video, Slides]
  9. Decomposability of representations (8:41) [Video, Slides]
  10. Complete reducibility of representations (7:08) [Video, Slides]
  11. Schur's lemma (7:53) [Video, Slides]
  12. 1-dimensional representations and abelian groups (12:10) [Video, Slides]
  13. Trace of a linear operator (8:28) [Video, Slides]
  14. Character of a representation (9:12) [Video, Slides]
  15. Class functions (10:18) [Video, Slides]
  16. Orthogonality relations for characters (16:25) [Video, Slides]
  17. Character table for Z4 and Z2 × Z2 (5:39) [Video, Slides]
  18. Character table for S3 (6:15) [Video, Slides]
  19. Proof of orthogonality relations, part 1 (14:04) [Video, Slides]
  20. Proof of orthogonality relations, part 2 (7:13) [Video, Slides]
  21. Regular representation (10:34) [Video, Slides]
  22. Number of irreducible representations (10:28) [Video, Slides]
  23. Character table for D4 (4:46) [Video, Slides]
  24. Second orthogonality relations (9:11) [Video, Slides]
  25. Frobenius divisibility, part 1 (8:30) [Video, Slides]
  26. Frobenius divisibility, part 2 (13:35) [Video, Slides]
  27. Induced characters (12:28) [Video, Slides]
  28. Character table for D5 (2:03) [Video, Slides]
  29. Construction of induced representations (11:53) [Video, Slides]
  30. Representations of products of groups (11:23) [Video, Slides]

Last modified 27 March 2021 by Charles Rezk. Email: rezk@illinios.edu

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