MAU34104 - Group representations

Course information

References

• Linear representations of finite groups by J.-P. Serre is an excellent reference on this topic. The contents of this module roughly correspond to part I of this book.


• Alex Bartel's lecture notes also cover (quite a little more than) the contents of this module, and give a slightly different point of view.

Summary of contents

Previous recorded lectures

• Chapter 1: Introduction and first concepts
  Notes
  Lecture videos:
  • Group actions. The concept of group representations. (26:54)
  • Morphisms of representations (16:50)
  • New representations from old (09:29)
  • Irreducibility, indecomposability (25:09)


• Chapter 2: The module point of view
  Notes
  Lecture videos:
  • Modules over a ring (20:46)
  • Representations as modules (18:58)
  • Semi-simplicity (19:19)
  • Maschke's theorem (23:18)
  
  
• Chapter 3: Characters
  Notes
  Lecture videos:
  • Schur's lemma (11:27)
  • Reminders on dot products (15:10)
  • The character of a representation (24:37)
  • The character of Hom (28:37)
  • Characters know everything (24:15)
  • Isotypical components (32:26)


• Chapter 4: Character tables
  Notes
  Lecture videos:
  • Decomposition of the regular representation (17:45)
  • Wedderburn on C[G] (39:37)
  • The number of irreducible representations (38:43)
  • The character table (55:23)
  • Second orthogonality relations & extra results (13:22)


• Chapter 5: Induced representations
  Notes
  Lecture videos:
  • Restriction and induction (07:07)
  • Definition(s) of the induced representation (38:56)
  • Example: permutation representations (17:42)
  • Induced characters (35:54)
  • Frobenius reciprocity (26:57)

Homework

Final exam