Math 4140 Spring 24

MATH 4140/5140: Abstract Algebra 2 (Spring 2024)

Representation Theory

Syllabus

Office hours

M 12:30 - 2 pm, F 10 - 11 am at Math 310

Final Exam:

Sunday 05/05, 7:30 - 10 pm, in class

Schedule

Numbers are for orientation and refer to sections with related material in the book by James and Liebeck.
  1. 01/17: review groups, subgroups (1)
  2. 01/19: review homomorphisms, kernels, normal subgroups (1)
  3. 01/22: review vector spaces, direct sums (2)
  4. 01/24: review matrices of linear transformations, projections (2)
  5. 01/26: group representations, equivalence, faithfulness (3)
  6. 01/29: F-algebras (6)
  7. 01/31: algebra representations, FG-modules (4)
  8. 02/02: permutation representations (4)
  9. 02/05: equivalent representations and isomorphic modules (7), simple modules (5)
  10. 02/07: direct sums of modules, completely reducible (7), regular FG-module (6)
  11. 02/09: Maschke's Theorem (8)
  12. 02/12: Schur's Lemma (9)
  13. 02/14: characterizing irreducible representations (9)
  14. 02/16: representations of abelian groups (9, p 81-82)
  15. 02/19: REVIEW
  16. 02/21: MIDTERM 1
  17. 02/23: discussion of midterm, irreducible CD_6-modules (10)
  18. 02/26: irreducible CG-modules (10)
  19. 02/28: space of CG-homomorphisms, multiplicity of simple modules in CG (11)
  20. 03/01: CG as direct product of matrix rings
  21. 03/04: number of simple CG-modules, center of CG (12)
  22. 03/06: conjugacy classes of Sn and An (12)
  23. 03/08: characters (13)
  24. 03/11: kernel, regular character, permutation character (13)
  25. 03/13: Irr(G) is an orthormal basis for the space of class functions (14)
  26. 03/18: orthogonal decomposition of CG, idempotents e1,...,ek
  27. 03/20: inner product on class functions (14)
  28. 03/22: orthogonality relations (16)
  29. 03/24: lifting characters from quotients (17)
  30. 04/01: products of characters (19)
  31. 04/03: symmetric and antisymmetric tensors, characters of direct products (19)
  32. 04/05: REVIEW
  33. 04/08: MIDTERM 2
  34. 04/10: characters of direct products (19), subgroups (20)
  35. 04/12: discussion of midterm, Prop 20.5
  36. 04/15: restriction to normal subgroups, Clifford's Thm (20)
  37. 04/17: Inducing characters from H to G, Frobenius reciprocity (21)
  38. 04/19: algebraic integers (22)

Assignments

  1. due Friday 01/26 [pdf] [tex]
  2. due Wednesday 01/31 [pdf] [tex]
  3. due Monday 02/05 [pdf] [tex]
  4. due 02/12 [pdf] [tex] [solutions]
  5. due 02/19 [pdf] [tex]
  6. due 02/26 [pdf] [tex]
  7. due 03/04 [pdf] [tex]
  8. due 03/11 [pdf] [tex]
  9. due 03/21 [pdf] [tex]
  10. due 04/03 [pdf] [tex] [solutions]
  11. due 04/15 [pdf] [tex] [solutions]
  12. due 04/22 [pdf] [tex]

Reading

  1. G. James and M. Liebeck. Representations and characters of groups, second edition, Cambridge University Press, 2001. (available electronically via the CU library)