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

Representation Theory

Office hours

Final Exam:

• Sunday 05/05, 7:30 - 10 pm, in class
• Topics
• Practice problems
• Discussion of results: Wednesday 05/08, 3-4 pm in Math 310

Schedule

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)
39. 04/22: algebraic integers are closed under +,-,*, 22.8, 22.10
40. 04/24: degree of irreducible character divides |G| (22.11)
41. 04/26: splitting fields, Galois groups, Fundamental Thm of Galois Theory
42. 04/29: Burnside's p^aq^b Thm (31.2, 31.3, 31.4)
43. 05/01: REVIEW (24) [pdf]

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)