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

### Assignments

- due Friday 01/26 [pdf] [tex]
- due Wednesday 01/31 [pdf] [tex]
- due Monday 02/05 [pdf] [tex]
- due 02/12 [pdf] [tex] [solutions]
- due 02/19 [pdf] [tex]
- due 02/26 [pdf] [tex]
- due 03/04 [pdf] [tex]
- due 03/11 [pdf] [tex]
- due 03/21 [pdf] [tex]
- due 04/03 [pdf] [tex] [solutions]
- due 04/15 [pdf] [tex] [solutions]
- due 04/22 [pdf] [tex]

### Reading

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