Tuesdays and Thursdays 3:30-4:50pm, One Drexel Plaza GL41

*The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions*, 2nd Edition, Bruce E. Sagan.

Week 1: Jan 08 (Tue), Jan 10 (Thu)

Definitions and examples of rings and ideals, isomorphism theorems

Read 7.1--7.3 in Dummit and Foote

Problem Set 1, Due Thursday, Jan 17.

Week 2: Jan 15 (Tue), Jan 17 (Thu)

Prime ideals and maximal ideals, definitions and examples of modules

Read 7.4, 10.1--10.2 in Dummit and Foote

Problem Set 2, Due Thursday, Jan 24.

Week 3: Jan 22, Jan 24

Module isomorphism theorems, products, coproducts, bases, rank

Read 10.2--10.3

Problem Set 3, Due Thursday, Jan 31.

Problem 9 from Problem Set 2 was moved to Problem 5 on this set.

Week 4: Jan 29, Jan 31

Chinese Remainder Theorem, principal ideal domains

Read 7.6, skim 8.1--8.3

Problem Set 4, Due Thursday, February 7.

Week 5: Feb 05, Feb 07

Structure theorem for modules over a principal ideal domain

The Wikipedia page on Smith normal form is close to the proof given in class.

Read 12.1--12.2

Problem Set 5, Due Thursday, February 14.

Week 6: Feb 12, Feb 14

Rational canonical form, Jordan canonical form

Read 12.2--12.3

Problem Set 6, Due Tuesday, February 26.

There is an extension for Problem set 6. It is now due February 26. Problem set 7 will still be due
February 28.

Week 7: Feb 19, Feb 21

Representation Theory

Read 18.1

Problem Set 7, Due Thursday, February 28.

Week 8: Feb 26, Feb 28

Character Theory

Read 18.1, 18.3

Problem Set 8, Due Thursday, March 7.

Week 9: Mar 05, Mar 07

Character Theory

Read 18.3, 19.1

Problem Set 9, Due Thursday, March 14.

Week 10: Mar 12, Mar 14

Artin-Wedderburn Theorem

Read 18.2

We will cover some subset of the following topics, to be refined as the course progresses:

Definitions and examples of rings and ideals, isomorphism theoremsDefinitions and examples of modules, isomorphism theorems

Polynomial rings

Graded rings and modules

Jordan-Hölder Theorem

Products, coproducts, universal properties

Chinese Remainder Theorem

Structure theorem for modules over a principal ideal domain

Artin-Wedderburn Theorem

Introduction to Representation Theory: irreducible representations, Maschke's Theorem, Schur's Lemma, character table, orthogonality relations

Tensor products, symmetric and exterior algebra, determinants

Representations of the symmetric group