Tuesdays and Thursdays 9:30-10:50am, Korman 245

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

Week 1: Jan 06 (Tue), Jan 08 (Thu)

Definitions and examples of rings and ideals, isomorphism theorems

Read 7.1--7.4 in Dummit and Foote

Problem Set 1, Due Tuesday, Jan 13. Problem 8 will be moved to the next problem set since we have not yet discussed prime ideals.
Definition needed for Problem 2 (d) (see 10.1): Let F be a field.
An F-algebra is a ring A together with a (unital) ring homomorphism f: F --> A such that the image f(F) is contained in the center of A.
The center of a ring A is the set {a in A : ab = ba for every b in A}. It follows from this definition that an F-algebra is also an F-vector space. So a finite-dimensional F-algebra is an F-algebra that is also finite-dimensional as an F-vector space.

Week 2: Jan 13 (Tue), Jan 15 (Thu)

Prime ideals and maximal ideals, definitions and examples of modules, isomorphism theorems

Read 7.4, 10.1--10.2 in Dummit and Foote

Problem Set 2, Due Tuesday, Jan 20.

Week 3: Jan 20, Jan 22

Products, coproducts, Chinese Remainder Theorem, principal ideal domains

Read 10.3, 7.6, 12.1, skim 8.1--8.3

Problem Set 3, Due Tuesday, Jan 27.

Week 4: Jan 27, Jan 29

Structure theorem for modules over a principal ideal domain, determinants

Read 12.1, 11.4

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

Problem Set 4, Due Tuesday, Feb 3.

Correction: Problem 8 is false! So, instead of proving that x_1,...,x_n form a basis, find an example of
a commutative ring R and linearly independent elements x_1,...,x_n of R^n such that these elements do not form a basis of R^n.

Problem 9 is correct.

Week 5: Feb 03, Feb 05

Rational canonical form, Jordan canonical form

Read 12.1--12.3

Problem Set 5, Due Tuesday, Feb 10.

Week 6: Feb 10, Feb 12

Representation Theory

Read 18.1

Problem Set 6, Due Tuesday, Feb 17.

Note: For Problem 9, work over the complex numbers, i.e., determine the representations D_4n -> GL_1(C).

Week 7: Feb 17, Feb 19

Representation Theory, Character Theory

Read 18.1, 18.3

Problem Set 7, Due Tuesday, Feb 24.

Week 8: Feb 24, Feb 26

Character Theory

Read 18.3, 19.1

Problem Set 8, Due Tuesday, Mar 03.

Week 9: Mar 03, Mar 05

Artin-Wedderburn Theorem, Tensor Products

Read 18.2, 10.4

Problem Set 9, Due Tuesday, Mar 10.

Week 10: Mar 10, Mar 12

Tensor algebra, exterior algebra

Read 10.4, 11.5

Problem Set 10, do not turn in.

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

R-algebras

Graded rings and modules

Jordan-Hölder 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

Localization

Direct product, direct sum, universal properties

Representations of the symmetric group

Representations of GL_n, schur functions, Schur-Weyl duality