MATH 400 – Directed Readings in Module and Representation Theory Semester Plan Spring 2010

Description

In this directed readings course, we will examine some areas of abstract algebra not covered in MATH 244, 351, or 352, but of basic importance for the subject. The basic topics we will discuss include:

1) Modules over a ring

2) Finitely-generated modules over a PID, and the structure theorem

3) Applications to rational and Jordan canonical form for matrices

4) Linear representations of finite groups and modules over group rings

5) Character theory and applications for finite groups

I think the schedule laid out below is relatively “leisurely.” If we move faster than this, then some possible additional topics to consider would include

6) Modules over polynomial rings k[x_{1}, …, x_{n}] and Hilbert’s Syzygy Theorem, and/or 7) Representation theory for compact Lie groups, SL_{2}(R), etc.

Sources

(Almost) everything we will want to cover in topics 1 - 5 above is in Chapters 10, 11, 12, 15, 16 of Dummit and Foote. There is one proof (Burnside’s Theorem) that they omit but that we will will want to fill in from another source. The others are additional well-known places where this material is presented (among the many textbooks that cover this). It may be profitable to look beyond Dummit and Foote at times for additional perspectives, alternative proofs, different applications, etc.

1) Dummit, D. and Foote, R. Abstract Algebra, Prentice Hall.

2) Serre, J.-P. Linear Representations of Finite Groups, Springer.

3) Jacobson, N. Basic Algebra I, II, Freeman.

4) van der Waerden, Modern Algebra II, Ungar.

How the course will run

You should read through the relevant sections and do the problems on that topic from Dummit and Foote (or at least a good selection) each week. I would like to meet once each week for something like 1.0 to 1.5 hours to discuss the material for that week, clear up any questions that have come up, etc. Late afternoon on Wednesday or Thursday would be ideal for me, but we can discuss the time. In those meetings, I will ask you to sketch proofs from that week’s material and/or present some problem solutions from the sections assigned for that week (some basic ones, plus some “more interesting” ones) – like a practice PhD oral exam(!) I will also give you a more formal midterm and final problem set to work on and those will be graded and returned to you with comments.

Tentative Schedule

1) Week of January 25 – Basic module theory – Dummit and Foote, Chapter 10

2) Week of February 1 – Review of linear algebra (Modules over a field) – Dummit and Foote, Chapter 11

3) Week of February 8 – Modules over a PID and the structure theorem – Dummit and Foote, Section 12.1

4) Week of February 15 – Rational canonical forms of linear mappings and matrices – Dummit and Foote, Section 12.2

5) Week of February 22 – The Jordan canonical form and applications – Dummit and Foote, Section 12.3

6) Week of March 8 – Linear representations of groups and modules over group rings – Dummit and Foote, Section 15.1, Midterm Problem Set out

7) Week of March 15 – Projective and injective modules – Dummit and Foote, Section 15.2

8) Week of March 22 – Wedderburn’s Theorem – Dummit and Foote, Section 15.3

9) Week of March 29 – Consequences of Wedderburn – Dummit and Foote, Section 15.3 10) Week of April 5 – Character theory for finite groups and the orthogonality relations

– Dummit and Foote, Section 15.4

11) Week of April 12 – Characters of groups of small order – Dummit and Foote, Section 16.1

12) Week of April 19 – Applications to finite groups (theorems of Burnside and Hall) – Dummit and Foote, Section 16.2

13) Week of April 26 – Induced characters – Dummit and Foote, Section 16.3, Final Problem Set out