MATH3966 Modules and Group Representations

This page contains information on the Senior advanced Unit of Study MATH3966: Modules and Group Representations.

• Taught in Semester 2.
• Credit point value: 6.
• Classes per week: Three lectures and one tutorial.
• Lecturer(s): Anthony Henderson.

Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the MATH3966 handbook entry for further information relating to MATH3966.

You may also view the Faculty Handbook entry for MATH3966 from the central units of study database.

Lecturer

Name:   Dr Anthony Henderson
Room:   Carslaw 805
Phone:   9351 3881
Consultation time:   Thursday 1-2pm
Email:   A.Henderson@maths.usyd.edu.au
If you cannot come to the designated consultation time, you may make another appointment by email, or just check my office to see whether I am free.

Class times

After the initial lecture on Monday July 28 at 10am in Carslaw 351, lectures will be held at the following times:

• Wednesday 10am, Carslaw 351
• Wednesday 1pm, Carslaw 250 (from Week 2)
• Thursday 10am, Carslaw 351

Tutorials will be held on Mondays at 10am in Carslaw 356, in Weeks 2-12 (alternative arrangements for Week 10, when Monday is a public holiday, will be announced later).

Lecture notes

The lectures will closely follow the book of notes `Modules and Group Representations' by Anthony Henderson, which is available from KopyStop. Tutorial and assignment questions will also be drawn from the exercises in this book, solutions to which will be posted progressively through the semester at the bottom of this page.

Unit outline

This unit aims to introduce two concepts which are fundamental in algebra: that of a module over a ring, and that of a representation of a group. One of the most important points to be explained is that the latter is an example of the former. Indeed, the theory of modules has applications to all areas of mathematics, although in MATH3966 we will concentrate on the applications to linear algebra and group theory. The rough weekly plan, with references to the sections in the lecture notes, is as follows.

Week 1:  Basic definition and examples of modules (1.1,1.2)
Week 2:  Free modules, torsion modules, and direct sums (1.3,1.4,1.5)
Week 3:  Matrices over PIDs and column-equivalence (2.1,2.2,2.3)
Week 4:  Equivalence and torsion invariants (2.4,2.5)
Week 5:  Primary invariants and conjugation of matrices (2.6,3.1)
Week 6:  Canonical forms of matrices (3.2,3.3)
Week 7:  Basic definition and examples of group representations (4.1,4.2)
Week 8:  The group algebra and Maschke's Theorem (4.3,4.4)
Week 9:  Characters and Schur's orthogonality relations (5.1,5.2,5.3)
Week 10:  Character tables (5.4,5.5)
Week 11:  Real representations (6.1,6.2)
Week 12:  Conclusion and revision (6.3)
Week 13:  Revision

Learning outcomes

By the end of the unit, you should be able to:

• give a range of examples of modules;
• understand which constructions and properties of vector spaces hold for modules over general rings and which fail;
• prove easy results in module theory;
• given a set of elements of a free module over a PID, determine a basis for the submodule they generate;
• explain how and why a finitely generated module over a PID (e.g. a finite abelian group) can be decomposed using torsion invariants or primary invariants;
• find the rational and Jordan canonical forms of a matrix;
• give a range of examples of representations;
• understand why the classification of representations of a group over a field up to equivalence is the same as the classification of modules over the corresponding group algebra up to isomorphism;
• prove easy results in representation theory;
• find invariant subspaces of a representation;
• compute the character table of a small group and use it to classify complex representations;
• determine whether a complex representation can be defined over the real numbers.

Assessment

Your mark for MATH3966 will be calculated as follows.

• 10%: Tutorial participation, one mark for each week up to a maximum of 10. To get the mark you must contribute to the class's understanding of how to answer the questions. This need not mean that you have solved them, but you should at least come to the tutorial with ideas that you are prepared to discuss. (If you cannot attend the actual tutorial, see me for an alternative arrangement.)
• 20%: Two (group) assignments, worth 10% each, due on Wednesday August 27 (Week 5) and Wednesday October 15 (Week 11). You may work on the assignments in groups of size one, two, or three. Every member of the group must write the answer to at least one question, and all members receive the same mark for the assignment as a whole. No discussion with other groups is allowed.
• 70%: Final exam, in the examination period in November.

Tutorials

The tutorial questions are drawn from the exercises in the lecture notes. Here is the plan for which exercises are to be discussed in which week (this is subject to change):

Week 2:  1.55, 1.56, 1.57, 1.58, 1.59
Week 3:  1.60, 1.61, 1.80, 1.81, 1.114
Week 4:  2.26, 2.27, 2.28, 2.29, 2.61
Week 5:  2.91, 2.92, 2.93, 2.94, 2.95
Week 6:  2.111, 2.112, 2.113, 2.114, 2.126
Week 7:  3.37, 3.38, 3.56, 3.57, 3.58
Week 8:  4.37, 4.38, 4.39, 4.41, 4.58
Week 9:  4.59, 4.60, 4.90, 4.104, 4.106
Week 10:  5.11, 5.12, 5.13, 5.24, 5.25
Week 11:  5.47, 5.48, 5.61, 5.62, 5.63
Week 12:  6.31, 6.32, 6.33, 6.46, 6.47

Assignments

The assignment questions are drawn from the remaining exercises. See above for the rules about working in groups.

Assignment 1 (due 27/08/08):  1.113, 2.47, 2.96

Exam information

The exam will last two hours, plus 10 minutes reading time. No books, notes, or calculators will be allowed. The exam will be based on all the material covered in lectures, tutorials, and assignments, with particular emphasis on the stated learning outcomes (see above). More details will be posted here closer to the exam period.

The exam will be broadly similar to last year's exam:

2007 exam in MATH3966 and solutions.

Reference books

The content of the unit is defined by the lecture notes, but you are encouraged to consult other books for different points of view. The following are on closed reserve in the Scitech Library.

• Rings, modules and linear algebra by Brian Hartley and Trevor Hawkes (call number 512.897 35). This covers the first half of the course, at a quite accessible level. There are other copies in Fisher and Madsen; I believe it is out of print.
• Representations and characters of groups by Gordon James and Martin Liebeck (second edition, call number 512.2 51). This covers the second half of the course, with many interesting examples and exercises. There are multiple copies in the library, and the book is available and not too expensive.
• Representation theory of finite groups and associative algebras by Charles W. Curtis and Irving Reiner (call number 512.86 167). This is a classic textbook which is far more comprehensive than MATH3966. There are multiple copies in the library, and it was recently reissued by the American Mathematical Society.

Each chapter file will be updated as more solutions become available, so it would be wise to hold off printing the last page(s) until the chapter becomes complete. (The solutions to the assignment exercises will be posted separately.)

• Solutions to Chapter 1
• Solutions to Chapter 2