MTH 461 / 561: Introduction to Representation Theory
Fall 2018

Instructor: Alexandru Chirvasitu

Lectures:
• TR 8:00 - 09:20 AM, Math 150
Office: 216 Mathematics Building
Office hours: TuTh 12 - 1
Email: achirvas AT buffalo.edu
Representation theory is a field of mathematics that seeks to recast various algebraic objects such as algebras or groups or Lie algebras as collections of matrices operating on a vector space so as to preserve the inherent algebraic structure.

To illustrate the principle with a simple example, consider the additive group Z of integers equipped with its usual addition operation (if you are not familiar with groups have a look at the link; we will define these). Its structure is very simple: there's the element 1, and everything else is obtained from this element by addition and taking inverses for the addition operation: n=1+1+..+1 if n is positive and
• n=1+..+1 with n summands if n is positive (or zero);
• n=-(1+..+1) if n is negative.
To reflect this structure in matrix language we can attach an arbitrary invertible nxn matrix T (with complex entries, say) to 1 ∈ Z. Then the matrix associated to m ∈ Z will be Tm, where T0 means the identity matrix by convention and T-m is the inverse of Tm. Specifying T means giving an n-dimensional complex representation of Z (`n-dimensional' because we're working with nxn matrices and `complex' because they're matrices with complex entries).

Having re-imagined the integers as matrices and the addition operation as matrix multiplication, we can now use the tools familiar from linear algebra to pick apart the inner workings of Z equipped with its addition operation: matrices have eigenvalues and eigenvectors for instance, and these give you a handle on what Z ``is like''.

We'd need some familiarity with linear algebra (so say vector spaces, eigenvalues, eigenvectors), but I will try to provide as much background as I can, on an as-needed basis.
Textbook

I will pick and choose topics from various sources freely available online, with some flexibility as to precisely what we're covering. One reference is

I'll probably ask you to read some of the sections in the textbook before most lectures, so we're in sync. I'll post the reading assignments here, numbered as sections of the textbook. The date is that of the lecture, so please do the reading before that.

Due date Assignment Remarks
1 Tu Sep 4 2.2, 2.8
2 Th Sep 6 2.3, 2.4
3 Tu Sep 11 2.5, 2.6, 2.7
4 Th Sep 13 3.1
5 Tu Sep 18 3.6, 4.1, 4.2
6 Th Sep 20 4.3
7 Tu Sep 25 2.11, 4.4
8 Th Sep 27 4.5
9 Tu Oct 2 4.6, 4.7
10 Tu Oct 9 5.8, 5.9, 5.10
11 Tu Oct 23 5.2
12 Th Oct 25 5.3, 3.10
13 Tu Nov 6 Chapter 7 and 8.1, 8.2
from Serre's book
14 Th Nov 8 5.12, 5.13 from Etingof
15 Th Nov 15 5.14, 5.15 from Etingof
No classes the whole week of Nov 19
16

Supplementary material

On occasion, I'll post extra notes, comments, etc. in this space.
Homework

All assignments are due in class on the dates indicated in the table below.

Due date Assignment Remarks
1 Tu Sep 11 Hw 1
2 Th Sep 20 Hw 2
3 Th Sep 27 Hw 3
4 Th Oct 4 Hw 4
+
show that the fundamental representation
of a symmetric group is irreducible
Th Oct 11 Exam
5 Th Oct 18 Hw 5
6 Th Oct 25 Hw 6
7 Th Nov 1 Hw 7
8 Th Nov 15 Exercise 7.3 from Serre's book
9 Tu Nov 27 Hw 9
Th Dec 6 Exam

Do collaborate on the homework if you like, but write up your own solutions. I also strongly advise you to have a look at the UB Academic Integrity Policy, as it very much applies to this class.

And by all means drop by at office hours if you need a hand.
Exams

We're having two exams, both in class at the usual time. The dates are as follows

• Exam 1: Th Oct 11
• Exam 2: Th Dec 6

Exam policy:

No communication of any kind (i.e. it needs to be an individual effort), but you are allowed any reading material you think will be of use to you (including electronic or via the internet).

There will be no make-up exams. Should you miss one for a very serious reason (e.g. illness), you'll need documentation (such as a doctor's note) justifying the absence. Under such circumstances I will agree to substitute the other one for the one you missed.