MTH 461 / 561: Introduction to Representation Theory
Fall 2019
Instructor: Alexandru Chirvasitu

Lectures: Office: 216 Mathematics Building
Office hours: W 11-12
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.

The main characters in our class will be Lie algebras. These are vector spaces equipped with some additional structure that makes for a very rich theory, as we will see over the course of the semester.

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

The main resource will be with serving as a secondary source later on.
Reading

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.

Unless specified otherwise the sections are from Humphreys' book. Sections from the Fulton-Harris book will be marked with an 'FH' in the last column.

Due date Assignment Remarks
1 W Aug 28 1.1, 1.2
2 F Aug 30 1.3, 1.4
2.1, 2.2
3 W Sep 4 2.3, 3.1, 3.2
4 F Sep 6 3.3
5 M Sep 9 4.1
6 W Sep 11 4.2
7 F Sep 13 4.3
8 F Sep 20 5.1, 5.2, 5.3, 5.4
9 F Sep 27 6.1, 6.2, 6.3, 6.4
F Oct 4 return 1st take-home exam
No classes the whole week of Oct 7
10 F Oct 18 II.7
11 F Oct 25 II.8
12 F Nov 1 11.1, 11.2
FH
No classes F Nov 8
13 F Nov 15 VI.20
No classes the whole week of Nov 25
14 F Dec 6 III.9


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 F Sep 6 I.1: 2,6,7,8
I.2: 2,3,7,8
I.3: 5,6,8		 Disregard references to
positive-characteristic fields,
i.e. solve only the portions of those problems
that pertain to fields of characteristic zero.
2 F Sep 13 II.4: 5,7,8
3 F Sep 20 II.5: 1,3,4,5,8
4 F Sep 27 II.6: 1,3,6,7
Exam 1 F Oct 4 Exam
5 F Oct 18 II.7: 1,2,4,7
6 F Nov 1 Exercises 11.10 and 11.11 in Fulton-Harris, p.151 FH
7 F Nov 15 VI.20: 1, 3
Exam 2 F Dec 6 III.9: 2, 5, 6, 7 For problem 7 only solve the first part, before
the sentence starting `More generally [...]'


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 take-home exams. The dates are as follows


Exam policy:

Those are the due dates. I will post the problem lists a few days before, so you will have plenty of time to work on them.
Grading
We'll drop the two lowest homework scores.
If you have any questions, don't hesitate to email me.

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