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
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.
I will pick and choose topics from various sources freely available online, with some flexibility as to precisely what we're covering. One source is
All numbered references are to that book unless specified otherwise. The link opens a pdf file, which you might want to download for safekeeping. Another book I might occasionally refer you to is
For some of the more general ring-theoretic material discussed in class you might also consult
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. Unless specified otherwise the sections are those in Etingof.
The date is that of the lecture, so please do the reading before that.
| Due date | Assignment | Remarks | |
|---|---|---|---|
| 1 | Th Sep 11 | 2.2 - 2.8 |
the dash between 2.2 and 2.8 means a range: all of those sections, 2.2 and 2.8 inclusive |
| 2 | Th Sep 18 |
3.1, 3.6 4.1 - 4.3 |
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| 3 | Th Sep 25 |
2.11 4.4, 4.5 |
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| 4 | Th Oct 02 |
4.6, 4.7 |
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| 5 | Th Oct 09 | 5.8 - 5.10 |
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| 6 | Th Oct 23 |
5.2, 5.3 3.10 |
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| 7 | Th Nov 13 |
5.12 - 5.15 |
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No classes the whole week of Nov 24 |
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On occasion I'll post extra notes, comments, etc. in this space.
The classification of semisimple rings is carried out in Theorem 3.5 of [Lam]. More generally, Sections 2 and 3 of that book are all about semisimplicity, so you will see a number of alternative characterizations there. That text focuses on rings rather than algebras: instead of being vector spaces over some field, these are only abelian groups still equipped with an associative multiplication, etc.
We exchange homework via UBLearns: I post the assignments there through the 'Assignments' facility provided by the system and you'll similarly be able to upload files.
No late homework for any reason, but we are dropping the two lowest scores.
| Due date | Assignment | Remarks | |
|---|---|---|---|
| 1 | Th Sep 11 | ||
| 2 | Th Sep 18 | ||
| 3 | Th Sep 25 | ||
| 4 | Th Oct 02 |
+
show that the fundamental representation of a symmetric group is irreducible |
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| exam 1 | Th Oct 09 | No homework this week | |
| 5 | Th Oct 16 | ||
| 6 | Th Oct 23 | ||
| 7 | Th Oct 30 | ||
| 8 | Th Nov 13 | ||
| exam 2 | Mo Dec 08 | We're done! | |
The grader will assign each homework a grade out of 10 with only a few problems being graded (3 or 4 or so). There will also be points awarded for the overall completeness of the assignment. I won't be telling you in advance which problems are being graded though..
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.
We're having two take-home exams. The dates are as follows
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.
We'll drop the two lowest homework scores.
If you have any questions, don't hesitate to email me.
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