**Office Hours:** 2844 East Hall, by drop in or appointment. If there is
demand for it, I will schedule a regular time.

**Webpage:** `http://www.math.lsa.umich.edu/~speyer/665_2012`

**Level:** Graduate students who are very comfortable with abstract
linear algebra (vector spaces, tensor products, symmetric and wedge
product), who have some familiarity with groups and
representation theory, and a high level of mathematical maturity.
Towards the end of the term, I may start using the language of categories.

**Student work expected:** I will give problem sets every week, due
Mondays in class. I will also require students to take turns serving as
scribe for the course, meaning taking TeXed notes on what I have said
that day.

**Scribing:** I will require students to take turns signing up to
record, in TeXed format, what I said in lecture. The due date for
turning notes is **one week after the lecture**, sent to me by
e-mail. I highly encourage you
to be faster, as this will both be easier for you and more helpful for
you fellow students. You may want to bring a laptop to class and try live
TeXing (some useful tips here).
I will edit these notes before posting them.

If you are scribing and are
confused about some point from the lecture, please let me know so that
I can work with you. Of course, this also applies if you are confused
and are not scribing!

There are currently 13 students registered for a 40 lecture class, and
I will scribe the first lecture. If all of you remain for the
whole term, then you will each scribe 3 times. I expect a more likely
estimate is that you will wind up scribing 4–5 times.

Please use this LaTeX template, which provides some convenient
formatting and macros. I will add useful macros throughout
the term as I think of them.

If you have never learned how to use LaTeX, then you really need to,
as it is the standard tool of all mathematical publishing and will be
crucial to anyone pursuing a career as a mathematician. Please alert
me to your situation, and I will help you.
A useful website for getting help with LaTeX difficulties is tex.stackexchange.com.

Note that these notes will be posted publicly online, in both TeX and
PDF form, with your name attached to them, and that I may edit them to
remove errors in my lectures or your understanding of them.
You give me permission to do this, but otherwise retain copyright to
your notes.

I don't intend for you to need to consult books and papers outside your notes. If you do consult such, you should be looking for better/other understanding of the definitions and concepts, not solutions to the problems.

You

Problem Set 1, due Sept 17

Problem Set 2, due Sept 24

Problem Set 3, due Oct 1

Problem Set 4, due Oct 8

Problem Set 5, due Oct 17 (Wednesday) because of Fall break

Problem Set 6, due Oct 22

Problem Set 7, due Nov 5 (two week problem set)

Problem Set 8, due Nov 12

Problem Set 9, due Nov 19

Problem Set 10, due Dec 3 (two week problem set)

Problem Set 11, due Dec 10. This problem set works you through a beautiful application of crystals.

- Sept 7: Partitions, elementary symmetric functions (TeX) Scribe: Kevin Carde
- Sept 10: Complete homogenous symmetric functions, the Hall inner product (TeX) Scribe: Yi Su
- Sept 12: The Hall inner product, continued (TeX) Scribe: Jake Levinson
- Sept 14: Schur polynomials (TeX) Scribe: Charlotte Chan
- Sept 17: Jacobi-Trudi (Guest star! Jonah Blasiak lectures) (TeX) Scribe: Chris Fraser
- Sept 19: The ratio of alternants formula (TeX) Scribe: Stefan Froelich
- Sept 21: Schur functions are orthonormal (TeX) Scribe: Rachel Karpman
- Sept 24: The Pieri rule (TeX) Scribe: David Speyer
- Sept 26: Skew Schur functions (Guest star! Ricky Liu lectures) (TeX) Scribe: Aaron Pribadi

- Sept 28: Compact groups and Haar measure (TeX) Scribe: Charlotte Chan
- Oct 1: Decomposition of representations into simple reps, introduction to characters (TeX) Scribe : Yi Su
- Oct 3: More characters, the ring of matrix coefficients (TeX) Scribe: Chris Fraser
- Oct 5: The (weak) Peter-Weyl theorem (TeX) Scribe: Stefan Froelich
- Oct 8: Consequences of the (weak) Peter-Weyl theorem (TeX) Scribe: Rachel Karpman
- Oct 10: Relation between the general linear group and the unitary group (TeX) Scribe: Jake Levinson
- Oct 12: Schurs are characters of irreps (TeX) Scribe: David Speyer
- Oct 15: FALL BREAK — Jump in a leaf pile!

- Oct 17: Constructing the irreps of GL
_{n}, preview (TeX) Scribe: Charlotte Chan

This class was meant to be "Examples", but we sent more time on homework discussion than expected. - Oct 19: Schur functors (TeX) Scribe: Aaron Pribadi
- Oct 22: Realizing Schur functors using matrix minors (TeX) Scribe: Yi Su
- Oct 24: The semistandard basis for V
_{λ}(TeX) Scribe: Stefan Froelich - Oct 26: Invariants (TeX) Scribe: Rachel Karpman

Some basic points I think I presented badly.

- Oct 29: The perfect matching basis for SL(2) invariants (TeX) Scribe: Chris Fraser
- Oct 31: The web basis for SL(3) invariants (TeX) Scribe: Jake Levinson
- Nov 2: Proofs (TeX) Scribe: Aaron Pribadi

- Nov 5: Specht modules (TeX) Scribe: Jake Levinson
- Nov 7: Schur-Weyl duality (TeX) Scribe: Yi Su
- Nov 9: Applications of Schur-Weyl
duality (TeX) Scribe: Chris Fraser

The Frobenius character map, a topic I wanted to cover but couldn't - Nov 12: Catch up day (TeX) Scribe: Charlotte Chan

- Nov 14: Lie algebras and Lie groups (TeX) Scribe: Stefan Froelich
- Nov 16: The Lie algebra gl
_{n}(TeX) Scribe: Rachel Karpman - Nov 19: Interaction between gl
_{2}strings (TeX) Scribe: Aaron Pribadi - Nov 21: High weight vectors (TeX) Scribe: David Speyer

- Nov 26: Introduction to crystals (TeX) Scribe: Charlotte Chan
- Nov 28: Crystal structures on words and tableaux (TeX) Scribe: Yi Su
- Nov 30: Regular crystals (TeX) Scribe: Aaron Pribadi
- Dec 3: Uniqueness of regular crystals (TeX) Scribe: Jake Levinson
- Dec 5: Word crystals are regular (TeX) Scribe: Rachel Karpman
- Dec 7: The Littlewood-Richardson rule (TeX) Scribe: Stefan Froelich
- Dec 10: Jeu de taquin and growth diagrams Scribe: Chris Fraser