# Group rings arrr commutative

If you are familiar with group rings, you might think that the title of this post is false. If G is a nonabelian group, multiplying the basis elements g and h in $\mathbb{Z}G$ can yield $gh \neq hg$, so we have a problem. In general, if you have a problem that you can’t solve, you should cheat and change it to a solvable one (According to my advisor, this strategy is due to Alexander the Great). Today, we will change the definition of commutative to make things work.

# This paper was written for our blog

I’ve recently been reading a paper which ties together a number of this blog’s themes: Canonical Quantization of Symplectic Vector Spaces over Finite Fields by Gurevich and Hadani. I’m going to try to write an introduction to this paper, in order to motivate you all to look at it. It really has something for everyone: symplectic vector spaces, analogies to physics, Fourier transforms, representation theory of finite groups, gauss sums, perverse sheaves and, yes, $\theta$ functions. In a later paper, together with Roger Howe, the authors use these methods to prove the law of quadratic reciprocity and to compute the sign of the Gauss sum. For the experts, Gurevich and Hadani’s result can be summarized as follows: they provide a conceptual explanation of why there is no analgoue of the metaplectic group over a finite field. Not an expert? Keep reading!

# Request: Modular forms

There was a request containing the phrase, “theory of modular forms,” so I’ll write an introduction to that. Chris seems to be taking care of the rest of that paragraph.

Pretty much all of the material below is 50-150 years old. Don’t expect too much originality.

# How to write down the representations of GL_n

A few years ago, I gave a talk at NCSU on some work I had done on Littlewood-Richardson numbers, cluster algebras and such things. For the first half hour or so, I outlined the basic results I would be using about the representation theory of the group $GL_n$. Afterwards, I had a number of grad students thank me for this. So I’m going to try to turn that into a blog post (and enlarge it a little). The goal here is not to give you any proofs; rather, I want to get to the main results, show you how they connect and, above all, how to actually write down the representations of $GL_n$.

# Representations of the small quantum group

Dear Internets,

I’m sorry for having disappeared for so long, and I promise I’ll get back to my series on planar algebras soon. I’ve been busy writing papers and couldn’t really justify to my coauthors why I’m not writing our papers and instead writing math on the internet. However, Saturday I was in San Francisco without access to LaTeX, and Malia was at her spinning guild, so here’s a post!

Yours,

Noah

P.S. Below the break I’ll walk you through understanding much of the representation theory of the small quantum group for $\mathfrak{sl}_2$ at a third root of unity, assuming you already understand the representation theory of the usual lie algebra $\mathfrak{sl}_2$. It’s an interesting example to work through because the representation theory is not semisimple. There will be lots of fun pictures. The choice of a third root of unity isn’t important here, everything would work similarly for any odd root of unity.

# Hypertoric varieties and Koszul duality

So, on Wednesday, I gave a talk with the above title at IAS, about work in progress with Tom Braden, Tony Licata, and Nick Proudfoot.  I was hoping to get David Nadler to blog it for me, but he was *ahem* indisposed.  Failing that, I’ll direct you all to David Ben-Zvi’s notes (warning: freaking huge PDF).  Hopefully, that will whet your appetite for the forthcoming paper.

# Representations of reductive groups in characteristic p

I’ve been at a couple of interesting conferences lately and so I have a lot to talk about. I’ll start by summarizing an excellent expository talk by Jonathan Brundan which he gave at an MSRI introductory workshop last week.

Let G be a reductive group over an algebraically closed field k of characteristic p. The topic of this post is the algebraic representations of G. In other works, we want to study algebraic maps $G \rightarrow GL(V)$ where V is a finite dimensional vector space over k. Over the years, a few people (Soroosh, Carl, Alex Ghitza) have asked me what I knew about this theory and I’m afraid that I always gave them very incomplete or inaccurate answers. Now, that I’ve been to Brundan’s talk I think that I understand what is going on much better and I’d like to summarize it. Of course there will be nothing “new” in this post — I think that all the theory was worked out 20 years ago.

# Pre-Talbot seminar

John Francis and I are organizing a seminar at MIT to prepare for the Talbot workshop at the end of March. The aim of the seminar is to help people understand Gaitsgory’s preprint about quantum geometric Satake and quantum Langlands, which should play a significant role in the workshop. As it happens, Joel posted about this very result last August.

I gave the first talk on Wednesday, giving an overview of the subject, and I just put some notes up on the seminar page. I tried to gloss over as many technical details as possible, because of the time constraints, but I’d like to hear about any actual errors, significant or not. The talk ended up bearing some structural similarity to Joel’s post (pure coincidence – although I said a bit more about tori). Speaker recruitment went perhaps a little too successfully, since I forgot that one of the weeks was spring break.

Addendum: If you’re a fan of Koszul duality, the notes from the Chicago talk (on the seminar page) have a sketch of a proof of the equivalence between factorizable sheaves and quantum group representations using a rather odd manifestation of Koszul duality.

# The Cheewhye Diagram

About five years ago, Cheewhye Chin gave a great year-long seminar on Langlands correspondence for $GL_r$ over function fields, building up to a description of Lafforgue’s proof of the theorem. In the beginning, he drew a diagram that captured the general architecture of the proof, and I liked it so much that I stole it for a talk I gave at Talbot in 2005. It seemed to get a good reception, and Mark Behrens pointed out that the Eichler-Shimura correspondence also fits into the picture with minimal alteration.

If we remove all of the explanatory text, the diagram looks like this:

$. \qquad \quad \text{Geometric}$
$. \qquad \nearrow \qquad \qquad \searrow$
$\text{Spectral} \qquad \qquad \text{Algebraic}$
$. \qquad \nwarrow \qquad \qquad \swarrow$
$. \qquad \quad \text{Analytic}$

I was a bit hesitant to draw this, because my advisor once told me, “If you ever find yourself drawing one of those meaningless diagrams with arrows connecting different areas of mathematics, it’s a good sign that you’re going senile.” Anyway, I’ll explain roughly how it works.

# Kronheimer on “Knot Groups and Lie Groups”

So, I’m in lovely Edinburgh, Scotland (everyone I’ve told about this said “Scotland? In November?” but it’s not actually worse than New Jersey) in advance of the Maxwell Colloquium on Knot Homology.

By sheer luck, my trip here happened to overlap with the University of Edinburgh’s Whittaker Lecture which is a bit like the Bowen Lectures at Berkeley, except that there’s only one of them. By even more luck, the speaker with Prof. Peter Kronheimer (from Harvard) and the topic was “Knot Groups and Lie Groups.”