Sept 7, Stephen Doty (Loyola) 3:00PM: Generalized q-Schur algebras

Generalized q-Schur algebras are q-deformations of certain finite dimensional algebras (the generalized Schur algebras) introduced by S. Donkin in the late 1980s in order to study the representation theory of reductive algebraic groups. I will summarize some of the historical background and outline a recent approach to these algebras via generators and relations (joint work with Tony Giaquinto).
Reference: arxiv.org/pdf/1012.5983.pdf

Sept 12, Peter McNamara (Stanford): Finite dimensional representations of KLR algebras

Khovanov-Lauda-Rouqier algebras are a family of algebras that appear in categorifying quantum groups. I will talk about the category of finite-dimensional representations of these algebras - classifying the simple representations, giving some understanding of higher Ext groups, and the related combinatorial structures. No previous knowledge of KLR algebras will be assumed.

Sept 19, Cary Huffman (Loyola): The MacWilliams Identities in GF(q)-linear GF(q^t)-codes.

There are numerous classic results for linear block codes over finite fields. When a new class of codes is studied, it is natural to ask if these classical results extend to the new class of codes. Extending the MacWilliams Identities uses a bit of linear character theory combined with the trace function and Poisson summation. No coding theory knowledge required.

Sept 26, Christopher Drupieski (DePaul): Finite-generation problems for cohomology rings

Cohomology rings can be useful tools for making connections between representation theory and algebraic geometry, but first one must show that these rings are finitely-generated. In this talk I will describe some of the major results in this area, and some of the standard tools and techniques that are used to attack these types of problems. I will then describe a previously-unobserved proof for a special case of the famous Friedlander-Suslin finite-generation theorem for the cohomology of finite group schemes. (No previous expertise on, or love for, cohomology will be assumed in this talk.)

Oct 3: Evan Jenkin (U. of Chicago): Classical Analogues of Fusion Categories

The development of quantum mechanics in physics led mathematicians to formulate various notions of quantization, that is, ways of deforming algebraic and geometric objects associated with classical mechanics into quantum mathematical objects. Tensor categories are one incarnation of quantum algebra, and their theory has developed in recent years largely independently of any underlying classical objects. A better understanding of what the "classical limits" of tensor categories might be could help explain conjectural relationships between quantum and classical invariants in low-dimensional topology.

October 17, Oded Yacobi (Toronto): An overview of algebraic categorification through polynomial functors

We will motivate the basic ideas of categorical representations of the affine special linear Lie algebra by looking at examples. Our main focus will be the category of polynomial functors, which we will show categorifies a host of symmetries on the symmetric functions. We will also discuss how Schur-Weyl duality (over arbitrary characteristic) fits nicely into this story. This is based on joint work with Jiuzu Hong.

October 24, Peter Tingley: Lie groups, representations, and combinatorics.

Group theory is about understanding symmetries of (mathematical) objects. But sometimes it is useful to turn this around: instead of starting with an object and trying to understand its symmetries, one starts with a group and tries to understand all objects or spaces that have those symmetries. At this point, one is doing representation theory. In this talk I will discuss the representation theory of various infinite groups, mostly the group SL(n) of n by n matrices with determinant 1. In particular, I will explain how various natural questions can be answered using some nice combinatorics (Kashiwara's crystals). This is the first of a 2-part talk, and is intended to be accessible to anyone with some background in either linear algebra or abstract algebra (and perhaps to be boring to experts); I may mention some off my own work, but only briefly.

October 31, Peter Tingley: Studying affine algebras using crystals

In this second talk I will introduce affine Kac-Moody algebras and their representations, which are the subjects of much of my own work. These are closely related to the Lie algebras we saw in the first half, and can be studied with some of the same ideas. I'll talk about 2 types of representations: finite dimensional representations and highest weight representations (which are now infinite dimensional). We'll see some ways these can be understood combinatorially using crystals, and some applications to things like MacDonald polynomials. This will include some results from my thesis and from a recent paper with Anne Schilling, but will include lots of other people's work too. The story is ongoing, and I'll mention some current research at the end.

November 7, Dinakar Muthiah: Affine PBW Bases and MV Polytopes in Rank 2

MV (Mirkovic-Vilonen) polytopes appear in a variety of different guises. Anderson first defined them as the moment map images of MV cycles in the affine Grassmannian. Kamnitzer provided an explicit description of MV polytopes, and in giving this description, he discovered that MV polytopes encode precisely the reparameterization data of PBW bases. Later work of Baumann-Kamnitzer showed a connection with Lusztig's varieties of nilpotent preprojective algebra representations.

A natural question is to ask how much of this picture generalizes to the affine case. Beck-Chari-Pressley, Akasaka, and Beck-Nakajima define PBW bases for all affine types. Recent work of Baumann-Kamnitzer-Tingley give a geometric definition of affine MV polytopes in all untwisted affine types using Lusztig's nilpotent varieties. Also, work of Baumann-Dunlap-Kamnitzer-Tingley gives a combinatorial definition of MV polytopes for the two rank 2 affine cases. However, unlike in the finite-type case, it is not at all clear that these definitions coincide.

In this talk, I will discuss recent work (joint with P. Tingley) that addresses this question. We show that all three notions of MV polytopes coincide in two rank-2 affine cases. Perhaps even more interesting than the statement of the result is the method of proof. Explicitly checking the equations of affine MV polytopes is too unwieldy. Instead, we give a short list of axioms for affine MV polytopes and check these axioms in the three cases. In particular, our methods give a new proof of the analogous result in the finite-type case and shed some light on why MV polytopes are so prevalent.

November 14, Vinoth NandaKumar: Quiver varieties and the infinity crystal in non-symmetric type

Given a symmetric Kac-Moody algebra one can attach a preprojective algebra over the complex numbers. Lusztig's quiver varieties are the varieties of nilpotent representations of the preprojective algebra on a fixed vector space. Kashiwara and Saito show that the union of the irreducible components of these varieties naturally index the infinity crystal. Here we describe a generalization of this result to arbitrary symmetrizable Kac-Moody algebras. This is based on a definition of non-symmetric preprojective algebras due to Dlab-Ringel, where the main new technical difficulty is that one must use non-algebraically closed fields. The techniques employed by Kashiwara and Saito go through without much difficulty in the more general setting. Time permitting, I will mention an application to constructing MV polytopes. This is joint work with P. Tingley.

November 28, Aaron Lauve: A menagerie of "coinvariant" spaces: a survey of modern takes on a classical theorem of Chevalley-Shephard-Todd
(Dumbach Hall, Room 123)

We take a brief tour through the history of the invariant theory of the symmetric group, from Schur and Weyl, through Chevalley, to some modern masters. Scheduled stops include Macdonald's positivity conjecture, the n! conjecture of Garsia-Haiman, and a menagerie of coinvariant spaces (each indexed by famous combinatorial gadgets) studied by Francois Bergeron and friends. We close with some recent stability results for diagonal coinvariants due to Farb, Bergeron, and their collaborators.

(Note 1:   This is meant mainly as a "pre-talk" for a lecture by Jenny Wilson (U. Chicago) scheduled for next semester. Only brief mention of my own joint work with F. Bergeron and S. Mason work will be made.)
(Note 2:   A majority of the talk will be accessible to students having taken Math 314. This includes some open research questions, should a student wish to pursue any.)

Dec 5, Zajj Daugherty: The quasi-partition algebra

The partition algebra arises as a centralizer algebra for the symmetric group acting on the k-fold tensor product of its permutation representation. It also has a diagrammatic description, generated by set partitions of 2k elements (thus the name). The permutation representation is not in general irreducible though. In this talk, I will define a new related algebra, the quasi-partition algebra, which also arises as a centralizer algebra for the symmetric group, but now acting on the k-fold tensor product of the large irreducible submodule of the permutation representation. A similar diagrammatic description and some wonderful combinatorial results will be given. This work is joint with Rosa Orellana.