Organizers: Aaron Lauve, Peter Tingley
February  
3 
No talk 

10 
Organizational meeting 
(at Ireland's pub) 
17 
Asilata Bapat (U. of Chicago) 
The BernsteinSato polynomial and the Strong Monodromy Conjecture 
24 
Travis Scrimshaw (Minnesota) 
Star crystal structure on rigged configurations 
March  
2  Seckin Adali (UIC)  Singularities of Restriction Varieties 
9 
No talk (spring break) 

16 
No talk 

23 
Vasily Dolgushev (Temple) 
A manifestation of the GrothendieckTeichmueller group in geometry 
30 
Chris Drupieski (DePaul) 
Some geometric objects associated to representations of Lie (super)algebras 
April  
6  Peter Tingley  Deformation theory and the PeterWeyl theorem 
13 
Aaron Lauve 
The saga of irreducible representations of the symmetric group 
20 
Steve Doty 
Primitive idempotents in a multiplicityfree family of finite dimensional algebras 
27 
Aaron Lauve 
Primitive idempotents in a multiplicityfree family, Part II 
June  
15 ** 3:00 p.m. ** 
Hugh Thomas (UQAM, Montréal, CA) 
Nathan Reading's shards and representation theory of preprojective algebra 
Directions 

IES (the institute for environmental sustainability) is located at the corner of W. Sheridan and N. Kenmore Avenues, Chicago, IL (map)
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Abstracts 

Feb 17: Asilata Bapat. The BernsteinSato polynomial and the Strong Monodromy Conjecture
To a singularity of an algebraic hypersurface, one can associate an invariant called the BernsteinSato polynomial or the bfunction. Although the bfunction is important and interesting, it is usually difficult to compute. It is conjectured (Strong Monodromy Conjecture or SMC) that some roots of the bfunction can be obtained from the poles of another singularity invariant, the topological zeta function. I will sketch the proof of the SMC for the case of Weyl hyperplane arrangements, via the "n/d conjecture" of Budur, Mustaţă, and Teitler. I will also describe some results towards computing the bfunction of these arrangements, focusing on a special case (the Vandermonde determinant). This is joint work with Robin Walters.
Feb 24: Travis Scrimshaw. Star crystal structure on rigged configurations
We describe the crystal structure and *crystal structure on the crystal B(\infty) using rigged configurations. This is joint work with Ben Salisbury.
Mar 2: Seckin Adali. Singularities of Restriction Varieties
Restriction varieties in the orthogonal Grassmannian are subvarieties of OG(k, n) defined by rank conditions given by a flag that is not necessarily isotropic with respect to the relevant symmetric bilinear form. In particular orthogonal Schubert varieties are examples of restriction varieties. In this talk, I will describe a resolution of singularities for restriction varieties and give a smoothness criterion. The image of the exceptional locus of this resolution gives the singular locus of a restriction variety which allows an algorithm for the singular locus. I will illustrate these results with examples.
March 23: Vasily Dolgushev. A manifestation of the GrothendieckTeichmueller group in geometry
Inspired by Grothendieck's legogame, Vladimir Drinfeld introduced, in 1990, the GrothendieckTeichmueller group GRT. This group has interesting links to the absolute Galois group of rationals, finite type invariants of tangles, deformation quantization and theory of motives. My talk will be devoted to the manifestation of GRT in the extended moduli of algebraic varieties, which was conjectured by Maxim Kontsevich in 1999. My talk is partially based on the joint paper with Chris Rogers and Thomas Willwacher which can be found here.
March 30: Chris Drupieski. Some geometric objects associated to representations of Lie (super)algebras
To each representation of a finitedimensional restricted Lie algebra one can associate a corresponding algebrogeometric object, called its (cohomological) support variety. In this talk I'll summarize some of the foundational results in this area that were established by Friedlander and Parshall in the 1980s. I'll then discuss some of my recent work with Jon Kujawa, in which we have begun to investigate the situation for (restricted and nonrestricted) Lie superalgebras.
April 6: Peter Tingley. Deformation theory and the PeterWeyl theorem
I will give an expository explanation of how I like to think about defomration theory. This will be related to what Tony told us about last semester, but also kind of different. Then I'll explain some ideas Tony and I have been thinking about (along with our former student Alex Gilman) that use quantum PeterWeyl theorem to make deformations look nicer.
April 13: Aaron Lauve. The saga of irreducible representations of the symmetric group
After the work of Artin, Maschke, and Wedderburn, we know that the group algebra \(\mathbb CS_n\) of the
symmetric group is really just a collection of blockdiagonal matrices inside \(M_{d}(\mathbb C)\), with
block sizes \(d_i\) corresponding to the dimensions of the irreducible representations of \(S_n\) (and \(d=\sum_i d_i\)). But
how do you realize it as such, concretely? Young gave us one way, Frobenius another, VershikOkunkov another,
JucysCherednikNazarov yet another. That these are all the same is a small wonder. Each set of authors,
of course, presents their approach as "the right one." In this talk, I will explain most of these approaches before
giving you the simplest approach I know of; the ideas are due to our very own Tony Giaquinto. (I leave it
to the audience to decide whether it's another "the right one.") This is a mainly expository talk that is meant
as preparation for Steve's talk next week. He will speak on joint work (he, me, and recent Loyola graduate George
Seelinger).
Large stretches of this talk will be accessible to undergrads with a semester of abstract
algebra under their belt.
April 20: Steve Doty. Primitive idempotents in a multiplicityfree family of finite dimensional algebras
This talk continues the talk of Aaron Lauve from last week, and
generalizes it. In a sense this is also a continuation of Tony
Giaquinto's talk from 2014, where he first explained his recursive way
of constructing primitive idempotents in the Hecke algebra of type
A. What is really going on is that restriction from \(S_n\) to \(S_{n1}\)
(or from \(\mathcal{H}(S_n)\) to \(\mathcal{H}(S_{n1})\))
is multiplicityfree, which implies that the GelfandTsetlin basis
makes sense (it is a kind of "canonical" basis). I will explain why
there are "canonical" primitive idempotents in any such setting, and
how to find them. This also explains why Giaquinto's recursive
description is essentially implicit in the work of VershikOkounkov in
1996 & 2004. As a result, we obtain a way to compute primitive
idempotents in any multiplicityfree family, including more exotic
examples such as Brauer algebras and partition algebras.
(This is joint work with Aaron Lauve and George Seelinger.)
April 27: Aaron Lauve. Primitive idempotents in a multiplicityfree family, Part II
This talk continues the talk of Steve Doty from last week. We recall the new formula for primitive idempotents, valid in any multiplicityfree family, given last week. The formula reduces the problem of computing (an indecomposible family of) primitives to computing the central primitives; the latter are unique, but not necessarily any easier to get a handle on. After surveying what is known in the classical case of \(\mathbb kS_n\) (a combinatorial formula due to Young and a charactertheory formula due to Frobenius), we turn to our main results. Namely, two Lagrangeinterpolation type formulas, one recursive, relying on generalized JucysMurphy elements, the other relying on judicious choice of generators for the centers of each member \(\mathcal A_n\) in the multiplicityfree family \(\left\{\mathcal A_n\right\}_{n\geq1}\).
If there is time (I expect there actually will be), we will do some computations
in Sage to illustrate the results. If there is still more time (I don't think there
will be), we mention a third formula for the primitive central idempotents in
the spirit of Frobenius' formula.
(This is joint work with Steve Doty and George Seelinger.)
June 15: Hugh Thomas. Nathan Reading's shards and representation theory of preprojective algebras
For a finite Coxeter group W, Nathan Reading gave a beautiful construction of the joinirreducibles of weak order on W, in terms of the "shards" of the reflection arrangement. I will recall this construction, and also explain how it arises in another context: that of categories of semistable modules for preprojective algebras of Dynkin type. This is joint work with David Speyer, and also draws on previous joint work with Osamu Iyama, Nathan Reading, and Idun Reiten, arXiv:1604.08401.