—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Spring 2016

Wednesdays 4-5. Location: IES 111

Organizers:  Aaron Lauve, Peter Tingley

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

Parking is available on-campus for $7 in the Parking Garage (building P1 on the Lake shore campus map). To get to the Parking Garage, enter campus at the corner of West Sheridan Road and North Kenmore Avenue.

Feb 17: Asilata Bapat. The Bernstein-Sato polynomial and the Strong Monodromy Conjecture

To a singularity of an algebraic hypersurface, one can associate an invariant called the Bernstein-Sato polynomial or the b-function. Although the b-function is important and interesting, it is usually difficult to compute. It is conjectured (Strong Monodromy Conjecture or SMC) that some roots of the b-function 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 b-function 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 Grothendieck-Teichmueller group in geometry

Inspired by Grothendieck's lego-game, Vladimir Drinfeld introduced, in 1990, the Grothendieck-Teichmueller 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 finite-dimensional restricted Lie algebra one can associate a corresponding algebro-geometric 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 non-restricted) Lie superalgebras.

April 6: Peter Tingley. Deformation theory and the Peter-Weyl 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 Peter-Weyl 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 block-diagonal 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, Vershik-Okunkov another, Jucys-Cherednik-Nazarov 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 multiplicity-free 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 \(S_{n-1}\) (or from \(\mathcal{H}(S_n)\) to \(\mathcal{H}(S_{n-1})\)) is multiplicity-free, which implies that the Gelfand-Tsetlin 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 Vershik-Okounkov in 1996 & 2004. As a result, we obtain a way to compute primitive idempotents in any multiplicity-free 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 multiplicity-free 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 multiplicity-free 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 character-theory formula due to Frobenius), we turn to our main results. Namely, two Lagrange-interpolation type formulas, one recursive, relying on generalized Jucys-Murphy elements, the other relying on judicious choice of generators for the centers of each member \(\mathcal A_n\) in the multiplicity-free 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 join-irreducibles 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.