—  Mathematics & Statistics

Algebra and Combinatorics Seminar, Fall 2015

Wednesdays 4-5 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.

I will discuss a way to calculate root multiplicities for indefinite Kac-Moody algebras by counting rational Dyck paths subject to various conditions. The conditions are very complicated, so exact calculation is non-trivial, but the method gives good asymptotics, at least in small rank. Well, that last is partially conjecture, but it is based on solid heuristics and computer evidence, provided by Colin Williams. The mathematical justification for all this goes through quiver varieties, and I'll explain some of that as well.

Sept 17: Ben Webster (Virginia), Gradings on (q-)Schur algebras and quiver representations

I'll explain a method for obtaining a surprising grading on (q-)Schur algebras. This grading is quite boring in the case where this algebra is semi-simple, but over a finite field, or when q is a root of unity, it's very interesting. As time allows, I'll discuss how this presentation is inspired by the geometry of quiver varieties, and how it relates to Kazhdan-Lusztig polynomials, Fock space and the representation theory of affine Lie algebras.

Oct 1: Jean-Baptiste Priez (Université Paris-Sud), Generalized parking functions and enumeration of minimal acyclic automata

We define generalized parking functions by a functional equation using the characteristic species of sets. As applications, firstly we obtain a non-commutative Frobenius characteristic over the 0-Hecke algebra action; and secondly we define bijection between those functions and (non-initial) acyclic automata. This bijection translates on parking function many informations of structure about automata. From those informations, we extract an enumeration formula of the minimal acyclic automata, by a double counting technique.

In 1937 Brauer diagrammatically defined an algebra and proved that his eponymous algebra gives the endomorphisms of the tensor powers of the natural representation for the symplectic and orthogonal groups. We now know that Brauer's results have a common generalization in which the Brauer algebra provides the endomorphisms of the tensor powers of the natural representation for the orthosymplectic Lie supergroup. This is the case when the underlying bilinear form is even. When it is odd we are instead studying the type P Lie supergroup. Moon described the relevant endomorphism algebras by generators and relations. We show that they also admit a natural diagrammatic description which generalizes Brauer's original construction. This is joint work with Ben Tharp. There will be pictures and the talk should be accessible to a general mathematical audience.

Given an \(n\times n\) matrix \(M\) over a (not necessarily commutative) field \(F\) and a candidate inverse \(X\), the system of \(n^2\) equations \(M\cdot X = I\) is satisfied iff \(X\) is indeed an inverse for \(M\) in \(\mathrm{End}_F(F^n)\). For us, it is a small wonder that

• the solution is unique, and
• the solution is the same as one would reach in solving the \(n^2\) different equations \(X\cdot M = I.\)

With students Josephine Wood and Adrienne Brackey, we are now diving heads-first into the case \(n=3\). I'll report on our progress.

The upper half of the quantum group associated to a Kac-Moody algebra is categorified by a family of graded associative algebras called Khovanov-Lauda-Rouquier (or KLR) algebras. Under a certain assumption on the characteristic of the ground field, KLR algebras R of affine ADE type are properly stratified. Roughly, this means that R-mod is stratified by categories of modules over much simpler algebras. There are actually many such stratifications of R, determined by a choice of convex order on the positive roots of the affine root system, each related via categorification to PBW bases for the quantum group. Understanding the stratum categories reduces to understanding real and imaginary strata. The real case being well-understood, we show that the imaginary strata are Morita equivalent to affine versions of Huerfano-Khovanov's zigzag algebra. This is joint work with Alexander Kleshchev.

A connection relating the Tamari lattices on symmetric groups, regarded as lattices under the weak Bruhat order, to the positive monoid P of Thompson's group F is presented. The Tamari congruence classes correspond to classes of equivalent elements in P. The two well known normal forms in P correspond to endpoints of intervals in the weak Bruhat order that determine the Tamari classes. In the monoid P these correspond to lexicographically largest and lexicographically smallest form, while on the level of permutations they correspond to 132-avoiding and 231-avoiding permutations. Forests appear naturally in both contexts as they are used to represent both permutations and elements of Thompson's monoid.

Dec 2: Tony Giaquinto, Deformation theory and quantum groups

This will be an introductory talk on deformations of algebras, with a focus on examples. Topics will include cohomology, continuous deformations, jump deformations, rigid algebras, and explicit deformation formulas. In the setting of quantum groups, frequent existence proofs regarding deformations are known, but explicit formulas are generally difficult to extract, even for the case of SL(2). For the standard quantum groups, some, but not all such formulas, can be easily deduced from the Peter Weyl theorem. I will explain these connections in the lecture.