Organizers: Aaron Lauve, Peter Tingley
September  
2 


9 
Peter Tingley 
Root multiplicities and Dyck paths 
17 *Thursday* 
Ben Webster (Virginia) 
Gradings on (q)Schur algebras and quiver representations 
23 


October  
1 *Thursday* 
JeanBaptiste Priez (ParisSud)  Generalized parking functions and enumeration of minimal acyclic automata 
7 
No talk (AMS meeting recovery) 

14 
Jonathan Kujawa (Oklahoma) 
The marked Brauer algebra 
21 
No meeting 

28 


November  
4  Aaron Lauve  Matrix Madness 
11 
Robert Muth (Oregon) 
Affine zigzag algebras and imaginary strata for KLR algebras 
18 
Zoran Sunic (Texas A&M) 
Tamari lattices and the Thompson monoid 
25 
No talk (thanksgiving) 

December  
2 

Directions 

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

I will discuss a way to calculate root multiplicities for indefinite KacMoody algebras by counting rational Dyck paths subject to various conditions. The conditions are very complicated, so exact calculation is nontrivial, 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 semisimple, 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 KazhdanLusztig polynomials, Fock space and the representation theory of affine Lie algebras.
Oct 1: JeanBaptiste Priez (Université ParisSud), 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 noncommutative Frobenius characteristic over the 0Hecke algebra action; and secondly we define bijection between those functions and (noninitial) 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.
Oct 14: Jonathan Kujawa, The marked Brauer algebraIn 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.
Nov 4: Aaron Lauve, Matrix MadnessGiven 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
With students Josephine Wood and Adrienne Brackey, we are now diving headsfirst into the case \(n=3\). I'll report on our progress.
Nov 11: Robert Muth, Affine zigzag algebras and imaginary strata for KLR algebrasThe upper half of the quantum group associated to a KacMoody algebra is categorified by a family of graded associative algebras called KhovanovLaudaRouquier (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 Rmod 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 wellunderstood, we show that the imaginary strata are Morita equivalent to affine versions of HuerfanoKhovanov's zigzag algebra. This is joint work with Alexander Kleshchev.
Nov 18: Zoren Sunic, Tamari lattices and the Thompson monoid
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 132avoiding and 231avoiding permutations. Forests appear naturally in both contexts as they are used to represent both permutations and elements of Thompson's monoid.