The Applied Algebra Seminar
A Monday afternoon research seminar

About the seminar (click here to see more)

The seminar is currently organized by Lucas Gagnon and Nantel Bergeron.


Dates are listed in reverse-chronological order. Unless otherwise indicated, all in person talks will take place on Monday from 15:00-16:00 in N638 Ross Building (York University).

Spring 2024

Date Speaker Title (click titles for abstract)
15 April 2024
Kartik Singh
U. Waterloo
Closure of Deodhar components
Deodhar decomposition of the Grassmannian is a refinement of the positroid decomposition, which is very interesting combinatorially. The components are indexed by Go-diagrams, which are pipe dreams such that the subexpression corresponding to the pipe dream is a distinguished subexpression. These objects also have a nice parametrization, Talaska and Williams use Go-diagrams to construct a network which can be used to parametrize these components. However, unlike the positroid decomposition of the Grassmannian, this decomposition is not a stratification. This makes the problem of describing the intersection of the closure of two Deodhar components a very interesting and hard problem. Marcott introduced corrective flips which allow us to convert any pipe dream into a Go-diagram. He used this give a conjecture which describes the intersection in a particular case. In joint work with Olya Mandelshtam and Kevin Purbhoo, we proved one side of the conjecture for a special case. I will be discussing an alternative parametrization which we will use to give a proof for that case.
8 April 2024
Blessing Oni
(Fields Inst.)
Pointed Hopf algebra co(actions) on function fields
Studies have shown that certain singular affine curves (e.g nodal cubic, lemniscate) admit a quantum homogeneous space structure. One can demand like their classical analogues the extension of the quantum symmetries given by quantum groups to the field of rational functions of these singular curves. In this talk, we see the construction of Hopf algebras acting on a given algebra in terms of algebra morphisms and the application of this theory to the field of rational functions containing the coordinate ring of the cusp. We also describe an explicit example of this theory and show the cusp is a quantum homogeneous space. This talk is based on a preprint with Ulrich Kraehmer.
1 April 2024 Cancelled TBA
25 March 2024
Sooyeong Kim
(York U.)
Kemeny’s constant and Matrix-tree theorem
Kemeny's constant, a fundamental parameter in the theory of Markov chains, has recently received significant attention within the graph theory community. Kemeny's constant gives a measure of how quickly a random walker can move around a graph and is thus a good measure of the connectivity of a graph. This graph invariant can be understood by counting spanning trees and spanning 2-forests. In this talk, we understand how graph structures provide insights into Kemeny’s constant, providing combinatorial inequality.
18 March 2024 Eric Sommers
(UMass Amherst)
Some Slodowy Slices Associated to Special Nilpotent Orbits
Among the nilpotent orbits in a simple Lie algebra are the special nilpotent orbits, which play an important role in representation theory. Some of the geometry of the closure of a nilpotent orbit can be understood by taking a transverse slice to a smaller orbit in the closure. This talk concerns a classification of two types of such transverse slices: (1) those between adjacent special nilpotent orbits; and (2) those between a special nilpotent orbit and a certain non-special nilpotent orbit in its closure. The slices in part (1) exhibit a duality, which extends an observation of Kraft and Procesi for type A. The slices in part (2) are related to a conjecture of Lusztig on special pieces. This talk is based on two preprints with Baohua Fu, Daniel Juteau, and Paul Levy.
11 March 2024
Travis Scrimshaw
(Hokkaido University)
Introduction to Cellular Algebras
The symmetric group is one of the first groups discussed in an algebra course. It has a rich representation theory in characteristic 0 from its connections with Lie groups. To construct all of the irreducible representations, we can use the seminormal basis of the group algebra, which is most naturally indexed by pairs of semistandard tableaux. However, this does not work over (small) positive characteristic fields because of the fractions in the construction. In the seminal paper by Graham and Lehrer, they introduce a new class of algebras called cellular algebras that have at least one special basis, called a cell basis, that abstracts the properties of the seminormal basis and allows one to explicitly construct all irreducible representations. In this talk, we will discuss one cell basis for the symmetric group algebra, partition algebra, and Temperley-Lieb algebra with applications to their representation theory and examples computed using SageMath.
4 March 2024 Vasu Tewari
(U. Toronto)
Tutte polynomial and a superspace quotient
I'll introduce superspace analogues of certain quotients by power ideals of Ardila--Postnikov. The bigraded Hilbert series will be shown to equal the Tutte polynomial of an appropriate matroid. If time permits, I'll discuss additional enumerative and representation-theoretic consequences. Joint work with Brendon Rhoades and Andy Wilson.
26 Feb. 2024
Timothy Miller
(U. Waterloo)
Vertex models for the product of a Schur and Demazure polynomial
Demazure atoms and characters are polynomials that each form a Z-basis for polynomials in n variables. The product of a Schur polynomial with a Demazure atom (resp. character) expands into a linear combination of Demazure atoms (resp. characters) with positive integer structure coefficients. There are known combinatorial rules that compute these coefficients using "skyline tableaux" given by Haglund, Luoto, Mason and Willigenburg. I have found alternative rules using the theory of integrable vertex models, inspired by a technique introduced by Zinn-Justin. I use "coloured" vertex models for atoms and characters obtained from Borodin and Wheeler's models for non-symmetric Macdonald polynomials (setting q=t=0). The structure coefficients are then obtained as the number of fillings of a "diamond" vertex model that is compatible with both Schur (uncoloured) and Demazure (coloured) vertex models. The proof is completely combinatorial and very pretty.
19 Feb. 2024 Reading Week
12 Feb. 2024
No Monday seminar See talk in Algebraic Combinatorics working seminar
5 Feb. 2024 George Seelinger
(U. Michigan)
Raising operator formulas for Macdonald polynomials and other related families
Macdonald polynomials are a basis of symmetric functions with coefficients in $\mathbb{Q}(q,t)$ exhibiting deep connections to representation theory and algebraic geometry. In particular, specific specializations of the $q,t$ parameters recover various widely studied bases of symmetric functions, such as Hall-Littlewood polynomials, Jack polynomials, q-Whittaker functions, and Schur functions. Central to this study is the fact that the Schur function basis expansion of the Macdonald polynomials have coefficients which are polynomials in $q,t$ with nonnegative integer coefficients, which can be realized via a representation-theoretic model. Surrounding this line of inquiry, a rich theory of combinatorics emerged, encoded in symmetric functions with expansions of the Macdonald polynomials into LLT polynomials via the work of Haglund-Haiman-Loehr, among others. LLT polynomials were first introduced by Lascoux-Leclerc-Thibbon as a q-deformation of a product of Schur polynomials and have appeared in various related contexts since their introduction. In this talk, I will explain this background and provide a new explicit "raising operator" formula for Macdonald polynomials, proved via an LLT expansion, with a detour showing how similar raising operator formulas can provide a bridge between algebraic and combinatorial formulations of some other symmetric functions. This work is joint with Jonah Blasiak, Mark Haiman, Jennifer Morse, and Anna Pun.

Fall 2023


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