The Applied Algebra Seminar

A
Monday afternoon research seminar

During 2023-24, the seminar is

IN PERSONat 15:00-16:00 EDT (GMT -4).

The seminar room is Ross Building room N638. If you come by public transportation, there is a York University subway station on the TTC Line 1 Yonge-Univerity route. If you come by car, you can find the available parking lots here.

The seminar has been running since 1997. The topics of talks have typically been any mixture of algebra with any other field: combinatorics, geometry, topology, physics, etc. Further down this page you will find links to the seminar webpages for previous years. The audience usually consists of 6–12 people, including several graduate students and post-docs. For this reason, speakers are encouraged to devote a portion of their talk to the suggestion of open problems and the directions for research in their area. If you are interested in speaking at the seminar, contact the organizers.

You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

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).

Date Speaker Title (click titles for abstract) 15 April 2024

Kartik Singh

U. WaterlooClosure 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 CancelledTBA TBA25 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

OnlineTravis 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 Week12 Feb. 2024

No Monday seminar See talk in Algebraic Combinatorics working seminar TBA5 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.

Date Speaker Title (click titles for abstract) 11 Dec. 2023 Jason Bell

University of WaterlooFiltered deformations of commutative algebras We’ll look at different ways of deforming the multiplicative structure of “classical” algebras to obtain new algebras and explain how this algebraic structure can often be understood combinatorially. We’ll then look at a special class of algebras one can produce this way called filtered deformations and we’ll discuss a conjecture of Etingof which asserts that in positive characteristic that filtered deformations of commutative rings should be in some natural sense very close to being commutative themselves. Not much background will be assumed.4 Dec. 2023 No Seminar (CMS meetings)27 Nov. 2023 Leo Jiang

University of TorontoReal matroid Schubert varieties, zonotopes, and virtual Weyl groups Every linear representation of a matroid determines a matroid Schubert variety whose geometry encodes combinatorics of the matroid. We compute topological invariants of real matroid Schubert varieties explicitly in terms of the combinatorics of zonotopes. When the real matroid Schubert variety comes from a Coxeter arrangement, the equivariant fundamental group is a “virtual” analogue of the corresponding Weyl group.20 Nov. 2023

Bruce Sagan

Michigan State UniversityChromatic symmetric functions and change of basis For a graph $G$, let $X(G)$ be Stanley's chromatic symmetric function. The famous Stanley-Stembridge conjecture states that for a certain family of graphs $G$, the coefficients of the expansion of $X(G)$ in the basis of elementrary symmetric functions $e_\lambda$ are all nonnegative. We show how progress on this conjecture can be made by first expressing $X(G)$ in some other basis and then doing a basis change. In particular, expanding first in the monomial basis gives connections between the independence and clique numbers of $G$ and the shapes $\lambda$ where $e_\lambda$ appears in $X(G)$. And using the Schur basis as intermediary gives a new interpretation for the coefficient of $e_n$ where $n$ is the number of vertices of $G$. All necessary terms concerning symmetric functions will be defined. This is joint work with Foster Tom.13 Nov. 2023 Mohamed Omar

York UniversityUsing slice-rank and partition-rank Recent breakthroughs in combinatorics, especially on bounds of sizes of sets avoiding particular configurations, have been afforded by the slice-rank and partition-rank methods. In this talk we introduce these concepts and the challenges that arise when using them, in hopes that audience members have access to a new tool they may find useful in their own work. Furthermore we discuss the work of the speaker in integrating partition lattices into the theory.6 Nov. 2023

Li Yu

University of TorontoIntegrable systems on the dual of nilpotent Lie subalgebras and $T$-Poisson cluster structures Let $\mathfrak g$ be a semisimple Lie algebra and $\mathfrak g = \mathfrak n \oplus \mathfrak h \oplus \mathfrak n_-$ a triangular decomposition. Motivated by a construction of Kostant-Lipsman-Wolf, we construct an integrable system on the dual space of $\mathfrak n_-$ equipped with the Kirillov-Kostant Poisson structure. The Bott-Samelson coordinates on the open Bruhat cell (equipped with the standard Poisson structure) makes it into a symmetric Poisson CGL extension, hence giving rise to a $T$-Poisson seed on it. Our integrable system is obtained from the initial cluster variables in the $T$-Poisson seed by taking their lowest degree terms with respect to the Bott-Samelson coordinates, and can be regarded as an analogue of taking the initial ideal in Groebner theory. This is joint work in progress with Yanpeng Li and Jiang-Hua Lu.30 Oct. 2023 Lucas Gagnon

York UniversityFrom supercharacters to characters of the Hecke algebra II This talk will be an independent companion to my talk two weeks ago. I will consider inducing certain characters of the unipotent upper triangular group UTn(Fq) to the general linear group GLn(Fq) as a problem in symmetric function theory, giving rise to some more concrete connections to characters of the Hecke algebra. Time permitting, I may show how this gives rise to a point-counting heuristic for the relevant characters.23 Oct. 2023

Alexander Yong

U. Illinois at Urbana-ChampaignCombinatorial commutative algebra rules: what is the degree of a projective variety? The notion of degree is fundamental to a projective variety X. Some algebraic combinatorics problems in algebraic geometry or Lie theory ask for "counting rules" for degrees. Using three standard ideas in combinatorial commutative algebra, we suggest a simple but general finite-time algorithm to generate visual sets whose unweighted count is equal to the degree of X. This talk is based on joint work arXiv:2306.00737 with Ada Stelzer (UIUC).16 Oct. 2023 Lucas Gagnon

York UniversityFrom supercharacters to characters of the Hecke algebra Finding the irreducible modules of the unipotent upper triangular group $\mathrm{UT}_{n}(\mathbb{F}_{q})$ is known to be “wild,” a mathematical synonym for “hopeless.” To work around this a theory of reduciblesupermoduleshas been developed, in which well-behaved reducible modules stand in for irreducibles. These mosules come from a generalization of the Kirillov orbit method and are combinatorially interesting. I have been interested in how these modules induce to $\mathrm{GL}_{n}(\mathbb{F}_{q})$ for a while, but haven’t been able to directly compute the answer. In this talk, I will describe a recursive algorithm for induction, and a surprising connection to evaluating characters in the Hecke algebra that comes up along the way.

Below you will find links to the seminar webpages for previous years.