The Applied Algebra Seminar

A
Monday afternoon research seminar

During 2022-23, 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) 3 April 2023 Olya Mandelshtam

Univeristy of WaterlooMacdonald polynomials and the multispecies zero range process Macdonald polynomials are a family of symmetric functions that are known to have remarkable connections to a well-studied particle model called the asymmetric simple exclusion process (ASEP). The modified Macdonald polynomials are obtained from the classical Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials specialize to the partition function of some other particle system. We answer this question in the affirmative with a certain multispecies totally asymmetric zero-range process (TAZRP). This link motivated a new tableaux formula for modified Macdonald polynomials. We present a Markov process on those tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations, proving a remarkable symmetry property. This talk is based on joint work with Arvind Ayyer and James Martin.27 Mar. 2023 Franco Saliola

UQAMLeft Regular Bands of Groups and the Mantaci-Reutenauer Algebra The talk will begin by defining semigroups called left regular bands (LRBs) and explain how these arise geometrically from hyperplane arrangements. Then, we will explore a generalization called LRBs of Groups (LRBGs) that delightfully mixes LRBs and group theory. The principal example throughout will be a LRBG defined by S. Hsiao, which links the braid arrangement with the Mantaci-Reutenauer algebra. The talk is based on joint work with Jose Bastidas (LACIM/UQAM) and Sarah Brauner (U Minnesota).20 Mar. 2023 Antoine Abram

UQAMStylic Monoid We will look on the left action of the free monoid A* on columns by Schensted left insertion. This defines a finite monoid, Styl(A), called the stylic monoid. To study this monoid, we introduce a set of representatives called N-tableaux, along with left and right insertions, and use these to show some nice properties on Styl(A); like the fact that it is a quotient of the plactic monoid by the relations a2=a for all letters a in A or it’s cardinality being the number of partition of a set on |A|+1 elements. The canonical involutive anti-automorphism on A*, which reverses the order on A, induces an involution on Styl(A), which is similar to the corresponding involution of the plactic monoid. This may be computed by an evacuation-like operation (Schützenberger involution on tableaux) on so-called standard immaculate tableaux. This monoid happens to be J-trivial. Using N-tableaux, we will study its J-order. Based on joint work with Christophe Reutenauer.13 Mar. 2023 Sergio Da Silva

Virginia State UniversityApplications of Geometric Vertex Decomposability Geometric vertex decomposition was originally used by Knutson, Miller and Yong to study degenerations of Schubert varieties, but has more recently been found to have broader applications. Because of its connection to liaison theory (due to work of Klein and Rajchgot), geometric vertex decomposability can be used to construct Groebner bases, or to check properties like being glicci or Cohen-Macaulay. Examples of each use can be found in studying toric ideals of graphs and Hessenberg patch ideals. I will highlight various results specific to these two topics, including a height formula for toric ideals of graphs and an alternate proof that regular nilpotent Hessenberg varieties have an affine paving. I will also provide a brief overview of geometric vertex decomposability as well as vertex decomposability for a simplicial complex.6 Mar. 2023 Félix Gélinas

(University of Toronto)Proof of a conjecture of Matherne, Morales, and Selover on encodings of unit interval orders There are two bijections from unit interval orders on n elements to Dyck paths from (0,0) to (n,n). One is to consider the pairs of incomparable elements, which form the set of boxes between some Dyck path and the diagonal. Another is to find a particular part listing (in the sense of Guay-Paquet) which yields an isomorphic poset, and to interpret the part listing as the area sequence of a Dyck path. Matherne, Morales, and Selover conjectured that, for any unit interval order, these two Dyck paths are related by Haglund's well-known zeta bijection. In this talk I prove their conjecture, based on joint work with Adrien Segovia and Hugh Thomas.27 Feb. 2023 Nick Olson-Harris

University of WaterlooBinary tubings and Dyson-Schwinger equations Dyson-Schwinger equations are integro-differential equations satisfied by correlation functions in quantum field theory, which play the role of the "equations of motion" of the theory. They have a recursive, tree-like structure which enables these equations and their solutions to be studied combinatorially. Marie and Yeats showed that in a special case, the solution could be expanded as a sum over connected chord diagrams; this was generalized to many more cases by Hihn and Yeats. Using Hopf algebra techniques we give new combinatorial expansions for a much larger class of Dyson-Schwinger equations and systems as sums over rooted trees equipped with a kind of recursive decomposition we call a "binary tubing". This talk is based on joint work with Paul-Hermann Balduf, Amelia Cantwell, Kurusch Ebrahimi-Fard, Lukas Nabergall, and Karen Yeats.Reading Week13 Feb. 2023 Jenna Rajchgot

McMaster UniversitySymmetric quivers and symmetric varieties Since the 1980s, mathematicians have found connections between orbit closures in type A quiver representation varieties and Schubert varieties in type A flag varieties. For example, singularity types appearing in type A quiver orbit closures coincide with those appearing in Schubert varieties in type A flag varieties; combinatorics of type A quiver orbit closure containment is governed by Bruhat order on the symmetric group; and formulas for classes of type A quiver orbit closures in torus equivariant cohomology and K-theory can be expressed in terms of Schubert polynomials, Grothendieck polynomials, and other objects from Schubert calculus. In this talk, I will motivate and recall some of this story. I will then discuss the related setting of H. Derksen and J. Weyman's symmetric quivers and their representation varieties. I will show how one can adapt results from the ordinary type A quiver setting to unify aspects of the equivariant geometry of type A symmetric quiver representation varieties with Borel orbit closures in corresponding symmetric varieties G/K (G = general linear group, K = orthogonal or symplectic group). This is joint work with Ryan Kinser and Martina Lanini.6 Feb. 2023 Rafael González D'León

Loyola University ChicagoFlow polytopes as a unifying framework for some familiar combinatorial objects Flow polytopes are a family of beautiful mathematical objects which have connections to many areas in mathematics including optimization and representation theory. Finding the volumes and enumerating lattice points of some flow polytopes turns out to be a combinatorially interesting problem that involves beautiful enumeration formulas and many familiar combinatorial objects. Baldoni and Vergne find a series of formulas, which they call Lidskii formulas, that are combinatorially pleasing. Together with Benedetti et al., we provide combinatorial interpretations to the Lidskii formulas in terms of familiar combinatorial objects similar to parking functions. A more recent proof of the Lidskii formulas has been achieved by Mészáros and Morales, following the ideas of Postnikov and Stanley, using polytopal subdivisions. For a smaller class of flow polytopes, these subdivisions are triangulations that coincide with a family of framed triangulations defined by Danilov, Karzanov, and Koshevoy. These triangulations turn out to have interesting hidden combinatorial structures. In joint work with von Bell, Mayorga, and Yip we characterize the combinatorial structures arising from two triangulation strategies on a family of polytopes which provide a surprising unifying framework for the Young lattice and the Tamari lattice.30 Jan. 2023

Oliver Pechenick

University of WaterlooGeometry of quasisymmetric functions The combinatorics of symmetric function theory plays a central role both in combinatorial representation theory (of symmetric and general linear groups) and in enumerative geometry (through the cohomology of Grassmannians). The latter connection yields "K-analogues" of the classical symmetric function bases and their combinatorics by enriching the cohomology of Grassmannians to their K-theory rings. Quasisymmetric functions (QSym) are analogues of symmetric functions introduced by Stanley and Gessel in the 70s for primarily enumerative reasons, but also with a key role in the representation theory of 0-Hecke algebras. However, analogous connections to geometry and topology have been missing. In particular, although there has significant interest in "K-analogues" of quasisymmetric functions, there has been no known space whose K-theory they describe. We build on work of Baker & Richter (2008) to identify a loop space with a cellular cohomology basis corresponding to a classical basis of QSym. We then introduce an instance of "cellular K-theory," yielding the first geometrically-interpreted K-basis of QSym. Our polynomials are similar to ones introduced by Lam & Pylyavskyy (2007) and yet are new. This is joint work with Matt Satriano (arXiv:2205.12415).23 Jan. 2023 Farid Aliniaeifard

University of British ColumbiaGeneralized chromatic functions We define vertex-colourings for edge-coloured digraphs, which unify the theory of $P$-partitions and proper vertex-colourings of graphs. Furthermore, we use our vertex-colourings to define generalized chromatic functions, which merge the chromatic symmetric and quasisymmetric functions and generating functions of $P$-partitions. We also discuss the relations between generalized chromatic functions, Schur functions in noncommuting variables, and the well-known Stanley-Stembridge $(3+1)$-free conjecture.Winter Break12 Dec. 2022

Unusual time: 4-5PMKaveh Mousavand

Queen's UniversityCombinatorial and Geometric treatments of Schur representations of finite dimensional algebras Schur representations (also known as bricks) over finite dimensional algebras are known to form an important subfamily of indecomposable modules. Study of these representations has received fresh impetus from various domains, including $\tau$-tilting theory, stability conditions, lattice theory of torsion classes, etc. In particular, for a given algebra A, it is natural to ask whether A admits infinitely many Schur representations. Furthermore, distribution of these representations with respect to their length is a phenomenon that relates to several interesting problems that we discuss in this talk. I will take an elementary approach to the study of Schur representations, where I only assume familiarity with the notion of modules, to state an open conjecture on the behavior of Schur representations. Then, I give more conceptual incarnations of this open conjecture in terms of combinatorics, algebraic geometry and torsion theory. Finally, we verify the conjecture for some families of algebras and reduce the general case to a specific family of algebras.Part of this talk is based on my joint work with Charles Paquette.

5 Dec. 2022

Jose Bastidas

UQAMThe Primitive Eulerian Polynomial We introduce the Primitive Eulerian polynomial $P_{\mathcal{A}}(z)$ of a central hyperplane arrangement $\mathcal{A}$, a reparametrization of its cocharacteristic polynomial. Previous work on the polytope algebra of deformations of a zonotope (2021) implicitly showed that this polynomial has nonnegative coefficients whenever $\mathcal{A}$ is a simplicial arrangement, but a combinatorial interpretation of the coefficients was only found for reflection arrangements of type A and B.We discuss the relation between the Primitive Eulerian polynomial and the usual Eulerian polynomial. We also present a geometric/combinatorial interpretation for the coefficients of $P_{\mathcal{A}}(z)$ for all simplicial arrangements $\mathcal{A}$, along with some real-rootedness results and conjectures. Based on joint work with Christophe Hohlweg and Franco Saliola.

28 Nov. 2022 GaYee Park

UQAMMinimal skew semistandard Young tableaux and the Hillman--Grassl correspondence Standard tableaux of skew shape are fundamental objects in enumerative and algebraic combinatorics and no product formula for the number is known. In 2014, Naruse gave a formula as a positive sum over excited diagrams of products of hook-lengths. In 2018, Morales, Pak, and Panova gave a $q$-analogue of Naruse's formula for semi-standard tableaux of skew shapes. They also showed, partly algebraically, that the Hillman-Grassl map restricted to skew shapes gave their $q$-analogue. We study the problem of making this argument completely bijective. For a skew shape, we define a new set of semi-standard Young tableaux, called theminimal SSYT, that are equinumerous with excited diagrams via a new description of the Hillman--Grassl bijection and have a version of excited moves. Lastly, we relate the minimal skew SSYT with the terms of the Okounkov-Olshanski formula for counting SYT of skew shape.21 Nov. 2022

Logan Crew

University of WaterlooChromatic Multisymmetric Functions The chromatic symmetric function of a graph refines the chromatic polynomial by enumerating proper colorings not only by number of colors used, but also by how many times each color is used. While trees with the same number of vertices all have identical chromatic polynomial, a perhaps surprising major open question asks whether nonisomorphic trees are distinguished by their chromatic symmetric function, and more generally exactly which graphs are not so distinguished. Another major open problem, the Stanley-Stembridge conjecture, asks (in its most refined form) whether unit interval graphs have e-positive chromatic symmetric function.In this talk, we introduce a chromatic multisymmetric function, a generalization in which the vertex set is partitioned, and each part is equipped with its own variable set. This function lets us to more easily and directly take advantage of nice structural properties in graphs, allowing us to generalize previous results by giving an algorithmic way to generate new relationships among chromatic symmetric functions from any given one. Furthermore, it provides a new perspective on Guay-Paquet's famous result reducing the Stanley-Stembridge conjecture from claw-free incomparability graphs to unit interval graphs, and suggests potential ways to apply Guay-Paquet's style of argument to other settings.

This is joint work with Evan Haithcock, Josephine Reynes, and Sophie Spirkl.

14 Nov. 2022 Taylor Brysiewicz

Western UniversityTrace Tests in Numerical Algebraic Geometry At its core, the computational framework of numerical algebraic geometry involves computing floating point approximations of points on linear intersections of varieties. Symbolic techniques exist for certifying that our approximations indeed correspond to true solutions, whereas numerical trace tests establish the other direction: that every solution has been (approximately) computed. I will give an overview of the classical trace test and discuss joint work with Michael Burr (Clemson) on our extension to zero-dimensional, Bernstein-generic, polynomial systems: the sparse trace test.7 Nov. 2022

Alistair Savage

University of OttawaDiagratification We will explain how one can construct diagrammatic presentations of categories of representations of Lie groups and their associated quantum groups using only a small amount of information about these categories. To illustrate the technique in concrete terms, we will focus on the exceptional Lie group of type $F_4$.31 Oct. 2022 Tim Miller

University of WaterlooVertex models applied to symmetric function theory Vertex models are a tool from statistical mechanics used to model systems of particles in a lattice such as ice crystals. In recent years, they have shown promise in the theory of (non)symmetric polynomials and there has been a flurry of research reworking the theory from this perspective. For example, thinking of polynomials as vertex models can yield conceptually simple proofs of properties like symmetry, obtain structure constants and show Cauchy identities. I will give an overview of these proof techniques and talk about my research in this direction.24 Oct. 2022

Nantel Bergeron

York UniversitySchubert Polynomials This talk will be about the combinatorics of Schubert polynomials and their multiplicative structure constants.17 Oct. 2022 Lucas Gagnon

York UniversityUnipotent Symmetric Functions The aim of this talk is to introduce a new game for producing interesting symmetric functions. Combinatorial Hopf algebras (CHAs) give a way of taking a family of objects with a Hopf structure and producing combinatorial invariants in the ring of (quasi-)symmetric functions. A fairly boring example of this is the complex characters of the general linear groups $\mathrm{GL}_{n}(\mathbb{F}_{q})$ over a finite field: one recovers the classical association between symmetric functions and $\mathrm{GL}_{n}(\mathbb{F}_{q})$-characters found in the work of Green and Zelevinsky. However, things get much more interesting when we also consider the maximal unipotent subgroup $\mathrm{UT}_{n}(\mathbb{F}_{q})$ of upper triangular matrices in $\mathrm{GL}_{n}(\mathbb{F}_{q})$: with the right Hopf structure, computing the induction of a $\mathrm{UT}_{n}(\mathbb{F}_{q})$-character to $\mathrm{GL}_{n}(\mathbb{F}_{q})$ can be reinterpreted as a CHA-type construction. The upshot is a set of combinatorial rules for making symmetric functions that have representation theoretic meaning, without having to do any representation theory. After explaining all of this, I will mention some interesting cases where this approach has already worked, and a few more which I hope to solve soon.Reading Week3 Oct. 2022 Kevin Purbhoo

University of WaterlooThe MTV Machine Mukhin, Tarasov and Varchenko developed a "machine" for solving certain algebraic systems of equations, which arise in several places, including: Schubert calculus, algebraic curves, linear series, Wronskians of polynomials, differential equations, and control theory. The machine is quite remarkable. Essentially, it transforms a HARD system of algebraic equations into an EASY system of differential equations. In this talk, I will attempt to explain how the machine works, and why it works.

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