The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar is currently organized by Aram Dermenjian and Nantel Bergeron.

During 2020-21, the seminar is

ONLINEat 15:00-16:00 EDT (GMT -4).

When we resume in person, we will continue to broadcast. The seminar room is RossBuilding 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 Aram Dermenjian or Nantel Bergeron.You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

Dates are listed in reverse-chronological order.

All talks will be online with Zoom link, generaly at 15:00-16:00 (EDT [GMT-4]) on Monday

When we resume in person, we will continue to broadcast. 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) 10 May 2021

(Virtual seminar on Zoom)Anna Vanden Wyngaerd

Université Libre de BruxellesTwo Delta conjecture implications The famous shuffle theorem is a combinatorial formula for a Schur positive symmetric function, nabla(e_n). Since its formulation, a number of variations and generalisations of the shuffle formula have been proposed, many of which remain open problems today. In this talk we present some of these conjectures and discuss two logical implications between them we recently established (joined work with Alessandro Iraci). The relevant combinatorial objects are decorated labelled lattice paths.3 May 2021

(Virtual seminar on Zoom)McCabe Olsen

Rose-Hulman Institute of TechnologyUnconditional Reflexive Polytopes A convex body is unconditional if it is symmetric with respect to reflections in all coordinate hyperplanes. We investigate unconditional lattice polytopes with respect to geometric, combinatorial, and algebraic properties. In particular, we characterize unconditional reflexive polytopes in terms of perfect graphs. As a prime example, we study the signed Birkhoff polytope. Moreover, we derive constructions for Gale-dual pairs of polytopes and we explicitly describe Gröbner bases for unconditional reflexive polytopes coming from partially ordered sets. This is joint work with Florian Kohl (Aalto University) and Raman Sanyal (Goethe Universität Frankfurt).26 Apr. 2021

(Virtual seminar on Zoom)Nancy Wallace

Université du Québec à MontréalUnexpected relations between parking function formulas, pattern avoiding permutations and the Robinson-Schensted algorithm The Shuffle theorem of Carlsson and Mellit, states that $\nabla(e_n)$ is given by parking function formulas. These formulas are symmetric in the variables q and t. More preciously, for all n, $\nabla(e_n)$ can be seen as a $GL2×Sn-module$. In this talk we will put forth a partial expansion in terms of the irreducible bicharacters of these modules. Namely we will expand a subset of the parking function formulas as products of Schur functions in the variables q and t and the usual Schur functions in the variables $X={x1,x2,…}$. Part of these formulas are uncovered using a bijection between a subset of paths of area $0$ and standard Young tableaux that sends the dinv statistic to the major index. The Robinson-Schensted algorithm associates a pair of standard Young tableaux $(P,Q)$ to a given permutation. We will end by showing how the previous bijection is linked to the $Q$-tableau of some pattern avoiding permutations that is unrelated to the word of the parking function.19 Apr. 2021

(Virtual seminar on Zoom)Jonah Blasiak

Drexel UniversityA raising operator formula for $\nabla$ on an LLT polynomial The symmetric function operator $\nabla$ arose in the theory of Macdonald polynomials and its action on various bases has been the subject of numerous conjectures over the last two decades. It developed that $\nabla$ is but a shadow of a more complete picture involving the elliptic Hall algebra of Burban and Schiffmann. This algebra is generated by subalgebras $\Lambda(X^{m,n})$ isomorphic to the ring of symmetric functions, one for each coprime pair of integers $(m,n)$. We identify certain combinatorially defined rational functions which correspond to LLT polynomials in any of the subalgebras $\Lambda(X^{m,n})$. As a corollary, we deduce an explicit raising operator formula for $\nabla$ on any LLT polynomial.

This is joint work with Mark Haiman, Jennifer Morse, Anna Pun, and George Seelinger.12 Apr. 2021

(Virtual seminar on Zoom)Laura Colmenarejo

UMass AmherstChromatic symmetric functions for Dyck paths and $q$-rook theory Given a graph and a set of colors, a coloring is a function that associates each vertex in the graph with a color. In 1995, Stanley generalized this definition to symmetric functions by looking at the number of times each color is used and extending the set of colors to $\mathbb{Z}^+$. In 2012, Shareshian and Wachs introduced a refinement of the chromatic functions for ordered graphs as $q$-analogues.

In the particular case of Dyck paths, Stanley and Stembridge described the connection between chromatic symmetric functions of abelian Dyck paths and square hit numbers, and Guay-Paquet described their relation to rectangular hit numbers. Recently, Abreu-Nigro generalized the former connection for the Shareshian-Wachs $q$-analogue, and in unpublished work, Guay-Paquet generalized the latter.

In this talk, I want to give an overview of the framework and present another proof of Guay-Paquet's identity using $q$-rook theory. Along the way, we will also discuss $q$-hit numbers, two variants of their statistic, and some deletion-contraction relations. This is recent work with Alejandro H. Morales and Greta Panova.5 Apr. 2021

(Virtual seminar on Zoom)Ezgi Kantarcı Oğuz

Royal Institute of Technology, StockholmPromotion and cyclic sieving on families of SSYT Cyclic sieving phenomenon is a connection between cyclic actions on a set with a polynomial evaluated at roots of unity that is surprisingly ubiquitous in the context of algebraic combinatorics. In this talk, we will consider some new instances of this phenomenon on families of tableaux under the promotion action. Based on work with Per Alexandersson and Svante Linusson.29 Mar. 2021

(Virtual seminar on Zoom)Maxime Bergeron

RiskfuelHilbert, Deep Neural Nets and Empirical Moduli Spaces The motivation behind Hilbert's 13th problem is often overlooked. In his original statement, he opens with: "nomography deals with the problem of solving equations by means of drawing families of curves depending on an arbitrary parameter". The question he posed sought to identify a family of functions amenable to such graphical solvers that were essential tools of his time. More formally, he asked if it was possible to solve algebraic equations in terms of towers of algebraic functions of a single parameter. While the question in its original form remains open to this day, in the continuous realm it turns out that there is no such thing as a truly multivariate function. In this talk, we will see how these ideas fit into the modern deep learning framework, forming a bridge between algebra and analysis.22 Mar. 2021

(Virtual seminar on Zoom)Hugo Mlodecki

Université Paris-SaclayA bidendriform automorphism of WQSym By Foissy's work, the bidendriform structure of the Word Quasisymmetric Functions Hopf algebra (WQSym) implies that it is isomorphic to its dual. In this talk, we present the construction of an explicit combinatorial bidendriform isomorphism. We represent two recursive decompositions of packed words by two new combinatorial families called red and blue biplan forests. We then obtain two bases of WQSym and its dual. The advantage of these bases is that by taking explicit subsets, we obtain bases of primitive elements and totally primitive elements. We then carefully combine red and blue forests to get bicolors forests. A simple re-coloring of the edges allows us to obtain the first explicit bidendriform automorphism of WQSym.15 Mar. 2021

(Virtual seminar on Zoom)Rosa Orellana

Dartmouth CollegeRestricting Howe Duality Classical Howe duality provides a representation theoretic framework for classical invariant theory. In the classical Howe duality, the general linear group, $GL_n(\mathbb{C})$, is the dual to $GL_n(\mathbb{C})$ when acting on the polynomial ring of the variables $x_{i,j}$ where $1\leq i \leq n$ and $1\leq j \leq k$. In this talk, I will introduce a multiset partition algebra, MP_k(n), as the Howe dual to the action of the symmetric group, $S_n$, on the polynomial ring.8 Mar. 2021

(Virtual seminar on Zoom)Daniel Tamayo

Université Paris-SaclayPermutree Sorting, Lattice Quotients, and Automata We define permutree sorting which generalizes Knuth's stack sorting and Reading's Coxeter sorting algorithms. (U,D)-permutree sorting consists of an algorithm that succeeds or fails for a permutation depending if it contains or avoids certain patterns determined by the sets U and D. We present this algorithm through a family of automata that read reduced words and show that the accepted reduced words form a search-tree structure related to lattice quotients of the weak order. This is joint work with Vincent Pilaud and Viviane Pons.1 Mar. 2021

(Talk Video)Foster Tom

University of California, BerkeleyA combinatorial Schur expansion of triangle-free horizontal-strip LLT polynomials In recent years, Alexandersson and others proved combinatorial formulas for the Schur function expansion of the horizontal-strip LLT polynomial $G_\lambda(x;q)$ in some special cases. We associate a weighted graph $\Pi$ to $\lambda$ and we use it to express a linear relation among LLT polynomials. We apply this relation to prove an explicit combinatorial Schur-positive expansion of $G_\lambda(x;q)$ whenever $\Pi$ is triangle-free. We also prove that the largest power of $q$ in the LLT polynomial is the total edge weight of our graph.22 Feb. 2021

(Talk Video)Anna Weigandt

Univeristy of MichiganThe Castelnuovo-Mumford Regularity of Matrix Schubert Varieties The Castelnuovo-Mumford regularity of a graded module provides a measure of how complicated its minimal free resolution is. In work with Rajchogt, Ren, Robichaux, and St. Dizier, we noted that the CM-regularity of matrix Schubert varieties can be easily obtained by knowing the degree of the corresponding Grothendieck polynomial. Furthermore, we gave explicit, combinatorial formulas for these degrees for symmetric Grothendieck polynomials. In this talk, I will present a general degree formula for Grothendieck polynomials. This is joint work with Oliver Pechenik and David Speyer.15 Feb. 2021

(Virtual seminar on Zoom)No SeminarFamily Day / Reading Week8 Feb. 2021

(Talk Video)Matteo Mucciconi

Tokyo Institute of TechnologySymmetric polynomials in Integrable Probability A number of solvable stochastic processes can be described in terms of notable families of symmetric functions. Classical models as the last passage percolation (LPP) or the totally asymmetric simple exclusion process (TASEP) sample measures built on Schur polynomials. Analogously, Whittaker functions are related to solvable models of random polymers as the O’Connell-Yor Polymer (OYP). In 2015 Corwin and Petrov introduced the higher spin vertex model, a family of stochastic processes sitting on top of a hierarchy of models including TASEP, LPP, OYP and of many other interesting systems including random walkers in random environment. We find that the higher spin vertex model and all of its degenerations can be solved using a unifying family of symmetric functions, the spin q-Whittaker (sqW) polynomials, a version of which was defined first by Borodin and Wheeler in 2017. Probabilistic intepretation of sqW allows us to establish a number of interesting combinatorial properties along with surprising conjectural relations. Studying scaling limits of sqW we recover classical objects as Schur and Grothendieck polynomials along with new families of symmetric functions. Based on a joint work with Leonid Petrov.1 Feb. 2021

(Talk Video)Andrés Vindas Meléndez

University of KentuckyDecompositions of Ehrhart h*-Polynomials for Rational Polytopes The Ehrhart quasipolynomial of a rational polytope P encodes the number of integer lattice points in dilates of P, and the h* -polynomial of P is the numerator of the accompanying generating function. We provide two decomposition formulas for the h*-polynomial of a rational polytope. The first decomposition generalizes a theorem of Betke and McMullen for lattice polytopes. We use our rational Betke--McMullen formula to provide a novel proof of Stanley's Monotonicity Theorem for the h*-polynomial of a rational polytope. The second decomposition generalizes a result of Stapledon, which we use to provide rational extensions of the Stanley and Hibi inequalities satisfied by the coefficients of the h*-polynomial for lattice polytopes. Lastly, we apply our results to rational polytopes containing the origin whose duals are lattice polytopes. This is joint work with Matthias Beck (San Francisco State Univ. & FU Berlin) and Ben Braun (Univ. of Kentucky).25 Jan. 2021

(Talk Video)Olya Mandelshtam

Brown UniversityThe multispecies TAZRP and modified Macdonald polynomials Recently, a formula for the symmetric Macdonald polynomials $P_{\lambda}(X;q,t)$ was given in terms of objects called multiline queues, which also compute probabilities of a statistical mechanics model called the multispecies asymmetric simple exclusion process (ASEP) on a ring. It is natural to ask whether the modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ can be obtained using a combinatorial gadget for some other statistical mechanics model. We answer this question in the affirmative. In this talk, we will give a new formula for $\widetilde{H}_{\lambda}(X;q,t)$ in terms of fillings of tableaux called polyqueue tableaux. We define a multispecies totally asymmetric zero range process (TAZRP) on a ring with parameter $t$, whose (unnormalized) stationary probabilities are computed by polyqueue tableaux, and whose partition function is equal to $\widetilde{H}_{\lambda}(X;1,t)$. This talk is based on joint work with Arvind Ayyer and James Martin.18 Jan. 2021

(Talk Video)Sunita Chepuri

University of MichiganKazhdan-Lusztig Immanants for $k$-positive Matrices Immanants are matrix functionals that generalize the determinant. One notable family of immanants are the Kazhdan-Lusztig immanants. These immanants are indexed by permutations and are defined as sums involving Kazhdan-Lusztig polynomials specialized at $q=1$. Kazhdan-Lusztig immanants have several interesting combinatorial properties, including that they are nonnegative on totally positive matrices. We give a condition on permutations that allows us to extend this theorem to the setting of $k$-positive matrices.Winter Break14 Dec. 2020

(Talk Video)Luis Serrano

Zapata ComputingFundamentals and recent advances in machine learning and neural networks In this talk we'll first go over the fundamental notions of machine learning such as linear regression and the perceptron algorithm. Then we'll delve into neural networks and some of their applications. If time permits, we'll get into some cutting edge material such as generative machine learning (the algorithms that draw faces as in www.thispersondoesnotexist.com) and some of their extensions.7 Dec. 2020

(Talk Video)Emily Gunawan

Oklahoma UniversityCambrian combinatorics on quiver representations We will present a geometric model of the Auslander-Reiten quiver of a type A quiver together with a stability function for which all indecomposable modules are stable. We also introduce a new Catalan object which we call a maximal almost rigid representation. We define a partial order on the set of maximal almost rigid representations and use our new geometric model to show that this partial order is a Cambrian lattice. If there is time, we will discuss a connection to the biCambrian lattice of diagonal rectangulations (which are counted by the Baxter numbers). This is based on joint work with E. Barnard, E. Meehan, and R. Schiffler.30 Nov. 2020

(Talk Video)Christian Gaetz

MITStable characters from permutation patterns We study the expected value (and higher moments) of the number of occurrences of a fixed permutation pattern on conjugacy classes of the symmetric group S_n. We prove that this virtual character stabilizes as n grows, so that there is a single polynomial computing these moments on any conjugacy class of any symmetric group. Our proof appears to be the first application of partition algebras to the study of permutation patterns. This is joint work with Christopher Ryba.23 Nov. 2020

(Talk Video)Nicolle Gonzalez

UCLAAffine Demazure crystals for nonsymmetric Macdonald polynomials Macdonald polynomials have long been hailed as a breakthrough in algebraic combinatorics as they simultaneously generalize both Hall-Littlewood and Jack symmetric polynomials. The nonsymmetric Macdonald polynomials $E_a(X;q,t)$ are a further generalization which contain the symmetric versions as special cases. When specialized at $t =0$ the nonsymmetric Macdonald polynomials were shown by Bogdon and Sanderson to arise as characters of affine Demazure modules, which are certain truncations of highest weight modules. In this talk, I will describe a type A combinatorial crystal which realizes the affine Demazure module structure and recovers the results of Bogdon and Sanderson crystal-theoretically. The construction yields a filtration of these affine crystals by finite Demazure crystals via certain embedding operators that model those of Knop and Sahi for nonsymmetric Macdonald polynomials. Thus, we obtain an explicit combinatorial expansion of the specialized nonsymmetric Macdonald polynomials as graded sums of key polynomials. As a consequence, we derive a new combinatorial formula for the Kostka-Foulkes polynomials. This is joint work with Sami Assaf.16 Nov. 2020 Aida Maraj

Max Planck InstituteReciprocal ML-degree of Brownian Motion Tree Models Brownian Motion Tree Models (BMTM) are multivariate Gaussian models that arise in phylogenetics when studying the evolution of species through time. They are realized by rooted directed trees. BMTM are wonderful as the space of their covariance matrices is a linear space of symmetric matrices, and the space of their concentration matrices is a toric variety. In applications, one is interested in computing the point in a model that is more probable for the observed data. The (reciprocal) Maximum Likelihood degree of the model gives an insight on the complexity of this problem. In BMTM the reciprocal ML-degree can be nicely computed from the structure of the tree. To prove this result we require help from toric geometry. This is based on joint work with T. Boege, J.I. Coons, C. Eur, and F. Röttger.9 Nov. 2020

(Talk Video)Galen Dorpalen-Barry

University of MinnesotaCones of Hyperplane Arrangements through the Varchenko-Gea'lfand Ring The coefficients of the characteristic polynomial of an arrangement in a real vector space have many interpretations. An interesting one is provided by the Varchenko-Gel'fand ring, which is the ring of functions from the chambers of the arrangement to the integers with pointwise multiplication. Varchenko and Gel'fand gave a simple presentation for this ring, along with a filtration whose associated graded ring has its Hilbert function given by the coefficients of the characteristic polynomial. We generalize these results to cones defined by intersections of halfspaces of some of the hyperplanes. Time permitting, we will discuss Varchenko-Gel'fand analogues of some well-known results in the Orlik-Solomon algebra regarding Koszulity and supersolvable arrangements.2 Nov. 2020

(Talk Video)Sophia Elia

FU BerlinCongruence Normality for Simplicial Hyperplane Arrangements Simplicial hyperplane arrangements still have much to reveal. In rank 3, it is not known whether the list of simplicial hyperplane arrangements is complete. We determine whether the associated posets of regions possess the combinatorial property of "congruence normality" for arrangements with up to 37 hyperplanes. We use methods from oriented matroids, which make the computations possible. This refines the structure of the list, breaking it into three separate combinatorial categories. In particular, we show that arrangements stemming from finite Weyl groupoids have congruence normal posets of regions. This is joint work with Jean-Philippe Labbé and Michael Cuntz.26 Oct. 2020

(Talk Video)Alex McDonough

Brown U.A Higher-Dimensional Sandpile Map Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of a class of orientable arithmetic matroids. I provide a family of combinatorially meaningful maps between these two sets. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin. I will not assume any background beyond undergraduate linear algebra.19 Oct. 2020

(Talk video)Yibo Gao

MITThe 1/3-2/3 Conjecture for Coxeter groups The 1/3-2/3 Conjecture, originally formulated in 1968, is one of the best-known open problems in the theory of posets, stating that the balance constant of any non-total order is at least 1/3. By reinterpreting balance constants of posets in terms of convex subsets of the symmetric group, we extend the study of balance constants to convex subsets C of any Coxeter group. Remarkably, we conjecture that the lower bound of 1/3 still applies in any finite Coxeter group, with new and interesting equality cases appearing. We generalize several of the main results towards the 1/3-2/3 Conjecture to this new setting: we prove our conjecture when C is a weak order interval below a fully commutative element in any acyclic Coxeter group (a generalization of the case of width-two posets), we give a uniform lower bound for balance constants in all finite Weyl groups using a new generalization of order polytopes to this context, and we introduce generalized semiorders for which we resolve the conjecture. We hope this new perspective may shed light on the proper level of generality in which to consider the 1/3-2/3 Conjecture, and therefore on which methods are likely to be successful in resolving it. This is joint work with Christian Gaetz.12 Oct. 2020 No SeminarThanksgiving/Reading Week5 Oct. 2020

(Talk video)Yusra Naqvi

U. of SydneyA gallery model for affine flag varieties Galleries, which are special sequences of elements in a Coxeter group, provide a nice combinatorial way of studying flag varieties. In this talk, we will discuss what these objects are, how they relate to each other, and how this relationship gives us a convenient recursion for computing certain double coset intersections in affine flag varieties. This talk is based on joint work with Elizabeth Milicģevicģ, Petra Schwer and Anne Thomas.28 Sep. 2020

(Talk video)Wencin Poh

UC DavisA crystal for stable Grothendieck polynomials We construct a type A crystal, which we call the *-crystal, whose character is the stable Grothendieck polynomials for fully-commutative permutations. This crystal is a K-theoretic generalization of Morse-Schilling crystal on decreasing factorizations. Using the residue map, we showed that this crystal intertwines with the crystal on set-valued tableaux given by Monical, Pechenik and Scrimshaw. We also proved that this crystal is isomorphic to that of pairs of semistandard Young tableaux using a newly defined insertion called the *-insertion. The insertion offers a combinatorial interpretation to the Schur positivity of the stable Grothendieck polynomials for fully-commutative permutations. Furthermore, the *-insertion has interesting properties in relation to row Hecke insertion and the uncrowding algorithm. This is joint work with Jennifer Morse, Jianping Pan and Anne Schilling.21 Sep. 2020 Logan Crew

U. of WaterlooEdge Deletion-Contraction in the Chromatic and Tutte Symmetric Functions "We consider symmetric function analogues of the chromatic and Tutte polynomials on graphs whose vertices have positive integer weights. We show that in this setting these functions admit edge deletion-contraction relations akin to those of the corresponding polynomials, and we use these relations to give enumerative and/or inductive proofs of properties of these functions. In particular we note that the Tutte symmetric function in this form is related to a family of vertex-weighted graph functions, from which we derive a recipe theorem and a spanning-tree expansion.

This is joint work with Sophie Spirkl."

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