The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar is currently organized by Laura Colmenarejo and Nantel Bergeron.

During 2017-18, the seminar takes place from 15:00-16:00 in Ross Building room N638. If you come by bus, the route 196A, 196B takes you to campus from Downsview subway station. 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 Laura Colmenarejo or Nantel Bergeron.You may also be interested in the Algebraic Combinatorics Seminar at the Fields Institute.

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

Date Speaker Title (click titles for abstract) 26 Mar. 2018 Carolina Benedetti (Universidad de Los Andes) Volumes of flow polytopes In this talk we give a happy ending to a conjecture posed last year in this very seminar, related to volumes of flow polytopes. In order to do this, we introduce new families of combinatorial objects whose enumeration computes volumes of flow polytopes. These objects provide an interpretation of Baldoni and Vergne's volume formula. A highlight of our model is an elegant formula for the flow polytope of a graph we call the caracol graph, hence, solving the aforementioned conjecture. As by-products of our work, we uncover a new triangle of numbers that interpolates between Catalan numbers and the number of parking functions. This is joint work with R. Gonzalez, C. Hanusa, P. Harris, A. Khare, A. Morales, M. Yip.23 Mar. 2018 at YorkU, 2:30 pm! François Bergeron

(UQAM)Why should we care about (diagonal) coinvariant spaces? One of the most striking results of finite group representation theory (in characteristic zero) is the characterization of groups generated by reflexions in terms of the simplicity of the structure of their space of invariant polynomials. Indeed, the Chevalley-Shephard-Todd theorem (1954-55) states that this ring is a polynomial ring if and only if the group is generated by reflexions. The group coinvariant space is obtained by considering polynomials modulo the ideal generated by constant term free symmetric polynomials. Chevalley-Shephard-Todd's theorem is equivalent to saying that this coinvariant (vector) space is of dimension equal to the order of the group, exactly in the case of reflexion groups. We will explain in a broadly accessible manner (including for graduate students) how this striking result has sparked many lines of inquiry, including extensions to groups that are not generated by reflexions such as in the seminal work of Garsia and Haiman on diagonal coinvariant spaces (started in the 1990's). More recent extensions along these lines make apparent deep ties with many areas such as: Rectangular Catalan Combinatorics, Homology of $(m,n)$-Torus Knots, Algebraic Geometry (Hilbert Scheme of points in the plane), and Theoretical Physics.12 Mar. 2018 Angele Hamel

(Wilfrid Laurier U.)Chromatic Symmetric Functions and H-Free Graphs Chromatic symmetric functions are defined in terms of colourings of particular graphs. Some key conjectures in this area concern whether chromatic symmetric functions of claw-free graphs can be written in terms of other symmetric functions with positive coefficients. Here we extend the claw-free idea to consider the e-positivity question for chromatic symmetric functions of H-free graphs with H ={claw, F}, where F is a four-vertex graph. We settle the question for all cases except H={claw, co-diamond}, and we provide some partial results in that case. This is joint work with Chinh Hoang and Jake Tuero.5 Mar. 2018 Aaron Lauve

(Loyola U. Chicago)Transfer of Structure for Hopf Algebras Having arrived on the combinatorial scene in the late '70s, there is by now an overabundance of Hopf algebras built on combinatorial gadgets: partitions, compositions, permutations, planar binary trees, Dyck paths, Feynman graphs, rooted forests, and posets only begins to scratch the surface. And many boast both (co)commutative and non(co)commutative versions. The reason is plain enough to state: if your favorite gadgets carry with them natural ways to break and combine, then you should treat this as a notion of coproduct and product and build yourself a Hopf algebra. One reward for your effort will be Formulas! I'll give a few examples. In this talk, reporting on joint work with Mitja Mastnak, we take a step back and ask for some way of organizing the study of this menagerie. We introduce the notion of Hopf algebra coverings and highlight the transfer of structure they provide. In particular, we advertise a way to find primitives and antipode in any graded connected cocommutative Hopf algebra.26 Feb. 2018 Cameron Marcott

(U. of Waterloo)Adding gauge data to the positive grassmannian The Grassmannian Gr(n,k) is the set of k dimensional subspaces of an n dimensional space. We often represent such a subspace as the row span of a n x k matrix. Introducing an extra column vector to such a matrix, we obtain a point in Gr(n+1,k). This talk will explore some interesting features of this construction.19 Feb. 2018 No seminar!

READING WEEK 12 Feb. 2018 Nat Thiem

(U.of Colorado Boulder)A q-partition algebra and splatter combinatorics The partition algebra is a classical diagram algebra arising out of Schur—Weyl duality in representation theory. While there are numerous approaches to constructing a q-deformation of the partition algebra, they each seem to have had different obstructions. This talk will review the partition algebra, its possible generalizations, and then focus on one particular version that has seen some recent progress. We recover some possibly new combinatorial objects called splatters, which are a generalization of set partitions. This work is joint work with Tom Halverson.05 Feb. 2018 Nantel Bergeron

(York U.)Hypergraphical polytope and antipode 29 Jan. 2017 Praise Adeyemo

(U. of Ibadan)On The Geometry of Springer Varieties There are quite a number of subvarieties that can be defined to sit inside flag varieties. One of such is the family of Hessenberg varieties. In this talk, I will give combinatorial and geometric descriptions of the computation of the Betti numbers of a member of this family known as Springer varieties and discuss the current problem in this direction.22 Jan. 2017 Mike Zabrocki (York Univeristy)

Products of characters of the symmetric groups In joint work with Rosa Orellana, we defined a basis of the symmetric functions (arXiv:1605.06672) indexed by a partition $\lambda$ which are the irreducible characters indexed by the partition $(n -|\lambda|, \lambda)$ of the symmetric group $S_n$ (as permutation matrices).

In a recent paper (arXiv:1709.08098) we were able to find combinatorial interpretations for certain products of symmetric functions expanded in the character basis in terms of multiset valued tableaux. I'll explain the relationship to representation theory and how this is related to finding a combinatorial interpretation for the reduced Kronecker product and the $Gl_n$ to $S_n$ restriction problem.15 Jan. 2017 Alexander Nenashev

(York U.)

Homological algebra for pointed sets Although the category of pointed sets is not additive, it turns out to be possible to develop quite a good homological algebra for it. We introduce such notions as complexes and their homology objects and prove analogues of some classical statements of homological algebra in the context of pointed sets. This includes the long exact sequence in homology, the 5-Lemma, and the Mayer-Vietoris sequence.11 Dec. 2017 at Fields, 2pm! Alejandro Morales

(UMass Amherst)Volume and lattice point formulas for flow polytopes The Lidskii formulas of Baldoni and Vergne show that the Kostant partition function of type A flow polytopes can be deduced from a formula for their volume. The formulas that Baldoni and Vergne give for the number of lattice points and the volume function of type A flow polytopes are combinatorial in nature, yet the proofs are via computations with residues. We prove these formulas via subdivisions and apply them to families of flow polytopes like the Pitman-Stanley polytope, the Chan-Robbins-Yuen polytope and a new family of polytopes coming from tridiagonal matrices. This is joint work with Karola Meszaros.04 Dec. 2017

Taylor Brysiewicz

(Texas A&M U.)Counting polynomially parametrized interpolants via Necklaces We consider the problem of locally approximating an analytic curve in the complex plane plane by a polynomial parametrization t -> (x_1(t),x_2(t)) of bidegree (d_1,d_2). Contrary to Taylor approximations, these parametrizations can achieve a higher order of contact at the cost of losing uniqueness and possibly the reality of the solution. We study the extent to which uniqueness fails by counting the number of such curves as the number of aperiodic combinatorial necklaces on d_1 white beads and d_2 black beads. We analyze when this count is odd as an initial step in studying when real solutions exist.27 Nov. 2017 Rafael S. González D'León

(York U.)The $\gamma$-coefficients of the tree Eulerian polynomials. We consider the generating polynomial $T_n(t)$ of the number of rooted trees on the set $\{1,2,\dots,n\}$ counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. In particular it has palindromic coefficients and hence it can be expressed in the the basis $\left \{ t^i(1+t)^{n-1-2i}\,\mid\, 0\le i \le \lfloor \frac{n-1}{2}\rfloor\right \}$, known as the $\gamma$-basis. We show that $T_n(t)$ has nonnegative $\gamma$-coefficients and we present various combinatorial interpretations for them.20 Nov. 2017

Samantha Dahlberg

(U. of British Columbia)Chromatic symmetric functions and e-positivity Richard Stanley introduced the chromatic symmetric function X_G of a simple graph G, which is the sum of all possible proper colorings with colors {1,2,3,... } coded as monomials in commuting variables. These formal power series are symmetric functions and generalize the chromatic polynomial. Soojin Cho and Stephanie van Willigenburg found that, given a sequence of connected graphs G_1, G_2, ... where G_i has i vertices, { X_{G_i} } forms a basis for the algebra of symmetric functions. This provides a multitude of new bases since they also discovered that only the sequence of complete graphs provides a basis that is equivalent to a classical basis, namely the elementary symmetric functions. This talk will discuss new results on chromatic symmetric functions using these new and old bases, and additionally we will also resolve Stanley's e-Positivity of Claw-Contractible-Free Graphs. This is joint work with Angele Foley and Stephanie van Willigenburg.13 Nov. 2017 Nantel Bergeron

(York U.)New invariants on graphs We define new invariants on graphs and study their properties.06 Nov. 2017

Josh Hallam

(Wake Forest U.)Whitney duals of graded partially ordered sets To each graded poset one can associate two sequences of numbers; the Whitney numbers of the first kind and the Whitney numbers of the second kind. One sequence keeps track of the Möbius function at each rank level and other keeps track of the number of elements at each rank level. The Whitney numbers appear in many areas of combinatorics. For example, the Stirling numbers of the first and second kind are the Whitney numbers of the first and second kind of the partition lattice. We say two posets P and Q are Whitney duals of each other if the absolute value of the Whitney numbers of the first kind of P is the Whitney numbers of the second kind of Q and vice-versa. In this talk, we will describe a method to construct Whitney duals. This method uses edge labelings and quotient posets. We will also discuss some examples of posets which have Whitney duals. The edge labelings we use to construct Whitney duals allow us to define an action of the 0-Hecke algebra on the maximal chains of posets with these labelings. Time permitting, we will discuss this action. No prior knowledge of Whitney numbers, edge labelings, or quotient posets will be assumed. This is joint work with Rafael S. Gonz\'alez D'le\'on.30 Oct. 2017 No Seminar 23 Oct. 2017

Laura Colmenarejo

(York U.)Factorization formulas for singular Macdonald polynomials We prove some factorization formulas for singular Macdonald polynomials indexed by particular partitions called quasistaircases. We also show some applications of these factorizations and some conjectures we are working on. This is joint work with Charles F. Dunkl and Jean-Gabriel Luque.16 Oct. 2017 Hugh Thomas

(UQAM)Robinson-Schensted-Knuth via quiver representations The Robinson-Schensted-Knuth correspondence is a many-faceted jewel of algebraic combinatorics. In one variation, it provides a bijection between permutations of n and pairs of standard Young tableaux with the same shape, which is a partition of n. In another (more general) version, it provides a bijection between fillings of a partition lambda by arbitrary non-negative integers and fillings of the same shape lambda by non-negative integers which weakly increase along rows and down columns. I will discuss an interpretation of RSK in terms of the representation theory of type A quivers (i.e., directed graphs obtained by orienting a path graph). This allows us to generalize RSK to other Dynkin types (plus a choice of minuscule weight), and is related to periodicity results for piecewise-linear toggling. I will not assume familiarity with either RSK or with quiver representations. This is joint work with Al Garver and Becky Patrias.13 Oct. 2017

Friday, York U.Jerzy Wejman

(U. of Connecticut)Finite free resolutions and root systems. In this talk I will discuss the structure of free resolutions of length 3 over Noetherian rings. Associate to a triple of ranks (r_3, r_2, r_1) in our free complex a triple (p,q,r)=(r_3+1, r_2-1, r_1+1). Associate to (p,q,r) the graph T_{p,q,r} (three arms of lengths p-1, q-1, r-1 attached to the central vertex). The main result is the explicit construction of a generic ring R_{gen} for resolutions of the format with the differentials of ranks r_1, r_2, r_3. This ring carries an action of a Kac-Moody Lie algebra associated to the graph T_{p,q,r}. In particular the ring R_{ten} is Noetherian if and only if T_{p.q.r} is a Dynkin graph. I will discuss the structure of the ring R_{gen} and possible consequences for the structure of perfect ideals of codimension 3.09 Oct. 2017 Thanksgiving Day! 02 Oct. 2017

At 4pm!Chi-Kwong Li

(College of William and Mary)Preserver Problems and Quantum Information Science Preserver problems concern the study of transformations on matrices or operators with special properties. In this talk, we will describe the history and recent advance of the topic. In particular, we will mention some results and problems related to quantum information science. No quantum mechanics background is needed. Undergraduates who have taken an linear algebra course are welcomed to attend.25 Sep. 2017 Rafael S. González D'León

(York U.)A family of symmetric functions associated with Stirling permutations We present exponential generating function analogues to two classical identities involving the ordinary generating function of the complete homogeneous symmetric function. After a suitable specialization the new identities reduce to identities involving the first and second order Eulerian polynomials. These results led us to consider a family of symmetric functions associated with the Stirling permutations introduced by Gessel and Stanley.18 Sep. 2017 Shu Xiao Li

(York U.)Structure of R-Polytopes on Relations We introduce a class of {0,1} polytopes, assiciated to relations on a finite set, called R-polytopes. The R-polytopes generalize a number of classic polytopes including non-attacking rook placement polytopes and Stanley's chain polytopes. We have a general theorem that describes their 1-skeleton. The R-polytopes also appear in graph theory, finding their facets is central in vertex packing problem. We will discuss and give description of some of the facets. This is joint work with Aliniaeifard, Benedetti, Bergeron & Saliola.

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