The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar is currently organized by John Machacek and Nantel Bergeron.

During 2018-19, the seminar takes place from 15:00-16:00 in 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 John Machacek 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) 1 Apr. 2019

Ahmed Umer Ashraf

(Western University)Alternative approach to matroid base polytope volumes Ardila, Benedetti and Doker gave an expression of volume of a matroid base polytope, in terms of a sum of products of Crapo's beta invariants of matroid minors corresponding to Dragon marriage condition. This was based on work of Postnikov. Derksen (and Fink) have defined a universal valuative invariant for all matroids. Using these and related ideas, we describe another approach to finding volume of a base polytope from lattice of cyclic flats of its respective matroid. As an application, we give a closed formula for the volumes of connected (sparse) paving matroid polytopes.25 Mar. 2019

Nantel Bergeron

(York University)Polytopes of independent sets of relations, their 1-skeleta and quasi-matroids We characterize the edges of two classes of 0/1-polytopes whose vertices encode theindependent setsof a relation on a finite set. The first class includes poset chain polytopes, the vertex packing polytopes from graph theory, some instances of matroid independence polytopes, as well as newly-defined polytopes whose vertices correspond to non- crossing set partitions. In analogy with matroid basis polytopes, the second class is obtained by considering the independent sets of maximal cardinality. We show that the maximal independent set form a quasi-matroid, a new notion we will introduce during the talk. This is joint work with F. Aliniaeifard, C. Benedetti, S.X. Li, and F. Saliola.18 Mar. 2019

Mike Zabrocki

(York University)A combinatorial model for multivariate polynomials as an $S_n$ module Let $S_n$ act on $m$ sets of commuting varables and $m'$ sets of anti-commuting variables. We are interested in the polynomial ring in these sets of variables as an $S_n$ module. In this talk I will give a combinatorial model for the mulitplicity of an irreducible of a given shape. The special case of the trivial representation is a combinatorial description of the invariants of the polynomial ring. This is joint work with Rosa Orellana.11 Mar. 2019

Ba Nguyen

(Queen's University)Broken Lines and Theta Basis for Type A & D Cluster Algebras Discovered by M. Gross, P. Hacking, S. Keel and M. Kontsevich, Broken Lines and Theta Functions are the most general combinatorial models so far. They are so powerful that can be used to proved several well-known conjectures including the positivity conjecture. In this talk, we will focus on the combinatorics behind broken lines of cluster variables of type A cluster algebras. For type D case, we will look at some types of arcs in a triangulation of an once-punctured n-gon and construct the corresponding sets of broken lines.4 Mar. 2019

(In BC 325)Carolina Benedetti

(Universidad de los Andes)Concordance of positroids Positroids are a subclass of representable matroids that have gain lots of attention in the recent years, in particular, due to their rich combinatorics. In this talk we will show partial results answering the following questions: Given a family of positroids P_1,..., P_k on the same ground set, can we determine combinatorially when the family P_1,..,P_k is concordant? We will provide the background necessary to understand this question, its significance, and results that answer it in specific cases. This is joint work with A. Chavez and D. Tamayo.25 Feb. 2019

(In BC 325)Mark Skandera

(Lehigh University)Total nonnegativity and induced sign characters of the Hecke algebra Abstract: Gantmacher's study of totally nonnegative (TNN) matrices in the 1930's eventually found applications in many areas of mathematics. Descending from his work are problems concerning TNN polynomials, those polynomial functions of n^2 variables which take nonnegative values on TNN matrices. Closely related to TNN polynomials are functions in the Hecke algebra trace space whose evaluations at certain Hecke algebra elements yield polynomials in N[q]. In all cases, it would be desirable to combinatorially interpret the resulting nonnegative numbers. In 2017, Kaliszewski, Lambright, and the presenter found the first cancellation-free combinatorial formula for the evaluation of all elements of a basis of V at all elements of a basis of the Hecke algebra. We will discuss a recent improvement upon this result which also advances our understanding of TNN polynomials. This is joint work with Adam Clearwater.18 Feb. 2019

Reading WeekNo Seminar11 Feb. 2019

(In BC 325)Justin Troyka

(York University)Exact and asymptotic enumeration of cyclic permutations according to descent set The number of permutations that are n-cycles is (n-1)!, and the number of permutations with a given descent set is also well-understood. In this talk, I will explain my research on counting n-cycles with a given descent set (joint work with Sergi Elizalde, preprint arxiv.org/abs/1710.05103, to appear in JCTA). Using a result of Gessel and Reutenauer, we find a simple formula for the number of cyclic permutations with a given descent set, by expressing it in terms of ordinary descent numbers. We then use this formula to show that, for almost all descent sets I, the fraction of size-n permutations with descent set I which are n-cycles is asymptotically 1/n. We also use our formula to count the cycles that do not have two consecutive descents. Time permitting, I will also talk about the role of quasisymmetric functions in this research.4 Feb. 2019

(In BC 325)Daping Weng

(Michigan State)Cluster Duality of Grassmannian and a Cyclic Sieving Phenomenon of Plane Partitions Fix two positive integers $a$ and $b$. Scott showed that the homogeneous coordinate ring of the Grassmannian $Gr_{a, a+b}$ has the structure of a cluster algebra. This homogeneous coordinate ring can be decomposed into a direct sum of irreducible representations of $GL_{a+b}$ which correspond to non-negative integer multiples of the fundamental weight $w_a$. We introduce a periodic configuration space $Conf_{a+b,a}$ equipped with a natural potential function $W$ and prove that the tropicalization of $(Conf_{a+b,a},W)$ canonically parametrizes bases for the irreducible summands of the homogeneous coordinate ring of $Gr_{a,a+b}$, as expected by the cluster duality conjecture of Fock and Goncharov. We identify the parametrizing set of each irreducible summand with a collection of plan partitions of size $a\times b$. As an application, we use this identification to show a cyclic sieving phenomenon of plane partitions under a certain sequence of toggling operations. This is joint work with Linhui Shen.1 Feb. 2019

(Friday at Fields)

Changjian Su

(University of Toronto)Motivic Chern classes and Iwahori invariants of principal series Let G be a split reductive p-adic group. In the Iwahori-invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove a conjecture of Bump--Nakasuji--Naruse about certain transition matrix between these two bases. The idea of the proof is to use the two geometric realizations of the affine Hecke algebra, and relate the Iwahori invariants to Maulik--Okounkov's stable envelopes and Brasselet--Schurmann--Yokura's motivic Chern classes for the Langlands dual groups. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.21 Jan. 2019

Nicholas Ovenhouse

(Michigan State)Pentagram Maps and Integrability The pentagram map is a discrete dynamical system defined on the space of polygons. It was proved to possess a property called "complete integrability". We will discuss one method of proof of this fact, due to Gekhtman, Shapiro, Tabachnikov, and Vainshtein, using the combinatorics of certain weighted directed graphs. Recently, there was a generalization of the pentagram map defined on the Grassmann manifold. We will discuss how the same combinatorial approach, using weighted graphs, can be modified for this generalized Grassmann version.14 Jan. 2019

John Machacek

(York University)Strong factorization and the braid arrangement fan We establish strong factorization for pairs of smooth fans which are refined by the braid arrangement fan. This includes normal fans of generalized permutahedra. We show that the realization of the toric variety of the permutahedron as an iterated blow-up of projective space can be achieved by considering a sequence of polytopes known as hyper-permutahedra. To any poset we associate a cone which is the union of some Weyl chambers of type A. We give conditions for when a toric variety defined by such a cone is Gorenstein and for the existence of a crepant resolution.

Winter Break3 Dec. 2018 Gabriel Frieden

(CRM-ISM)An affine generalization of evacuation The Robinson--Schensted correspondence is a bijection between permutations and pairs of standard Young tableaux of the same shape. Under this bijection, the reverse complement of a permutation corresponds to the evacuation of the two tableaux. The number of standard tableaux of shape \lambda which are fixed by evacuation is equal to f^\lambda(-1), where f^\lambda(q) is the q-analogue of the hook-length formula. Building on work of Shi, Chmutov--Pylyavskyy--Yudovina recently introduced a generalization of Robinson--Schensted which maps elements of the affine symmetric group to pairs of tabloids (= standard row-strict tableaux) of the same shape. There is a natural involution of the affine symmetric group that generalizes the reverse complement of permutations. In this talk, we give an explicit description of the corresponding ``affine evacuation'' map on tabloids, and we show that the number of tabloids fixed by this map is equal to the evaluation of a certain Green's polynomial at q = -1. Along the way, we discover a combinatorial interpretation of the evaluation of the Kostka--Foulkes polynomials at q = -1. This is based on joint work with Mike Chmutov, Dongkwan Kim, Joel Lewis, and Elena Yudovina.26 Nov. 2018

CancelledNo Seminar19 Nov. 2018 Tianyuan Xu

(Queen's University)On the subregular J-rings of Coxeter systems Let $(W,S)$ be an arbitrary Coxeter system and let $G$ be its Coxeter diagram. We recall Lusztig's construction of the asymptotic Hecke algebra $J$ of $(W,S)$, an associative algebra closely related to the Iwahori--Hecke algebra of $(W,S)$, and present some results on a subalgebra $J_C$ of $J$ that we call the subregular $J$-ring. We show that while products in $J$ are defined in terms of Kazhdan--Lusztig polynomials, they can be computed in $J_C$ by a simple combinatorial algorithm centered around a certain truncated Clebsch--Gordan rule. As applications, we relate $J_C$ to the path algebra of a quiver whose underlying graph is $G$ and deduce some results on the structure and representations of $J_C$. This is joint work with Ivan Dimitrov, Charles Paquette, and David Wehlau.12 Nov. 2018

Pauline Hubert

(UQAM)The bi-characters of spaces of diagonal harmonic polynomials with inert variables The spaces of diagonal harmonic polynomials and some generalizations have been studied a lot since the 1990's. After a quick reminder of already known results, we will introduce a generalization of the space of diagonal harmonics with polynomials containing inert variables. Finnaly we will discuss the computation of the ($GL_r \times S_n$)-character of those spaces.5 Nov. 2018 Aram Dermenjian

(UQAM)Facial weak order in hyperplane arrangements We will discuss a poset structure that extends the poset of regions on a central hyperplane arrangement to the set of all faces of the arrangement. This order is the facial weak order and was first described on Coxeter groups where we showed it to be a lattice in the general case. We provide various characterizations of this poset including a local one, which was first studied by Krob, Latapy, Novelli, Phan, and Schwer in the case of symmetric groups, and a global one, that generalizes the notion of separation sets. These characterizations are the keys to show that for simplicial hyperplane arrangements the facial weak order is a lattice, generalizing a result by Björner, Edelman and Ziegler showing the poset of regions is a lattice for simplicial arrangements.29 Oct. 2018

Robin Sulzgruber

(York University)P-partitions and p-positivity Using the combinatorics of alpha-unimodal permutations we prove two new results on quasi-symmetric functions: (1) an expansion of Gessel's fundamental basis into quasi-symmetric power-sums, and (2) a positive expansion of Stanley's P-partition generating functions into quasi-symmetric power-sums. As a consequence we obtain a unified proof that many interesting families of (quasi-)symmetric functions expand positively into power-sums, for example chromatic quasi-symmetric functions and certain LLT polynomials.22 Oct. 2018 Hugo Mlodecki

(Paris-Sud)Auto-duality of WQSym, the Hopf algebra on packed words A Hopf algebra is a formalism that makes it possible to study algorithms on combinatorial object assembling and disassembling. Among these objects, we find the permutations where each number between 1 and n appears one and only once, or the packed words where each number between 1 and m appears at least once. We will study different ways of assembling and disassembling permutations and the relationships of duality and simple auto-duality that these operations verify. We will then try to generalize these relations to the packed words along a more complex path.15 Oct. 2018 John Machacek

(York University)Locally acyclic cluster algebras and their quivers We will dicuss locally acyclic cluster algebras and techniques for showing a cluster algebra is locally acyclic. We will also explore to realationship between the existence of a reddening sequence and equality of a cluster algebra and its upper cluster algebra.8 Oct. 2018 ThanksgivingNo Seminar1 Oct. 2018 Kelvin Chan

(York University)Induction Relations in the Symmetric Groups and Jucys-Murphy Elements Transitive factorizations faithfully encode many interesting objects. The well-known ones include ramified coverings of the sphere and hypermaps. Enumeration of specific classes of such objects have been known for quite some time now. Hurwitz numbers, monotone Hurwitz numbers and hypermaps numbers were discovered using different techniques. Recently, Carrell and Goulden found a unified algebraic approach to count these objects in genus 0. Jucys-Murphy elements and centrality play important roles in establishing induction relations. Such a method is interesting in its own right. Its corresponding combinatorial decomposition is however intriguingly mysterious. Towards a understanding of direct combinatorial analysis of multiplication of arbitrary permutations, we consider methods, especially operators on symmetric functions, and related problems in symmetric groups.24 Sep. 2018 LaCIM celebrationNo Seminar

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