The Applied Algebra Seminar

A
Monday afternoon research seminar

The seminar is currently organized by Cesar Ceballos and Nantel Bergeron.

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 Cesar Ceballos 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 14:00-15:00 in N638 Ross Building (York University).

Date Speaker Title (click titles for abstract) 14 Apr. 2014 Vincent Pilaud

(É. Polytechnique)Signed tree associahedra An associahedron is a polytope whose vertices correspond to the triangulations of a convex polygon and whose edges correspond to flips between them. Loday gave a particularly elegant realization of the associahedron which has been generalized in two directions: on the one hand by Hohlweg and Lange to obtain multiple realizations of the associahedron parametrized by a sequence of signs, and on the other hand by Postnikov to obtain a realization of the graph associahedra of Carr and Devadoss. The goal of this talk is to unify and extend these two constructions to signed tree associahedra. We will also present the rich combinatorial and geometric properties of the resulting polytopes. The talk is based on arXiv:1309.5222.7 Apr. 2014 Viviane Pons

(U. Vienna)Intervals of the Tamari lattice We present the Tamari lattice on binary trees and more specifically, the questions related to the intervals of the lattice. The purpose of the talk is to define a new combinatorial object called Tamari interval-posets which can be used to deal with those questions and give some enumeration results.31 Mar. 2014 Nantel Bergeron

(York U.)Non-commutative algebras and non-commutative Gröbner bases of homogeneous ideals Any finitely generated non-commutative algebra is the quotient of a (non-commutative) polynomial ring by an ideal. In the recent years, I have been interested by three such algebras:

(1) When the ideal is Symmetric polynomials in n (non-commutative) variables

(2) When the ideal is Quasisymmetric functions in n variables

(3) Fomin-Kirrilov algebra

The main question in the three cases is: Is the algebra finite dimensional? Very little is know in each case. [Mike, last week, also presented such an algebra] It would be very interesting to understand the Gröbner basis of the defining ideal in each case. Since the relations are homogeneous, the Gröbner basis is homogeneous as well. I will present the algebras above. Do a brief survey of Gröbner basis algorithm. Show that in the homogeneous case the algorithm can be modified to give the answer degree by degree. This allows us to have partial answer for the algebras above, and hopefully, understand them better.24 Mar. 2014 Mike Zabrocki

(York U.)Non-commutative Gröbner bases and self avoiding walks I will demonstrate a method of computing self avoiding walks using algebra by realizing paths in an n-dimensional lattice as the monomials of a 2n-variable non-commutative algebra. The quotient by the two sided ideal of monomials representing paths that start and end at the same point is an algebra whose graded dimensions are the number of self avoiding walks that end at a fixed point.

This is work in progress, but we found some surprising initial results. It turns out that the non-commutative Gröbner basis of the ideal is the set of monomials representing self-avoiding polygons plus those paths representing a step forward and back. In addition, the algebra seems to agree with factorization algorithms that appear in the self avoiding walk literature. I will try to demonstrate how these calculations can be done using a package in GAP for computing non-commutative Gröbner bases called "GBNP".

This is joint work with Andrew Rechnitzer17 Mar. 2014 Angela Hicks

(Stanford U.)Parallelogram Polyominoes and (Surprise!)-- The Diagonal Harmonics A recent paper of Dukes and Le Borgne studied two statistics on parallelogram polyominoes-- two nonintersecting paths, each composed of north and east steps and bounded by a rectangular m x n bounding box. Conjecturing that q,t-counting the polyominoes by the two statistics resulted in polynomials that were symmetric in q and t as well as m and n, they called the statistics "area"' and "bounce," in reference to the historic statistics on parking functions. This talk will discuss a following joint paper (with the original two authors, Aval, and D'Adderio) which introduced a third statistic, "dinv," on polyominoes and demonstrated the conjectured symmetries. In a surprising twist, the proof illuminates a direct link from polyominoes to parking functions and the famous space of diagonal harmonics.10 Mar. 2014 Darij Grinberg

(MIT)The order of birational rowmotion (joint work with Tom Roby) For any finite poset P, rowmotion is a certain permutation of the set of order ideals of P. Studied by various authors (sometimes under different names and in different guises), this permutation has proven to have interesting and nontrivial properties -- e. g., its order is p + q when P is the product [p] x [q] of two chains. In very recent work (inspired by discussions with Berenstein), Einstein and Propp describe a way to generalize rowmotion: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting.

In the latter setting, "birational rowmotion" is a birational self-equivalence of a certain algebraic variety, and no longer has finite order in the general case. Yet we were able to compute its order for several classes of posets, including the product [p] x [q] of two chains (here the order is the same as in the case of ordinary rowmotion, that is, p + q), some triangle-shaped posets and graded forests. Our methods are partly based on those used by Alexandre Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture, and the well-behavedness of birational rowmotion seems to be related to the combinatorics of root lattices.3 Mar. 2014 Marcelo Aguiar

(Cornell U.)The ring of graphs and the chromatic polynomial Certain basic operations among graphs resemble addition and multiplication of ordinary numbers. Formally, the species of graphs is a ring in an appropriate category. We explain this fact and employ it to obtain a novel understanding and a wide generalization of the chromatic polynomial (and the corresponding symmetric function of Stanley) in terms of ring theory. The talk is based on joint work with Swapneel Mahajan and Jacob White.24 Feb. 2014 Cesar Ceballos

(York U.)Dyck path triangulations and extendability We introduce the Dyck path triangulation of the cartesian product of two simplices. The maximal simplices of this triangulation are given by Dyck paths, and its construction partially generalizes to produce triangulations using rational Dyck paths. Our study of the Dyck path triangulation is motivated by extendability problems of partial triangulations of products of two simplices. We show that whenever m≥k>n, any triangulation of Δ_{m-1}^{k-1}x Δ_{n-1}extends to a unique triangulation of Δ_{m-1}x Δ_{n-1}. Moreover, with an explicit construction, we prove that the bound k>n is optimal. We also exhibit interesting interpretations of our results in the language of tropical oriented matroids, which are analogous to classical results in oriented matroid theory.

This is joint work with Arnau Padrol and Camilo Sarmiento.17 Feb. 2014

NO SEMINARFamily Day (University is closed) 10 Feb. 2014 Trueman Machenry

(York U.)Hessenberg-Stirling Matrices and divisible closure of the algebra of weighted Isobaric Polynomials Aura Conci, Huilan Li, T. MacHenry, Geanina Tudose

In a paper called "Reflections on Symmetric Polynomials and Arithmetic Functions", Geanina Tudose and I exhibited an embedding of the Weighted Isobaric Polynomials (WIPs) in their injective hull, that is, we adjoined all of the rational roots with respect to the convolution product of these polynomials to these polynomials. The WIP polynomials are just the Schur Hook Polynomials written on the elementary symmetric polynomial basis, and include the Generalized Fibonacci Polynomials (GFP) and the Generalized Lucas Polynomial (GLP). Because the Group of Multiplicative Arithmetic Functions can be faithfully represented using GFPs, and the Group of Additive Arithmetic Functions can be faithfully represented using GLPs, these groups of Arithmetic functions inherit this embedding, that is, are also explicitly embedded in their divisible closures, this time with respect to the Dirichlet product. In a recently published paper Huilan Li and I used Hessenberg Matrices to represent the GFP and GLP. In a current paper, Aura Conci and I have used Hessenberg matrices and some functions related to the Stirling Numbers of the first and second kind to give matrix representations for these embeddings. The Hessenberg matrices are especially suitable for such embeddings because of their computability properties, and the nice relation between their Determinants and Permanents, Permanents being of importance in particle physics.27 Jan. 2014 Jonathan Toledo

(Cinvestav)Strong persistence property of square-free monomial ideals We say that an ideal I of a commutative Noetherian ring A has thestrong persistence propertyif (I^{k+1}:I)=I^k for each k≥1. This concept was introduced by J. Herzog and A.A. Qureshi in their workPersistence and stability properties of powers of ideals, where they present an equivalence in terms of associated primes and show that each polymatroidal ideal satisfies this property. Nonetheless, it had been studied among others by S. Morey, J. Matínez-Bernal, R.H. Villarreal in their workAssociated primes of powers of edges ideals. The authors show that edge ideals of graphs have this property as a step towards obtaining other results, so we began to study the case of square-free monomial ideal. The results presented in this talk were obtained recently in my PhD. We will begin by showing some general results about this property, like how to study it through components of the minimal set of generators considering as a clutter, we will also show that for every square-free monomial ideal I holds (I^2:I)=I. After that, we will give examples and classes of monomial ideals which have the strong persistence property, like cases of path ideals and edge ideals of weighted graphs. Finally we will talk about target problem which we are working on.25 Nov. 2013 Drew Armstrong

(U. of Miami)Rational Catalan Numbers and Rational Associahedra For each rational number x outisde the interval [-1,0] I will define a positive integer Cat(x) called the "rational Catalan number". The classical Catalan number corresponds to x=n and the Fuss-Catalan number corresponds to x=n/((k-1)n+1). These numbers satisfy the symmetry Cat(x)=Cat(-x-1), which implies that Cat(1/(x-1))=Cat(x/(1-x)). I will call this common value the "derived Catalan number" Cat'(x):=Cat(1/(x-1))=Cat(x/(1-x)), and it follows that Cat'(x)=Cat'(1/x). Rational Catalan numbers are categorified by various generalizations of traditional Catalan structures. In particular, I will describe joint work with B. Rhoades and N. Williams in which we define a "rational associahedron". This is a pure simplicial complex with Cat(x) many maximal faces. It is not a polytope but it is homotopy equivalent to a wedge of Cat'(x) many spheres. We conjectured that the equality Cat'(x)=Cat'(1/x) is represented by Alexander duality of rational associahedra. This conjecture was recently proved by B. Rhoades.18 Nov. 2013 Nathan Williams

(LaCIM, UQAM)Cataland I will talk about two combinatorial miracles relating purely poset-theoretic objects with purely Coxeter-theoretic objects. The first miracle is that there are the same number of linear extensions of the root poset as reduced words of the longest element (occasionally), while the second is that there are the same number of order ideals in the root poset as certain group elements (usually). I will conjecturally place these miracles on remarkably similar footing and examine the generality at which we should expect such statements to be true.11 Nov. 2013 Laura Colmenarejo

(U. de Sevilla)Trying to prove stability We define the plethysm of two Schur symmetric functions as a new operation, which is more complicated and interesting than the Kronecker product. We will discuss this and other technics (like FI-modules and Vertex operators) which can be used to study stability problems that appear in different contexts.4 Nov. 2013 Christopher Hanusa

(Queens College)Applications of abacus diagrams: Simultaneous core partitions, alcoves, and a major statistic A t-core partition is a partition whose Young diagram has no hooks of length t. Partitions that are both s-core and t-core for integers s and t are called simultaneous core partitions. We will discuss the applications of simultaneous core partitions--we visit with lattice paths, alcoves in a hyperplane arrangement, and a "major index" statistic that recovers a q-analog for Catalan numbers. This is joint work with Brant Jones and Drew Armstrong.28 Oct. 2013 Laura Escobar

(Cornell U.)Bott-Samelson varieties, subword complexes and brick polytopes The Bott-Samelson varieties are a resolution of singularities for Schubert varieties. Intuitively, Bott-Samelson varieties factor G/B into a product of ℂℙ^{1}'s via a map into G/B. These varieties are mostly studied in the case in which the map into G/B is birational, however in this talk we will study fibers of this map when it is not birational. We will see that in some cases this fiber is a toric variety. In order to do so we will translate this problem into a purely combinatorial one in terms of subword complexes. These simplicial complexes, defined by Knutson and Miller, encode a lot of information about reduced words in a Coxeter system. Pilaud and Stump realized certain subword complexes as the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is generalized associahedron. These stories connect in a nice way: for certain words a fiber of the Bott-Samelson map is the toric variety of the Brick polytope.21 Oct. 2013 Farid Aliniaeifard

(York U.)The Zero-Divisor Graphs of Semigroups, Rings, and Group Rings We associate some graphs to a ring R and we investigate the interplay between the ring-theoretic properties of R and the graph-theoretic properties of the graphs associated to R. The Zero-divisor graph of a semigroup S is a graph with non-zero zero-divisors of S as vertex set and distinct vertices x and y are adjacent if xy = 0 or yx = 0. We investigate diameter, girth, and Isomorphism Problem for zero-divisor graphs of rings. Also, we show that the set of ideals of R is a semigroup. So we can define a zero-divisor graph for the set of ideals of R. At the end we investigate the genus of these graphs.14 Oct. 2013

NO SEMINARThanksgiving

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