The Applied Algebra Seminar

A
Monday afternoon research seminar

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

During 2016-17, 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) 10 Apr. 2017 Karen Yeats

(U. of Waterloo)★ 03 Apr. 2017 Praise Adeyemo

(Fields Institute)Cohomological consequences of the pattern map Billey and Braden defined maps on flag manifolds which are geometric counterpart of permutation patterns. A section of their pattern map is an embedding of the flag manifold of a Levi subgroup into the full flag manifold. We give two expressions for the induced map on cohomology. One is in terms of generators and the other is in terms of the Schubert basis. We show that the coefficients in the second expression are naturally Schubert structure constants and therefore positive. This is joint work with Frank Sottile, Texas A & M University.27 Mar. 2017 Martha Yip

(U. of Kentucky)Jordan forms of upper triangular matrices and combinatorics The set of n by n upper-triangular nilpotent matrices with entries in a finite field F_q has Jordan canonical forms indexed by partitions lambda of n. In this talk, I present a combinatorial formula for computing the number of matrices of fixed Jordan type as a sum over weighted paths in Young's lattice. We will also see how a natural refinement of this formula leads to a new bijection between rook placements and set partitions.20 Mar. 2017 Frank Sottile

(Texas A&M University)The Bergman fan and the coamoeba of a linear space The coamoeba of an algebraic subvariety of a complex torus is the set of arguments of its points. While this is an object from complex analysis, it has surprisingly combinatorial structures coming from its boundary, called the phase limit set. We illustrate this in the comaoeba and phase limit sets of a linear subspace, showing that its dimension and boundary components may be described largely using the Bergman fan of the corresponding matroid. All terms will be described, and some pictures will be shown. This is joint work with Mounir Nisse.13 Mar. 2017 Rosa Orellana

(Dartmouth College)The partition algebra, symmetric functions and Kronecker coefficients The Schur-Weyl duality between the symmetric group and the general linear group allows us to connect the representation theory of these two groups. A consequence of this duality is the Frobenius formula which connects the irreducible characters of the general linear group and the symmetric group via symmetric functions. The symmetric group is also in Schur Weyl duality with the partition algebra. This duality allows us to introduce a new Frobenius type formula that connects the characters of the symmetric group and those of the partition algebra. In this talk we introduce a new basis of the ring of symmetric functions which specialize to the characters of the symmetric group when evaluated at roots of unity. Furthermore, the structure coefficients for this new basis of symmetric functions are the stable (or reduced) Kronecker coefficients. We will also discuss how this new basis allows us to use symmetric functions to study the representation theory of the partition algebra and the Kronecker coefficients. This is joint work with Mike Zabrocki.06 Mar. 2017 Mike Zabrocki

(York U.)The Catalan version of the Delta Conjecture Haglund-Remmel and Wilson discovered a generalization of the shuffle conjecture that explains the action of \Delta_{e_k} on e_n. We will present this conjecture and an outline of a proof of the coefficient of s_{n-r,1^r} in this expression.27 Feb. 2017 Franco Saliola

(UQAM)Spectral analysis of random-to-random Markov chains "Pick a card (any card), remove it and put it back anywhere in the deck." Repeating this process leads to a card shuffling technique known as the random-to-random shuffle. An important open problem is to determine how many of these shuffles are needed to randomize the deck. This is controlled by the spectra of the associated transition matrices. By considering all the random-to-random shuffles simultaneously, we prove that the eigenspaces admit a beautiful recursive structure. This structure allows one to build eigenbases starting from bases for the kernels. Among other things, we obtain complete combinatorial descriptions of the eigenvalues of the transition matrices. The representation theory of the symmetric group features prominently in our analysis, but the results and the talk can be appreciated with no prior knowledge of representation theory. This talk is based on joint work with Ton Dieker.13 Feb. 2017 Elissa Ross

(MESH Consultants)Geometric challenges in digital design: Exact offsetting of polygon meshes Architectural designs are frequently represented digitally as plane-faced meshes, yet these can be challenging to translate into built structures. Offsetting operations may be used to give thickness to meshes, and are produced by offsetting the faces, edges or vertices of the mesh in an appropriately defined normal direction. In this talk I will describe an algorithm for precisely resolving the details of an offset mesh obtained by face-offsetting. I will focus on a recent extension to this algorithm that uses tools from algebraic graph theory to answer the question: under what conditions is a variable rate face-offset mesh also a uniform distance edge-offset mesh? I will also give an overview of the activities of MESH Consultants Inc., and describe an ongoing software development project to define the 3D modelling platform of the future.06 Feb. 2017 Carolina Benedetti

(York U.)A journey towards Catalan and Kostant partition function. In this talk I will introduce an ongoing project aiming to provide a combinatorial proof of the volume of a polytope associated to the complete graph on n vertices. Zeilberger proved that such volume is the product of Catalan numbers. We aim to make use of the relation between Kostant partition functions and flow polytopes to understand this problem. I will introduce Caracol diagrams and how they may allow us to progress toward the solution of the main problem. This is current work with R. Gonzalez, C. Hanusa, P. Harris, M. Yip.30 Jan. 2017 John Machacek

(Michigan State U.)The chromatic symmetric function, hypertrees, and generalized graphs Stanley's chromatic symmetric function is a graph invariant which has been (and still is) the subject of much research. We will (attempt to) make case for the study of a chromatic symmetric function in hypergraphs and other generalizations of graphs. The existence (or non-existence) of two non-isomorphic trees with equal chromatic symmetric functions is an open problem. Martin, Morin, and Wagner have shown that the chromatic symmetric function of a tree determines its degree sequence. We will show that the degree sequence of a uniform hypertree is determined by its chromatic symmetric function, but there do exist non-isomorphic pairs of 3-uniform hypertrees with the same chromatic symmetric function. A definition of "generalized graph coloring" will be given and will encompass graph and hypergraph coloring as well as oriented coloring and acylic coloring.23 Jan. 2017 Pamela E. Harris

(Williams College)A proof of the peak polynomial positivity conjecture We say that a permutation $\pi=\pi_1\pi_2\cdots \pi_n \in \mathfrak{S}_n$ has a peak at index $i$ if $\pi_{i-1} < \pi_i > \pi_{i+1}$. Let $\mathcal{P}(\pi)$ denote the set of indices where $\pi$ has a peak. Given a set $S$ of positive integers, we define $\mathcal{P}(S;n)=\{\pi\in\mathfrak{S}_n:\mathcal{P}(\pi)=S\}$. In 2013 Billey, Burdzy, and Sagan showed that for subsets of positive integers $S$ and sufficiently large $n$, $| \mathcal{P}(S;n)|=p_S(n)2^{n-|S|-1}$ where $p_S(x)$ is a polynomial depending on $S$. They gave a recursive formula for $p_S(x)$ involving an alternating sum, and they conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all nonnegative. In this talk we introduce a new recursive formula for $|\mathcal{P}(S;n)|$ without alternating sums and we use this recursion to prove that their conjecture is true.16 Jan. 2017 Franco Saliola

(UQAM)Combinatorial topology and homological invariants of monoid algebras arising in algebraic combinatorics This is an expository (and accessible!) talk presenting a beautiful interplay between combinatorial topology and homological algebra for a class of monoids that arose naturally in algebraic combinatorics and discrete Markov chains. We will survey several examples of the monoids and the original application to the analysis of certain Markov chains (including card-shuffling). These Markov chains can be realized as random walks on the faces of a hyperplane arrangement; and their analysis takes advantage of the monoid structure on the faces. The representation theory of monoids plays a prominent role in this analysis: it is used to compute the spectrum of the transition operators of the Markov chains; and to prove diagonalizability of the transition operators. Further developments have uncovered a close connection between algebraic and combinatorial invariants of these monoids. More precisely, certain homological invariants of the monoid algebras (Ext-spaces) coincide with the cohomology of order complexes of posets naturally associated with the monoids. We will explore several applications of this interplay. This talk is based on joint work with Stuart Margolis (Bar Ilan) and Benjamin Steinberg (CUNY).09 Jan. 2017 Steven Melczer

(U Waterloo & ENS Lyon)Effective Analytic Combinatorics in Several Variables The field of analytic combinatorics studies the asymptotic behaviour of sequences through analytic properties of their generating functions. In addition to the now classical univariate theory, recent work in the study of analytic combinatorics in several variables (ACSV) has shown how to derive asymptotics for the coefficients of certain D-finite functions by representing them as diagonals of multivariate rational functions. We detail the rich theory of ACSV from a computer algebra viewpoint, with an eye towards automatic implementations that can be used by those with no specialized knowledge in the field. Applications from several areas of combinatorics, number theory, and physics will be discussed.12 Dec. 2016

(3:30pm)Blake Madill

(U. of Waterloo)On some recent applications of combinatorics on words to noncommutative ring theory In this talk we will discuss some recent applications of combinatorics on words to noncommutative ring theory. We will use certain right-infinite words over finite alphabets to create interesting examples in graded ring theory. Namely, we will use right-infinite words obtained by iterating morphisms of the free monoid generated by an alphabet to produce a monomial algebra which is graded-nilpotent, semiprimitive, of Gelfand-Kirillov dimension 2, and finitely generated as a Lie algebra. This answers several questions of Greenfeld, Leroy, Smoktunowicz, and Ziembowski. Several open problems will be presented and discussed. This is joint work with Jason Bell.5 Dec. 2016 Alex Fink

(Queen Mary U. of London)The Tutte polynomial via Ehrhart theory and GKM Tam\'as K\'alm\'an discovered a pair of univariate polynomials associated to a hypergraph which enumerate its spanning trees by internal and external activity. In the first part of the talk I'll discuss joint work with Amanda Cameron in which we extend this to a bivariate polynomial of polymatroids enumerating both activity statistics at once, using lattice point enumeration. On matroids our invariant agrees with the Tutte polynomial, though not in its most obvious basis. In the second part I'll introduce two other sources of formulae for the Tutte polynomial and similar invariants, and speculate on the relationships between them. One of these comes from joint work with David Speyer using K-theory of the Grassmannian, and the other is due to Kristin Shaw using Chern-Schwartz-Macpherson classes of hyperplane arrangement complements.28 Nov. 2016 Li Ying

(Texas A&M U.)Stability of the Heisenberg Product on Symmetric Functions. The Heisenberg product is an associative product defined on symmetric functions which interpolates between the usual product and Kronecker product. I will give the definition of this product and describe some properties of it. One well known thing about the Kronecker product of Schur functions is the stability phenomenon discovered by Murnaghan in 1938. I will give an analogous result for the Heisenberg product of Schur functions.21 Nov. 2016 Farid Aliniaeifard

(York U.)Normal supercharacter theory, Dyck paths, and Hopf structures. Normal supercharacter theory is a mechanism to substitute normal subgroups by conjugacy classes in such a way that this swelling simulate some features of irreducible charactes. We construct a normal supercharacter theory for the group of square matrices with entries in a finite field and ones on the diagonal and zeroes in all entries below the diagonal. The supercharacters in this supercharacter theory are indexed by Dyck paths. We show that this construction is identical with Scott Andrews' construction after gluing by torus group. Then we build up Hopf monoid structures base on these supercharacters.14 Nov. 2016 (3:30pm) Mikhail Mazin

(Kansas State U.)Rational Dyck Paths in the Non-Relatively Prime Case In the relatively prime case, the rational (n,m)-Dyck paths are in bijection with the (n,m)-invariant subsets of integers, considered up to shifts. This bijection brings a connection between rational Catalan combinatorics and the geometry of certain algebraic varieties. In particular, it allows one to reinterpret the dinv statistic as the dimension of the corresponding complex affine cell in a certain affine Springer fiber. The non-relatively prime case is more complicated. Although on the combinatorial side many things can be generalized, including the dinv statistic and even Shuffle conjecture, there is no known generalization of the geometric interpretation of the dinv statistic. In this talk, I will explain how one can extend the bijection between rational Dyck paths and the invariant subsets in Z to the non-relatively prime case. The natural obstacle is that the set of invariant subsets is not finite in the non-relatively prime case. One has to consider certain equivalence relation on the invariant subsets to make the bijection work. The hope is that this construction will lead to a geometric or representation theoretic interpretation of the dinv statistic in the non-relatively prime case. This is a joint project with Eugene Gorsky and Monica Vazirani.14 Nov. 2016 (2:30pm) Alex Woo

(U. of Idaho)Depth in classical Coxeter groups Motivated by questions about sorting, Petersen and Tenner defined a notion of depth in an arbitrary Coxeter group and gave a combinatorial formula, with a constructive proof, for the depth of a permutation in terms of sums of sizes of exceedances. We do the same for the Coxeter groups of types B and D using the usual embedding into the symmetric group. We find that depth cannot always be realized with a minimal number of transpositions but can always be realized by a reduced product. We also characterize elements for which depth equals length and for which depth, length, and reflection length are all equal. This is joint work with Eli Bagno, Riccardo Biagioli, and Moti Novick.07 Nov. 2016 Zach Teitler

(Boise State U.)Waring ranks of homogeneous forms The Waring rank of a homogeneous form is the number of terms needed to write it as a sum of powers of linear forms. It is related to secant varieties, provides a measure of the complexity of polynomials, and has applications in statistics, sciences, and engineering. I will discuss three topics related to Waring rank. (1) Waring ranks of general forms have been known for some time, but it is also of interest to determine Waring rank of particular forms such as the generic determinant and permanent. I will describe some recent results obtained via algebraic and geometric lower bounds for Waring rank; this is joint work with Jaroslaw Buczynski and with Harm Derksen. (2) A variation of a conjecture of Strassen asserts that the Waring rank of the sum of two forms in independent variables is the sum of the ranks of the summands. I will describe an elementary sufficient condition for a strong version of Strassen's conjecture. (3) It is an open question to determine the maximum Waring rank occurring among forms of a given degree, in a given number of variables. I will describe an upper bound; this is joint work with Gregoriy Blekherman.31 Oct. 2016 Laura Colmenarejo

(York U.)Stability in the combinatorics of symmetric functions In this talk, I will introduce the h-plethysm coefficients, related to the usual plethysm coefficients. I will present a combinatorial description of them. This description will be the tool to give a combinatorial proof of some well-known stability properties of the plethysm coefficients. I will also introduce some families of reduced kronecker coefficients. For these families, I will give their generating function as well as combinatorial descriptions in terms of plane partitions and quasipolynomials. These results are included in my thesis.24 Oct. 2016 Dave Anderson

(Ohio State U.)Diagrams and essential sets for signed permutations The essential set of a permutation, defined via its Rothe diagram by Fulton in 1992, gives a minimal list of rank conditions cutting out the corresponding Schubert variety in the flag manifold. I will describe an analogous notion for signed permutations, giving minimal conditions for Schubert varieties in flag varieties for other classical groups. This is related to a poset-theoretic construction of Reiner, Woo, and Yong, and thus gives a diagrammatic method for computing the latter.17 Oct. 2016 Praise Adeyemo

(Fields Institute)The Three Presentations of T- Equivariant K- Theory of G/B Let G be a simple, simply connected, complex Lie group with a Borel subgroup B. G/B is the flag manifold associated to G. It is acted upon by maximal torus T contained in B by multiplication. We study the T- equivariant K-theory of G/B, the Grothendieck group of coherent sheaves on G/B. In this talk, I will give the three presentations of T-equivariant K-theory of this homogeneous space- Schubert presentation, Borel presentation and GKM ring. I will describe partially how to convert an element in one presentation to another. It represents the ongoing work with Shizuo Kaji.3 Oct. 2016 Georg Merz

(Math. Institut)Newton Okounkov bodies - A bridge between algebraic and convex geometry. Inspired by work of A. Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata independently introduced the notion of a Newton-Okounkov body which associates a convex body to a line bundle on a projective variety. This construction is a generalization of the toric case, and it allows to transport algebra geometric questions into such of convex geometry, and vice versa. In this talk I will give an outline of the construction, present some known results and give some generalizations of these in the setting of graded linear series.26 Sep. 2016 John Maxwell Campbell

(York U.)The evaluation of immaculate functions in terms of the ribbon basis There is an elegant combinatorial formula for evaluating elements of the ribbon basis of NSym in terms of immaculate functions, but there is no known cancellation-free formula for expanding elements of the immaculate basis in terms of ribbon functions in NSym. However, recent research suggests that many different classes of immaculate functions may be evaluated in terms of ribbon functions using cancellation-free Jacobi-Trudi-like formulas. In this talk, we offer a sketch of a proof of a surprising Jacobi-Trudi-like formula for expanding immaculate-rectangles as linear combinations of ribbon functions.12 Sep. 2016 Jake Levinson

(U. of Mich.)(Real) Schubert Calculus from Marked Points on P^1 I will describe a family of Schubert problems on the Grassmannian, defined using tangent flags to points of P^1 (or more generally, a stable curve) in its Veronese embedding. For Schubert problems having a finite set of solutions, the Shapiro-Shapiro Conjecture (later proven by Mukhin-Tarasov-Varchenko) proposed that, when the marked points are all real, the solutions would be as real as possible. More recently, Speyer gave a remarkable description of the real topology of the family in terms of Young tableaux and Schützenberger's jeu de taquin. I will go through this story, then give analogous results on the topology of one-dimensional Schubert problems (where the family consists of curves). In this case the combinatorics involves orbits of so-called "evacuation-shuffling", an algorithm related to tableau promotion and evacuation.

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