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 Jun. 2020

(Virtual seminar on Zoom)Digjoy Paul

(HBNI)Note: special time at 11am ESTNew approaches to the restriction problem

(Video) (slides)Given an irreducible polynomial representation $W_n$ of the general linear group $GL_n$, we can restrict it to the representations of the symmetric group $S_n$ that seats inside $GL_n$ as a subgroup. The restriction problem is to find a combinatorial interpretation of the restriction coefficient: the multiplicity of an irreducible $S_n$ modules in such restriction of $W_n$. This is an open problem (see OPAC 2021!) in algebraic combinatorics. In Polynomial Induction and the Restriction Problem, we construct the polynomial induction functor, which is the right adjoint to the restriction functor from the category of polynomial representations of $GL_n$ to the category of representations of $S_n$. This construction leads to a representation-theoretic proof of Littlewood's Plethystic formula for the restriction coefficient. Character polynomials have been used to study characters of families of representations of symmetric groups (see Garsia and Goupil), also used in the context of FI-modules by Church, Ellenberg, and Farb (see FI-modules and stability for representations of symmetric groups). In Character Polynomials and the Restriction Problem, we compute character polynomial for the family of restrictions of $W_n$ as $n$ varies. We give an interpretation of the restriction coefficient as a moment of a certain character polynomial. To characterize partitions for which the corresponding Weyl module has non zero $S_n$-invariant vectors is quite hard. We solve this problem for partition with two rows, two columns, and for hook-partitions.

20 Apr. 2020

(Virtual seminar on Zoom)Martin Rubey

(TU Wien)The existence of a cyclic sieving phenomenon for permutations via a bound on the number of border strip tableaux and invariant theory (Video) We consider permutations pi of {1,...,n} as chord diagrams, where the elements label the vertices of a regular n-gon, and there is a directed arc from i to pi(i) for each element i. We can "rotate" a permutation by rotating its chord diagram. As one of our main results we show that there must exist a map from permutations of {1,...,n} to integer partitions of n that has the same distribution as the Robinson-Schensted shape, but is invariant under rotation. The proof uses a little combinatorial representation and invariant theory, and some calculus. We are unable to exhibit the map explicitly.

joint work with Per Alexandersson, Stephan Pfannerer and Joakim Uhlin30 Mar. 2020

(Virtual seminar on Zoom)Florian Aigner

(UQAM)Alternating sign matrices and totally symmetric plane partitions (Video) In this talk, we study the Schur polynomial expansion of a family of symmetric polynomials related to the refined enumeration of alternating sign matrices. We prove that the expansion can be expressed as a sum over totally symmetric plane partitions and we are also able to determine the coefficients. This establishes a new connection between alternating sign matrices and a class of plane partitions. As a by-product we obtain an interesting map from totally symmetric plane partitions to Dyck paths. This is joint work with I. Fischer, M. Konvalinka, P. Nadeau and V. Tewari.23 Mar. 2019

(On Zoom)Robin Sulzgruber

(York)An explicit e-expansion of vertical strip LLT polynomials (Video) The vertical-strip LLT polynomials $G_P(x;q+1)$ are certain symmetric function. Recently it has been shown that they expand positively into elementary symmetric functions, however, an explicit formula in terms of the elementary basis has so far been missing. I will present such a formula which is governed by the combinatorics of orientations of unit interval graphs.16 Mar. 2020

(Virtual seminar on Zoom)Mike Zabrocki

(York)The multiset partition algebra and Kronecker product An analogue of the partition algebra is the centralizer algebra of the symmetric group $S_n$ acting on the polynomial ring ${\mathbb C}[ x_{ij} : 1 \leq i \leq n, 1 \leq j \leq k]$ where $\sigma(x_{ij}) = x_{\sigma(i)j}$. We propose a basis for this algebra indexed by multiset partitions as a starting point for studying this algebra. Our approach to studying the algebra and its representation theory is through combinatorics of multiset tableaux and symmetric functions. Like the partition algebra, the Kronecker coefficients are multiplicities of irreducibles in the restriction of an irreducible of this algebra.9 Mar. 2020 Steven Karp

(LaCIM, UQAM)Topology of totally positive spaces The classical example of a totally positive space is the set of n x n totally positive matrices, formed by matrices whose every submatrix has positive determinant. Total positivity has been studied extensively over the past century, and has seen a renewed interest in the past 30 years, initiated by work of Lusztig and of Fomin and Zelevinsky. My talk will focus on the topology of such spaces. Historically, the motivation for studying the topology of totally positive spaces is that they have cell decompositions which realize interesting posets in combinatorics, related to Bruhat orders. A new motivation comes from recent work in theoretical physics, which calls for understanding convex polytopes generalized from affine space into the Grassmannian, where the notion of convexity is replaced by total positivity. I will present new techniques, developed in joint work with Pavel Galashin and Thomas Lam, for establishing the homeomorphism type of totally positive spaces and their compactifications. In particular, we prove that the totally nonnegative part of a partial flag variety forms a regular CW complex, confirming conjectures of Postnikov and of Williams.2 Mar. 2020 John Machacek

(York)Filtrations of the symmetric group harmonics We will consider two filtrations of the symmetric group harmonics by defining families of ideals containing the invariant ideal. Conjecturally one filtration has dimension given by falling factorials while the other has dimension given by multinomial coefficients. We will verify our conjectures in some simpler cases and discuss approaches to the general situation.24 Feb. 2020

Harmony Zhan

(York)Continuous-time quantum walks on graphs A continuous-time quantum walk is determined by the transition operator $U(t) = \exp(itA)$, where $A$ is the adjacency matrix of a graph. Many interesting quantum phenomena, such as perfect state transfer, uniform mixing and fractions revival, translate into algebraic properties of the graph. I will give an overview of this area, and present some recent results. No knowledge of physics is needed for this talk.17 Feb. 2020 Reading WeekNo Seminar10 Feb. 2020 Freydoon Rahbarnia

(FUM)Domination in Graphs In this lecture we present a short course on domination in graphs and we offer some selected results on this topic. We consider two conjectures for the independent domination number of cubic graphs. We've observed that one conjecture is true when a given graph is 2-connected and planar. Finally, we disprove the other conjecture by providing a new infinite family of graphs.Winter Break2 Dec. 2019 Michael Weselcouch

(Wake Forest)The Uniqueness and Irreducibility of P-partition Generating Functions The $(P,\omega)$-partition generating function of a labeled poset $(P,\omega)$ is a quasisymmetric function enumerating certain order-preserving maps from $P$ to $\mathbb{Z}+$. We study the expansion of this generating function in the recently introduced type 1 quasisymmetric power sum basis $\{\Psi_{\alpha}\}$. Using this expansion, we show that connected, naturally labeled posets have irreducible P-partition generating functions. We conclude by giving a combinatorial interpretation for the coefficients of the $\Psi_{\alpha}$-expansion of the$ (P,\omega)$-partition generating function akin to the Murnaghan-Nakayama rule.25 Nov. 2019 Marcel Golz

(Waterloo)Graph polynomials in quantum field theory I will give an overview of the various graph polynomials that appear in quantum field theory (Kirchhoff, Symanzik, Dodgson, spanning forest, corolla polynomials, among others). After a broader discussion of their combinatorics and what they can tell us about physics I will focus on a new identity that relates large sums of Dodgson polynomials, indexed by chord diagrams, to much simpler expressions.18 Nov. 2019 John Machacek (York) New invariants from Hopf algebras We will review work of Aguiar-Bergeron-Sottile which gives a Hopf algebraic framework to produce quasisymmetric functions and polynomials associated to combinatorial objects. We will then focus of graphs and give generalizations of the chromatic symmetric function and chromatic polynomial. This talk is based on joint work with J.-C. Aval and N. Bergeron.11 Nov. 2019 Anna Weigandt

(Michigan)Diagonal Grobner geometry of matrix Schubert varieties Knutson and Miller (2005) studied the Grobner geometry of matrix Schubert varieties. They showed that the multidegree of a matrix Schubert variety is its corresponding Schubert polynomial. Furthermore, when degenerating with respect to any antidiagonal term order, the components of the limit are naturally labeled by pipe dreams. In subsequent work, Knutson, Miller, and Yong (2009) considered diagonal term orders. In the vexillary case, components of the limit are in bijection with flagged tableaux. We present further results and conjectures outside of the vexillary case. This is joint work with Zach Hamaker and Oliver Pechenik.8 Nov. 2019

(Friday at Fields)Sami Assaf

(USC)A Pieri rule for key polynomials Schur functions are an amazing basis of symmetric functions originally defined as characters of irreducible modules for GL_n. The Pieri rule for the product of a Schur function and a single row Schur function is a multiplicity-free branching rule with a beautiful combinatorial interpretation in terms of adding boxes to a Young diagram. Key polynomials are an interesting basis of the polynomial ring originally defined as characters of submodules for irreducible GL_n modules under the action of upper triangular matrices. In this talk, I'll present joint work with Danjoseph Quijada where we give a Pieri rule for the product of a key polynomial and a single row key polynomial. While this formula has signs, it is multiplicity-free and has an interpretation in terms of adding balls to a key diagram.4 Nov. 2019 Yulan Qing

(Toronto)Algorithms to find coarse lengths of geodesics in Out(F_n) We study the group Out(F_n), the group of outer automorphisms of the free group. We start with a discussion of an influential algorithm called the Stallings folding algorithm which solves a lot of the combinatorial problems in Out(F_n). However, we demonstrate that this algorithm fail to produce efficient paths in Out(F_n). In contrast, a similar algorithmic approach in the setting of mapping class groups produces quasi-geodesics (called Hierarchy paths) resulting in the celebrated Masur-Minsky distance formula for the word length of an element in the mapping class group. We show that a direct analogue of the Hierarchy paths does not work in the setting Out(F_n), i.e. there exist points in Out(F_n) where every quasi-geodesics between them backtracks in all of the Out(F_n) complexes .28 Oct. 2019 Aaron Lauve

(Loyola)Powersums in the Hopf algebra of quasisymmetric functions The classical (commutative) powersums wear a number different hats. To name a few, they: demonstrate the Cartier-Milnor-Moore isomorphism for the ring SYM of symmetric functions; provide for a recursive construction of character tables for the symmetric group; provide elegant combinatorial links to other classical bases of SYM; .... After a brief survey, we'll discuss the main problem: the limitless sea of generalizations to the Hopf algebra NSYM of noncommutative symmetric functions. In this talk, I offer a generalization to the ring QSYM of quasisymmetric functions that maintains an elegant combinatorial link to other bases. I'll close with a discussion of open questions and possible applications. (A report on work in progress with Anthony Lazzeroni.)21 Oct. 2018 Justin Troyka

(York)Combinatorial species and counting split graphs The theory of combinatorial species is a powerful framework for counting combinatorial objects acted on by permutations. I will give an introduction to species theory and share my recent results on the enumeration of split graphs (graphs whose vertices can be partitioned into a clique and a stable set).14 Oct. 2019 Reading WeekNo Seminar7 Oct. 2019 Nantel Bergeron

(York)On flag P-partitions 30 Sep. 2019 Emine Yildirim

(Queen's University)Rowmotion and Coxeter transformation Let $P$ be a cominuscule poset which can be thought of as a parabolic analogue of the poset of positive roots of a finite root system. Let $J(P)$ be the poset of order ideals of $P$. There is an action called \emph{rowmotion} on the set of order ideals which is defined purely combinatorially. By Rush and Shi, it is known that this action has order $h$ where h is the Coxeter number for the corresponding root system. On the other hand, we consider the Coxeter transformation $\tau$ on the poset $J(P)$. We showed that $\tau^{h+1}=\pm id$ for the two of the three infinite families of cominuscule posets, and the exceptional cases. Now, the question is whether these two actions related. In this talk, I will discuss the combinatorial relation between rowmotion and $\tau$ for a special case.

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