<   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

AlCoVE 2022 will be held virtually on Zoom on June 6 – 7, 2022 (Monday and Tuesday).


Click here for information about the 1st iteration of AlCoVE (2020) and the 2nd iteration (2021).


Organizers: Laura Colmenarejo, Maria Gillespie, Oliver Pechenik, and Liam Solus


AlCoVE aims to bring together researchers interested in algebraic combinatorics from around the world. Each talk will be 30 minutes and between talks, there will be casual social activities for spending time with your friends and making new friends.


To access the Zoom links, you must first register for the conference:

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List of Confirmed Speakers:


Poster Session

We will be holding a virtual poster session in Gather on June 7. Poster titles and abstracts are available here.



Schedule (subject to change, all times EDT):

JUNE 6 (MONDAY):
10:00 - 10:30 AM IntroductionZoom link
10:30 - 11:00 Jang Soo KimAffine Gordon-Bender-Knuth identities and cylindric Young tableauxslidesRecordingZoom link
11:00 - 11:30Ice breakersZoom link
11:30 - noon Peter CameronSynchronization: from automata to weakly perfect graphsslidesRecordingZoom link
noon - 12:30"Lunch" breakGather link
12:30-1:00 PM Cesar CeballosReconstruction of polytopes and Kalai's conjecture on reconstruction of spheresslidesRecordingZoom link
1:00 - 1:30Virtual expeditionZoom link
1:30 - 2:00 Bridget TennerPermutations, shapes, and fillingsslidesZoom link
2:00 - 2:30AlCoVE Puzzle Hunt: Alluring Conundrums for Vivacious ExplorersPuzzles by Solving FunZoom link
2:30 - 3:00 Daoji HuangBumpless pipe dream RSK, growth diagrams, and Schubert structure constantsslidesRecordingZoom link
3:00 - 3:30 Coffee breakGather link
3:30 - 4:00 Luis FerroniEhrhart polynomials of slices of rectangular prismsslidesRecordingZoom link
4:00 - 5:00Happy hour!Gather link

JUNE 7 (TUESDAY):
10:00 - 10:30 AM Coffee/gatheringGather link
10:30 - 11:00 Jessica StrikerPromotion, rowmotion, rotation, and websslidesRecordingZoom link
11:00 - 11:30Critter timeZoom link
11:30 - noon İrem PortakalRigid Gorenstein toric Fano varieties arising from directed graphsslidesZoom link
noon - 12:30 PM "Lunch" breakGather link
12:30 - 1:00 Antonio NigroSplitting the cohomology of regular semisimple Hessenberg varietiesslidesRecordingZoom link
1:00 - 1:30Minecraft expedition, led by Ben YoungRecordingZoom link
1:30 - 2:00 Sheila SundaramThe immaculate Hecke posetslidesRecordingZoom link
2:00 - 2:30AlCoVE Puzzle Hunt: Alluring Conundrums for Vivacious ExplorersPuzzles by Solving FunZoom link
2:30 - 3:00Jenna RajchgotGeometric vertex decomposition and Gorenstein liaison for toric ideals of graphsslidesRecordingZoom link
3:00 - 3:30Coffee breakGather link
3:30 - 4:00 Brendon RhoadesThe characters of local permutation statisticsslidesRecordingZoom link
4:00 - 5:00Poster sessionGather link

Abstracts of talks

Peter Cameron

Synchronization: from automata to weakly perfect graphs

A (finite-state) automaton is synchronizing if there is a reset word in its alphabet such that, if it reads this word, it ends in a known state independent of its starting state. Much of the interest in this question comes from the infamous Černý conjecture, stating that if an \(n\)-state automaton is synchronizing then it has a reset word of length at most \((n-1)^2\). I will talk not about this but about other aspects of synchronization.


Algebraically, an automaton can be regarded as a transformation monoid (acting on the set of states) with distinguished generating set (corresponding to the alphabet); it is synchronizing if it contains a transformation of rank 1. There is a single obstruction to synchronization; a transformation monoid \(M\) on \(\Omega\) is non-synchronizing if and only if it is contained in the endomorphism monoid of a non-trivial weakly perfect graph (one with clique number and chromatic number equal).


A lot of interest has centred on the permutation groups \(G\) such that, for any non-permutation \(t\), the monoid \(\langle G,t\rangle\) is synchronizing. Such groups must be primitive and, in terms of the O'Nan–Scott classification, are affine, diagonal (with two simple factors in the socle) or almost simple. The proof of this involves the famous Hall–Paige conjecture.


Cesar Ceballos

Reconstruction of polytopes and Kalai's conjecture on reconstruction of spheres

A famous theorem in polytope theory states that the combinatorial type of a simplicial polytope is completely determined by its facet-ridge graph. This celebrated result was proven by Blind and Mani in 1987, via a non-constructive proof using topological tools from homology theory. An elegant constructive proof (“A simple way to tell a simple polytope from its graph”), but requiring exponential time to compute, is due to Kalai shortly after. In their original paper, Blind and Mani asked whether their result can be extended to simplicial spheres, and a positive answer to their question was conjectured by Kalai. The purpose of this talk is to show that Kalai’s conjecture holds in the particular case of Knutson and Miller’s spherical subword complexes, a family of simplicial spheres of importance in the study of Gröbner geometry of Schubert varieties.


This talk is based on current joint work with Joseph Doolittle.


Luis Ferroni

Ehrhart polynomials of slices of rectangular prisms

Hypersimplices are ubiquitous within algebraic combinatorics. The problem of calculating their volumes, which happen to be the well-known Eulerian numbers, has motivated much research in the past decades. In this presentation we will address the Ehrhart theory of a much more general version of hypersimplices. We show that the enumeration of certain weighted permutations allows to give a complete description of the coefficients of the Ehrhart polynomial of these polytopes. We will also explore a nice extension of results by Nick Early and Donghyun Kim about the \(h^*\)-vectors of hypersimplices. We will discuss combinatorial interpretations of the entries of the \(h^*\)-vector of arbitrary slices of prisms. The coefficients of the \(h^*\)-vectors of these polytopes have been studied before from a point of view of commutative algebra, as they are precisely the numerators of the Hilbert series of all algebras of Veronese type. If time allows, we will also discuss how the combinatorial objects that we introduce in our calculations hint a natural extension of the flag Eulerian numbers, and how the volumes and \(h^*\)-polynomials of the slices of prisms allow to calculate them.


Daoji Huang

Bumpless pipe dream RSK, growth diagrams, and Schubert structure constants

We introduce analogs of left and right RSK insertion for Schubert calculus of complete flag varieties. The objects being inserted are certain biwords, the insertion objects are bumpless pipe dreams, and the recording objects are decorated chains in Bruhat order. As an application, we adopt Lenart's growth diagrams of permutations to give a combinatorial rule for Schubert structure constants in the separated descent case. This is joint work with Pavlo Pylyavskyy.


Jang Soo Kim

Affine Gordon-Bender-Knuth identities and cylindric Young tableaux

The Gordon-Bender-Knuth identities are determinant formulas for the sum of Schur functions of partitions with bounded length. There are interesting combinatorial consequences of the Gordon-Bender-Knuth identities, for instance, connections between standard Young tableaux of bounded height, lattice walks in a Weyl chamber, and noncrossing matchings. In this talk we prove an affine analog of the Gordon-Bender-Knuth identities and study their combinatorial properties. As a consequence we obtain an unexpected connection between cylindric standard Young tableaux and \(r\)-noncrossing and \(s\)-nonnesting matchings. This is joint work with JiSun Huh, Christian Krattenthaler, and Soichi Okada.


Antonio Nigro

Splitting the cohomology of regular semisimple Hessenberg varieties

For each indifference graph, there is an associated Hessenberg variety, whose cohomology recovers the chromatic symmetric function of the graph. The decomposition theorem applied to the forgetful map from the Hessenberg variety to the projective space describes the cohomology of the Hessenberg variety as a sum of smaller pieces. We give a combinatorial description of the Frobenius character of each piece. As a consequence, we can prove that the coefficient of \(e_{\lambda}\), where \(\lambda\) is a partition of length \(2\), in the \(e\)-expansion of the chromatic symmetric function of indifference graphs is non-negative. This is based on joint works with Alex Abreu.


İrem Portakal

Rigid Gorenstein toric Fano varieties arising from directed graphs

A directed edge polytope \(A_G\) is a lattice polytope arising from root system \(A_n\) and a finite directed graph \(G\). If every directed edge of \(G\) belongs to a directed cycle in \(G\), then \(A_G\) is terminal and reflexive, that is, one can associate this polytope to a Gorenstein toric Fano variety \(X_G\) with terminal singularities. It is shown by Totaro that a toric Fano variety which is smooth in codimension 2 and \(Q\)-factorial in codimension 3 is rigid. In this talk, we classify all directed graphs \(G\) such that \(X_G\) is a toric Fano variety which is smooth in codimension 2 and \(Q\)-factorial in codimension 3. This is joint-work with Selvi Kara and Akiyoshi Tsuchiya.


Jenna Rajchgot

Geometric vertex decomposition and Gorenstein liaison for toric ideals of graphs

Toric ideals of graphs are homogeneous binomial ideals defined from graphs. By results of Hochster and of Sturmfels, if \(J\) is a toric ideal of a graph in a polynomial ring \(R\) and \(J\) has a squarefree initial ideal, then \(R/J\) is both normal and Cohen-Macaulay. Consequently, toric ideals of graphs with squarefree initial ideals are a natural class of ideals to study in relation to one of the long-standing open questions in Gorenstein liaison theory, namely, "is every arithmetically Cohen-Macaulay subscheme of projective space in the Gorenstein liaison class of a complete intersection (abbreviated glicci)?".


In this talk, I'll review some of the basics of toric ideals of graphs and of Gorenstein liaison. Then I'll show that toric ideals of certain families of graphs are glicci because they can be Gröbner degenerated via a sequence of geometric vertex decompositions (in the sense of Knutson-Miller-Yong).


This talk is based on joint work with Mike Cummings, Sergio Da Silva, and Adam Van Tuyl and joint work with Patricia Klein.


Brendon Rhoades

The characters of local permutation statistics

A permutation statistic \(f\) on \(S_n\) is \(k\)-local if \(f(w)\) is determined by the restriction of \(w\) to \(k\)-element subsets of \([n]\). Many of the classical statistics (inv, des, maj, exc, ... ) are local, as are pattern counting statistics. We study the best class function approximation \(R \; f\) of local statistics \(f\) and give a general method for expanding \(R \; f\) into irreducible characters. The asymptotics of this rule generalizes work of Gaetz-Ryba on pattern enumeration in \(S_n\)-conjugacy classes. Along the way, we obtain a new path power sum basis of the ring of symmetric functions whose Schur expansions are governed by a novel combinatorics of monotonic ribbon tilings. Joint with Zach Hamaker.


Jessica Striker

Promotion, rowmotion, rotation, and webs

Many combinatorial objects with strikingly good enumerative formulae also have remarkable dynamical behavior. In this talk, we give examples of actions on combinatorial objects that have a small, predictable order because they are rotations in disguise. These include: promotion on standard Young and increasing tableaux of certain shapes and rowmotion on order ideals of specific planar and non-planar posets. We also discuss invariant polynomials for the related webs. This talk is based on joint works Kevin Dilks, Rebecca Patrias, Oliver Pechenik, and Nathan Williams.


Sheila Sundaram

The immaculate Hecke poset

The immaculate Hecke poset \(\mathcal{P}(\alpha)\) is defined by a partial order on standard immaculate tableaux \(\mathrm{SIT}(\alpha)\) of fixed composition shape \(\alpha\). These tableaux originally appeared in work of Berg-Bergeron-Saliola-Serrano-Zabrocki (2014-15), and give rise to the dual immaculate basis of the ring of quasisymmetric functions. I will describe four different descent sets associated to these tableaux, each of which determines a family of quasisymmetric functions, and in particular the new basis of row-strict dual immaculate functions. The four descent sets give rise to four different actions of the 0-Hecke algebra on the set \(\mathrm{SIT}(\alpha)\), all captured by the same immaculate Hecke poset \(\mathcal{P}(\alpha)\). This remarkable poset reveals many different cyclic and indecomposable 0-Hecke modules with combinatorially interesting quasisymmetric characteristics. This is joint work with Elizabeth Niese, Stephanie van Willigenburg, Julianne Vega and Shiyun Wang.


Bridget Tenner

Permutations, shapes, and fillings

Permutations \(w \in S_n\) are famously in bijection with pairs of standard Young tableaux having shape \(\lambda(w)\), where \(\lambda(w)\) is a partition of \(n\). That bijection can be exploited to reveal structural connections between permutations and their shapes, and those relationships have important combinatorial implications. We will talk about recent developments regarding these relationships, giving combinatorial meaning to the lengths of the rows below the top row. We will also reveal a connection between a permutation's reduced words and the contents of its corresponding tableaux. This work is in collaboration with Volodymyr Mazorchuk and with Emily Gunawan, Jianping Pan, and Heather Russell.