<   AlCoVE: an Algebraic Combinatorics Virtual Expedition   >

AlCoVE 2025 will be held virtually on Zoom on May 29 – 30, 2025 (Thursday and Friday).


Past conferences: AlCoVE 2020, AlCoVE 2021, AlCoVE 2022, AlCoVE 2023, and AlCoVE 2024.


Organizers: Laura Colmenarejo, Maria Gillespie, Oliver Pechenik, Liam Solus, Foster Tom, and Lorenzo Vecchi


AlCoVE brings 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.


Registration and poster session application

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


We will be holding virtual poster sessions in Gather on May 29 and May 30. Please submit a short application to present a poster HERE by the end of the day Eastern time on April 2.


List of Confirmed Speakers:



Schedule (subject to change, all times EDT):

TBA. The password for Zoom and Gather is the same, and will be sent to registered participants. Hint: How does Catalan count to 5? (only numbers)


MAY 29 (THURSDAY):
10:00 - 10:30 AMWelcomeZoom link
10:30 - 11:00 Tatsuyuki HikitaTBAZoom link
11:00 - 11:30Break/activityZoom link
11:30 - noon Mireille Bousquet-MélouThe ascent lattice on Dyck pathsZoom link
noon - 1:30Lunch and poster session AGather link
1:30 - 2:00 Darij GrinbergThe random-to-random shuffles and their q-deformationspaperslidesZoom link
2:00 - 2:30Break/activityZoom link
2:30 - 3:00 Yuhan JiangThe Ehrhart series of alcoved polytopesZoom link
3:00 - 4:00 Break/activityZoom link
4:00 - 4:30Byung-Hak HwangRefinement of Hikita's e-positivity theorem via Abreu–Nigro's g-functions and restricted modular lawZoom link
4:30 - 5:00Coffee breakGather link
5:00 - 5:30Megumi HaradaDegenerations and compactifications of varieties from mutations of polytopesZoom link
5:30 - 6:00Happy hourGather link

MAY 30 (FRIDAY):
10:00 - 10:30 AMWelcomeGather link
10:30 - 11:00 Lukas KühneWhen alcoved polytopes addZoom link
11:00 - 11:30Break/activityZoom link
11:30 - noon Emine YıldırımFrieze patterns of affine type DZoom link
noon - 1:30 PM Lunch and poster session BGather link
1:30 - 2:00 Federico CastilloTBAZoom link
2:00 - 2:30Break/activityZoom link
2:30 - 3:00Melody ChanCombinatorics of tropical abelian varietiesZoom link
3:00 - 4:00Break/activityZoom link
4:00 - 4:30 Joseph PappeWhen is the chromatic quasisymmetric function symmetric?Zoom link
4:30 - 5:00Coffee breakGather link
5:00 - 5:30Melissa Sherman-BennettPermutahedral subdivisions and toric degenerationsZoom link
5:30 - 6:00Happy hourGather link

Abstracts of talks

Tatsuyuki Hikita

TBA

TBA


Mireille Bousquet-Mélou

The ascent lattice on Dyck paths

Several posets defined on Dyck paths of length 2n have been studied, including in recent years: let us cite the Stanley lattice, the Tamari lattice and its greedy version... In particular, the enumeration of their intervals has revealed unexpected links with planar maps.


Here we consider a greedy version of Stanley's lattice, in which cover relations are obtained by exchanging a down step with the whole ascent that follows it. This order is connected to a more general lattice recently defined by Nadeau and Tewari, and it follows that it is also a lattice, the so-called "ascent lattice". One also considers the sub-posets induced on m-Dyck paths (all ascent lengths are multiples of m) and their mirrors (all descent lengths are multiples of m).


Counting intervals in these posets reveals links with plane walks confined to a cone. The associated generating function is algebraic for m=1, for mysterious reasons, but not algebraic nor D-finite for m>1. However, the numbers of intervals of mirrored m-Dyck paths occur in the OEIS, and one can establish a bijection between these intervals and congruence classes of the "sylvester monoid" introduced in 2005 par Hivert, Novelli and Thibon. This is a joint work with Jean-Luc Baril, Sergey Kirgizov (Université de Bourgogne, Dijon, France) and Mehdi Naima (Sorbonne Université, Paris, France).


Darij Grinberg

The random-to-random shuffles and their q-deformations

Consider a random shuffle acting on a deck of n cards as follows: Uniformly at random, we select k out of our n cards, remove them from the deck, and then move them back to k uniformly random positions. This shuffle — the so-called "k-random-to-random shuffle" — is a Markov chain that is given by a certain element of the group algebra of the symmetric algebra. A celebrated result of Dieker, Saliola and Lafrenière says that this shuffle is diagonalizable with all eigenvalues rational. Earlier, it was observed by Reiner, Saliola and Welker that two such shuffles for different k's always commute. Both results are deep and hard. I will discuss a new approach to these shuffles that has resulted in simpler proofs as well as a q-deformation — i.e., a generalization into the Hecke algebra of the symmetric group. Along the way, some properties of the Hecke algebras have been revealed, as well as some general results about integrality of eigenvalues. Joint work with Sarah Brauner, Patricia Commins and Franco Saliola.


Yuhan Jiang

The Ehrhart series of alcoved polytopes

Alcoved polytopes are convex polytopes that are the closure of a union of alcoves in an affine Coxeter arrangement. They are rational polytopes and, therefore, have Ehrhart quasipolynomials. Here we describe a method for computing the generating function of the Ehrhart quasipolynomial, or Ehrhart series, of any alcoved polytope via a particular shelling order of its alcoves. We also show a connection between Early's decorated ordered set partitions and this shelling order for the hypersimplex \(\Delta_{2,n}\).


Byung-Hak Hwang

Refinement of Hikita's e-positivity theorem via Abreu–Nigro's g-functions and restricted modular law

The e-positivity of chromatic quasisymmetric functions of unit interval orders remains a long-standing open problem in algebraic combinatorics. Recently, Hikita made a breakthrough by proving the e-positivity of chromatic symmetric functions. He assigned a rational function in q to each standard Young tableau and provided a probabilistic interpretation of this function. He then showed that the e-coefficient indexed by a partition \(\lambda\) is given by the sum of these rational functions over all standard Young tableaux of shape \(\lambda\), thereby establishing the positivity of the coefficients at \(q=1\). In this talk, I present a refinement of Hikita’s result. Specifically, I show that the sum of the rational functions over a certain subset of standard Young tableaux of shape \(\lambda\) equals the e-coefficient of Abreu–Nigro’s g-function. This coincidence yields the e-positivity of the g-functions at \(q = 1\), providing a partial answer to a conjecture of Abreu and Nigro. This is joint work with JiSun Huh, Donghyun Kim, Jang Soo Kim, and Jaeseong Oh.


Megumi Harada

Degenerations and compactifications of varieties from mutations of polytopes

Toric geometry provides a useful dictionary between combinatorics and (toric) algebraic geometry. The theory of Newton–Okounkov bodies allows the combinatorial techniques of toric geometry to be applied to more general projective varieties. In past joint work with Escobar, we described a phenomenon of wall-crossing for Newton–Okounkov bodies, which involves piecewise-linear mutation maps between different Newton–Okounkov bodies associated to the same variety. Similar phenomena appear in the work of Rietsch–Williams, as well as Bossinger–Cheung–Magee–Nájera Chávez on Newton–Okounkov bodies associated to compactifications of cluster varieties. In addition, Kaveh and Manon have analyzed the theory of valuations into semifields of piecewise linear functions, and explored their connections to families of toric degenerations. In these settings, the mutations between Newton–Okounkov bodies can reflect important aspects of the geometry and combinatorics of the associated variety. Inspired by these ideas, in an ongoing joint project with Escobar and Manon, we wrap the data of a collection of lattices related by piecewise-linear bijections together into a single semi-algebraic object, equipped with its own notions of convexity and polyhedra. In certain situations, such a (generalized) polytope encodes a compactification of an affine variety whose coordinate ring can be equipped with a valuation into one of these objects, and aspects of the geometry of the compactification (including some of its toric degenerations) can be understood combinatorially. In this talk, I will (very) briefly introduce elements of this construction, give examples, and — if time permits — point to some unanswered questions.


Lukas Kühne

When alcoved polytopes add

Alcoved polytopes are characterized by the property that all facet normal directions are parallel to the roots \(e_i - e_j\). This fundamental class of polytopes appears in several applications such as optimization, tropical geometry or physics. This talk focuses on the type fan of alcoved polytopes which is the subdivision of the metric cone by combinatorial types of alcoved polytopes. The type fan governs when the Minkowski sum of alcoved polytopes is again alcoved. We prove that the structure of the type fan is governed by its two-dimensional faces and give criteria to study the rays of alcoved simplices.


This talk is based on joint work with Nick Early and Leonid Monin.


Emine Yıldırım

Frieze patterns of affine type D

Friezes appear in the intersection of representation theory of algebras, cluster algebras and surface combinatorics. This leads to interesting questions that can be solved combining techniques from all these areas. In our joint work with K. Baur, L. Bittman, E. Gunawan, and G. Todorov, we prove a certain property "growth coefficient" of these friezes coming from affine type D.


Federico Castillo

TBA

TBA


Melody Chan

Combinatorics of tropical abelian varieties

I will discuss some of the polyhedral aspects of moduli spaces of abelian varieties in tropical geometry, assuming no prior knowledge of tropical geometry. An important role is played by work of Voronoi from the early 1900s and the beautiful combinatorics of Delaunay subdivisions, which are particular infinite periodic polyhedral tilings of \(\mathbb{R}^n\).


Joseph Pappe

When is the chromatic quasisymmetric function symmetric?

The chromatic quasisymmetric function(CQF), introduced by Shareshian and Wachs, has primarily been studied in the context of unit interval graphs. In this talk, we will take a broader perspective by examining which labelled graphs have a symmetric CQF. I will give several necessary conditions on the underlying graph in order to be symmetric, introduce a new symmetric family of graphs, as well as explain a possible connection between the symmetry and e-positivity of the chromatic quasisymmetric function. This is joint work with Maria Gillespie and Kyle Salois.


Melissa Sherman-Bennett

Permutahedral subdivisions and toric degenerations

I will discuss some regular subdivisions of the permutahedron, one for each Coxeter element in the symmetric group. These subdivisions are "Bruhat interval" subdivisions, meaning that each face is the convex hull of the permutations in a Bruhat interval (regarded as vectors). Bruhat interval subdivisions in general correspond to cones in the positive tropical flag variety by work of Joswig–Loho–Luber–Olarte and Boretsky–Eur–Williams. I will also mention some (related) formulas for the class of the permutahedral variety as a sum of Richardson classes. This is joint work with Allen Knutson and Mario Sanchez.