Schedule for: 20w5164 - Dynamical Algebraic Combinatorics (Online)

A brief introduction to BIRS with important logistical information and technology instruction.

In this talk, we introduce Dynamical Algebraic Combinatorics by investigating ever more general domains in which the actions of promotion on tableaux (or tableaux-like objects) and rowmotion on order ideals (or generalizations of order ideals) correspond. These domains include: (1) promotion on $2\times n$ standard Young tableaux and rowmotion on order ideals of the Type A root poset, (2) K-promotion on rectangular increasing tableaux and rowmotion on order ideals of the product of three chains poset, (3) generalized promotion on increasing labelings of a finite poset and rowmotion on order ideals of a corresponding poset, and, finally, (4) promotion on new objects we call P-strict labelings (named in analogy to column-strict tableaux) and piecewise-linear rowmotion on P-partitions of a corresponding poset. This talk will be accessible to those with little DAC background and of interest to those working in the field. It includes joint works with J. Bernstein, K. Dilks, O. Pechenik, C. Vorland, and N. Williams.

Homomesy is a phenomenon in which a statistic on a set under an action has the same average value over any orbit under as its global average. Homomesy results have been discovered among many combinatorial objects, such as order ideals of posets and various tableaux. In this talk, I will give a brief introduction to homomesy and explore some of these results. The main emphasis will be Propp and Roby’s homomesy results on order ideals of a product of two chains poset under rowmotion and promotion, along with my own results on order ideals of a product of three chains.

We study promotion as a piecewise-linear operation on reverse plane partitions. We prove that this version of promotion is periodic by presenting representation-theoretic incarnations of reverse plane partitions and promotion. This is joint work with Rebecca Patrias and Hugh Thomas.

Consider a plane partition $P$ as an order ideal in the product $[a] \times [b] \times [c]$ of three chain posets. The combinatorial rowmotion operator sends $P$ to the plane partition generated by the minimal elements of its complement. What is the orbit structure of this action? I will attempt to survey the state of this question. In particular, I will describe my recent work with Becky Patrias, showing that rowmotion exhibits a strong form of resonance with frequency $a+b+c-1$, in the sense that each orbit size shares a prime divisor with $a+b+c-1$. This confirms a 1995 conjecture of Peter Cameron and Dmitri Fon-Der-Flaass.

In 2008, Petersen--Pylyavskyy--Rhoades proved that promotion on 2-row and 3-row rectangular standard Young tableaux can be realized as rotation of certain planar graphs called webs, which were introduced by Kuperberg. In this talk, we will introduce webs and their result. We will then generalize it to a larger family of webs---webs with both black and white boundary vertices. Lastly, we discuss on-going work to generalize further to the setting of K-theory combinatorics. This on-going work is joint with Oliver Pechenik, Jessica Striker, and Juliana Tymoczko.

Frieze patterns were studied by Conway and Coxeter in the 1970s. More recently, in 2015, Baur, Parsons, and Tschabold introduced infinite friezes and related them to the once-punctured disk and annulus. In this talk, we will explain the connection between periodic infinite friezes and cluster algebras of type D and affine A (modeled by once-punctured disks and annuli, respectively). We will discuss an invariant called growth coefficients which correspond to bracelets on the surface. These growth coefficients may or may not be homomesy-like.

Maps and actions on sets of combinatorial objects often have interesting extensions to the piecewise-linear realm of order and chain polytopes These can be further lifted to the birational realm via detropicalization/geometricization, and even to a setting with noncommuting variables. Surprisingly often, properties shown at the "combinatorial shadow" level, such as homomesy and low-order periodicity, lift all the way up to these higher realms.

Musiker and Roby used an explicit formula for iterated actions of the birational rowmotion map on a product of two chains, a type A minuscule poset, to gave the first proof of the birational analogue of file homomesy. In this talk, we extend the file homomesy result to birational rowmotion on arbitrary minuscule posets and give an almost uniform proof. Also we discuss a similar result for Coxeter-motion, which is a generalization of promotion on a product of two chains.

(1) Carlos Alejandro Alfaro - The sandpile groups of outerplanar graphs (joint work with Ralihe Raúl Villagrán) (2) Joseph Bernstein - P-strict promotion and piecewise-linear rowmotion (joint work with Jessica Striker and Corey Vorland) (3) Colin Defant - Promotion Sorting (joint work with Noah Kravitz) (4) Ben Drucker, Eli Garcia, and Rose Silver - RSK algorithm and the box-ball system (joint work with Aubrey Rumbolt) (5) Noah Kravitz - Friends and strangers walking on graphs (joint work with Noga Alon and Colin Defant) (6) Matthew Macauley - Abstract Algebra through Cayley diagrams, actions, and lattices (7) Rene Marczinzik - Distributive lattices and Auslander regular algebras (joint work with Osamu Iyama) (8) Jaeseong Oh - Cyclic sieving and orbit harmonics (joint work with Brendon Rhoades) (9) GaYee Park - Naruse Hook length formula for linear extensions of mobile posets (10) Matthew Plante - Periodicity and Homomesy for the $V × [n]$ poset and center-seeking snakes (11) Samu Potka - Refined Catalan and Narayana Cyclic Sieving (joint work with Per Alexandersson, Svante Linusson, and Joakim Uhlin) (12) James Propp - A Spectral Theory for Combinatorial Dynamics (13) Bruce Sagan - Fences, unimodality, and rowmotion (includes joint work with Thomas McConville and Clifford Smyth) (14) Hugh Thomas - Independence posets (joint work with Nathan Williams)

Let G be an acyclic directed graph. For each vertex of G, we define an involution on the independent sets of G. We call these involutions flips, and use them to define the independence poset for G--a new partial order on independent sets of G. Our independence posets are a generalization of distributive lattices, eliminating the lattice requirement: an independence poset that is a graded lattice is always a distributive lattice. Many well-known posets turn out to be special cases of our construction.

For a certain class of finite lattices called semidistributive, there exists a map k which gives a bijection between the set of join-irreducible elements and meet-irreducible elements. In this talk, we begin by connecting the map k and the Kreweras complement defined on noncrossing partitions. Our goal is to describe the map k in the context of torsion classes and the Kreweras complement in the context of wide subcategories. Experience with torsion classes and wide subcategories will not be assumed, and many examples will be given.

Let h to be the Coxeter number of a root system. We show that the Coxeter transformation of the incidence algebra coming from the order ideals in a cominuscule poset is periodic of order 'h+1' (up to a sign) in most cases using tools from representation theory of algebras. On the other hand, there is a combinatorial action, called the Rowmotion, defined on cominuscule posets. It is well-known that this action has order 'h' on the order ideals of a cominuscule poset. In this talk, we will demonstrate combinatorial similarities of the orbits of these two actions.

I will discuss a novel (partial) symmetry of Littlewood-Richardson coefficients conjectured by Pelletier and Ressayre (arXiv:2005.09877), and its proof (arXiv:2008.06128). The proof proceeds by constructing a birational involution and applying it to the tropical semifield, making for a particularly wieldly example of how (de)tropicalization can be used to prove combinatorial results.

The Lalanne–Kreweras involution (LK) on Dyck paths yields a bijective proof of the symmetry of two statistics: the number of valleys and the major index. Equivalently, this involution can be considered on the set of antichains of the type A root poset, on which rowmotion and LK together generate a dihedral action (as first discovered by Panyushev). Piecewise-linear and birational rowmotion were first defined by Einstein and Propp. Moving further in this direction, we define an analogue of the LK involution to the piecewise-linear and birational realms. In fact, LK is a special case of a more general action, rowvacuation, an involution that can be defined on any finite graded poset where it forms a dihedral action with rowmotion. We will explain that the symmetry properties of the number of valleys and the major index also lift to the higher realms. In this process, we have discovered more refined homomesies for LK, and we will explain how certain statistics which are homomesic under rowvacuation are also homomesic under rowmotion. This is joint work with Sam Hopkins.

I will present a conjectural way that ideas from Dynamical Algebraic Combinatorics could be used to resolve a fundamental problem in Coxeter-Catalan combinatorics: bijectively demonstrating the symmetry of the nonnesting W-Narayana numbers. This continues a project of Panyushev, whose interest in this problem led him to study rowmotion for root posets, and thus initiated a lot of the recent activity in DAC. I hope that others will become interested in this problem, and that we can "bring DAC full circle."

Musiker and Roby used an explicit formula for iterated actions of the birational rowmotion map on a product of two chains, a type A minuscule poset, to gave the first proof of the birational analogue of file homomesy. In this talk, we extend the file homomesy result to birational rowmotion on arbitrary minuscule posets and give an almost uniform proof. Also we discuss a similar result for Coxeter-motion, which is a generalization of promotion on a product of two chains. (The first viewing of this talk was not properly recorded.)