scholarly journals The order of birational rowmotion

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Darij Grinberg ◽  
Tom Roby
Keyword(s):  
Recent Work ◽  
Finite Order ◽  
Piecewise Linear ◽  
Finite Poset ◽  
Natural Operation ◽  
International Audience ◽  
Order Ideals ◽  
Linear Setting

International audience Various authors have studied a natural operation (under various names) on the order ideals (equivalently antichains) of a finite poset, here called \emphrowmotion. For certain posets of interest, the order of this map is much smaller than one would naively expect, and the orbits exhibit unexpected properties. In very recent work (inspired by discussions with Berenstein) Einstein and Propp describe how rowmotion can be generalized: first to the piecewise-linear setting of order polytopes, then via detropicalization to the birational setting. In the latter setting, it is no longer \empha priori clear even that birational rowmotion has finite order, and for many posets the order is infinite. However, we are able to show that birational rowmotion has the same order, p+q, for the poset P=[p]×[q] (product of two chains), as ordinary rowmotion. We also show that birational (hence ordinary) rowmotion has finite order for some other classes of posets, e.g., the upper, lower, right and left halves of the poset above, and trees having all leaves on the same level. Our methods are based on those used by Volkov to resolve the type AA (rectangular) Zamolodchikov Periodicity Conjecture.

scholarly journals Antichain Toggling and Rowmotion

10.37236/7454 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Michael Joseph
Keyword(s):  
Piecewise Linear ◽  
Algebraic Combinatorics ◽  
Finite Poset ◽  
Order Ideals ◽  
Almost All ◽  
The Relationship ◽  
Linear Setting ◽  
Families Of Subsets

In this paper, we analyze the toggle group on the set of antichains of a poset. Toggle groups, generated by simple involutions, were first introduced by Cameron and Fon-Der-Flaass for order ideals of posets. Recently Striker has motivated the study of toggle groups on general families of subsets, including antichains. This paper expands on this work by examining the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups (for a finite poset). We also focus on the rowmotion action on antichains of a poset that has been well-studied in dynamical algebraic combinatorics, describing it as the composition of antichain toggles. We also describe a piecewise-linear analogue of toggling to the Stanley’s chain polytope. We examine the connections with the piecewise-linear toggling Einstein and Propp introduced for order polytopes and prove that almost all of our results for antichain toggles extend to the piecewise-linear setting.

scholarly journals Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets

10.37236/4334 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Darij Grinberg ◽  
Tom Roby
Keyword(s):  
Finite Order ◽  
Disjoint Union ◽  
Parallel Analysis ◽  
Birational Map ◽  
Finite Poset ◽  
Order Ideals ◽  
Set Up ◽  
Grafting Onto

We study a birational map associated to any finite poset $P$. This map is a far-reaching generalization (found by Einstein and Propp) of classical rowmotion, which is a certain permutation of the set of order ideals of $P$. Classical rowmotion has been studied by various authors (Fon-der-Flaass, Cameron, Brouwer, Schrijver, Striker, Williams and many more) under different guises (Striker-Williams promotion and Panyushev complementation are two examples of maps equivalent to it). In contrast, birational rowmotion is new and has yet to reveal several of its mysteries. In this paper, we set up the tools for analyzing the properties of iterates of this map, and prove that it has finite order for a certain class of posets which we call "skeletal". Roughly speaking, these are graded posets constructed from one-element posets by repeated disjoint union and "grafting onto an antichain"; in particular, any forest having its leaves all on the same rank is such a poset. We also make a parallel analysis of classical rowmotion on this kind of posets, and prove that the order in this case equals the order of birational rowmotion.

scholarly journals Piecewise-linear and birational toggling

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
David Einstein ◽  
James Propp
Keyword(s):  
Symmetry Property ◽  
Piecewise Linear ◽  
Young Tableaux ◽  
Fonction Transfert ◽  
International Audience ◽  
Reciprocal Symmetry ◽  
Transfer Map ◽  
Order Ideals

International audience We define piecewise-linear and birational analogues of toggle-involutions, rowmotion, and promotion on order ideals of a poset $P$ as studied by Striker and Williams. Piecewise-linear rowmotion relates to Stanley's transfer map for order polytopes; piecewise-linear promotion relates to Schützenberger promotion for semistandard Young tableaux. When $P = [a] \times [b]$, a reciprocal symmetry property recently proved by Grinberg and Roby implies that birational rowmotion (and consequently piecewise-linear rowmotion) is of order $a+b$. We prove some homomesy results, showing that for certain functions $f$, the average of $f$ over each rowmotion/promotion orbit is independent of the orbit chosen. Nous définissons et étudions certains analogues linéaires-par-morceaux et birationnels d’involutions toggles, rowmotion et promotion sur les idéaux d’un poset $P$, comme étudié par Striker et Williams. La rowmotion linéaire-par-morceaux est liée à la fonction transfert de Stanley pour les polytopes d’ordre; la promotion linéaire-par-morceaux se rapporte à la promotion de Schützenberger pour les tableaux semi-standards de Young. Lorsque $P = [a] \times [b]$, une propriété de symétrie réciproque récemment prouvée par Grinberg et Roby implique que la rowmotion birationnelle (et par conséquent la rowmotion linéaire-par-morceaux) est de l’ordre $a+b$. Nous démontrons quelques résultats d’homomésie, montrant que pour certaines fonctions $f$, la moyenne de $f$ sur chaque orbite de rowmotion/promotion est indépendante de l’orbite choisie.

scholarly journals Promotion and Rowmotion

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Jessica Striker ◽  
Nathan Williams
Keyword(s):  
Recent Work ◽  
Linear Extensions ◽  
Alternating Sign Matrices ◽  
Plane Partitions ◽  
International Audience ◽  
Order Ideals

International audience We present an equivariant bijection between two actions—promotion and rowmotion—on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and certain cases of recent work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation. We extend the same idea to give an equivariant bijection between alternating sign matrices under rowmotion and under B. Wieland's gyration. Lastly, we define two actions with related orders on alternating sign matrices and totally symmetric self-complementary plane partitions. Nous prèsentons une bijection èquivariante entre deux actions—promotion et rowmotion—sur les idèaux d'ordre dans certaines posets. Cette bijection gènèralise simultanèment un rèsultat de R. Stanley concernant la promotion sur les extensions linèaire de deux cha\^ınes disjointes et certains cas des travaux rècents de D. Armstrong, C. Stump, et H. Thomas sur les partitions noncroisèes et nonembo\^ıtèes. Nous appliquons cette bijection à plusieurs classes de posets pour obtenir des bijections èquivariantes a des diffèrents objets connus sous la rotation. Nous gènèralisons la même idèe pour donnè une bijection èquivariante entre les matrices à signes alternants sous rowmotion et sous la gyration de B. Wieland. Finalement, nous dèfinissons deux actions avec des ordres similaires sur les matrices à signes alternants et les partitions plane totalement symètriques et autocomplèmentaires.

scholarly journals Iterative Properties of Birational Rowmotion II: Rectangles and Triangles

10.37236/4335 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Darij Grinberg ◽  
Tom Roby
Keyword(s):  
Finite Order ◽  
P Element ◽  
Birational Map ◽  
Finite Poset ◽  
Finite Set ◽  
Element Chain ◽  
Order Ideals ◽  
Finite Posets

Birational rowmotion — a birational map associated to any finite poset $P$ — has been introduced by Einstein and Propp as a far-reaching generalization of the (well-studied) classical rowmotion map on the set of order ideals of $P$. Continuing our exploration of this birational rowmotion, we prove that it has order $p+q$ on the $\left(  p, q\right)  $-rectangle poset (i.e., on the product of a $p$-element chain with a $q$-element chain); we also compute its orders on some triangle-shaped posets. In all cases mentioned, it turns out to have finite (and explicitly computable) order, a property it does not exhibit for general finite posets (unlike classical rowmotion, which is a permutation of a finite set). Our proof in the case of the rectangle poset uses an idea introduced by Volkov (arXiv:hep-th/0606094) to prove the $AA$ case of the Zamolodchikov periodicity conjecture; in fact, the finite order of birational rowmotion on many posets can be considered an analogue to Zamolodchikov periodicity. We comment on suspected, but so far enigmatic, connections to the theory of root posets.

scholarly journals Promotion and Evacuation

10.37236/75 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
Richard P. Stanley
Keyword(s):  
Vector Space ◽  
Hecke Algebra ◽  
Linear Extensions ◽  
Linear Transformations ◽  
Basic Properties ◽  
Finite Poset ◽  
Order Ideals ◽  
Maximal Chains

Promotion and evacuation are bijections on the set of linear extensions of a finite poset first defined by Schützenberger. This paper surveys the basic properties of these two operations and discusses some generalizations. Linear extensions of a finite poset $P$ may be regarded as maximal chains in the lattice $J(P)$ of order ideals of $P$. The generalizations concern permutations of the maximal chains of a wider class of posets, or more generally bijective linear transformations on the vector space with basis consisting of the maximal chains of any poset. When the poset is the lattice of subspaces of ${\Bbb F}_q^n$, then the results can be stated in terms of the expansion of certain Hecke algebra products.

scholarly journals Boolean complexes and boolean numbers

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Bridget Eileen Tenner
Keyword(s):  
Coxeter Group ◽  
Bruhat Order ◽  
Graph Invariant ◽  
Coxeter System ◽  
Combinatorial Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Order Ideals ◽  
Simplicial Poset ◽  
The Ideal

International audience The Bruhat order gives a poset structure to any Coxeter group. The ideal of elements in this poset having boolean principal order ideals forms a simplicial poset. This simplicial poset defines the boolean complex for the group. In a Coxeter system of rank n, we show that the boolean complex is homotopy equivalent to a wedge of (n-1)-dimensional spheres. The number of these spheres is the boolean number, which can be computed inductively from the unlabeled Coxeter system, thus defining a graph invariant. For certain families of graphs, the boolean numbers have intriguing combinatorial properties. This work involves joint efforts with Claesson, Kitaev, and Ragnarsson. \par L'ordre de Bruhat munit tout groupe de Coxeter d'une structure de poset. L'idéal composé des éléments de ce poset engendrant des idéaux principaux ordonnés booléens, forme un poset simplicial. Ce poset simplicial définit le complexe booléen pour le groupe. Dans un système de Coxeter de rang n, nous montrons que le complexe booléen est homotopiquement équivalent à un bouquet de sphères de dimension (n-1). Le nombre de ces sphères est le nombre booléen, qui peut être calculé inductivement à partir du système de Coxeter non-étiquetté; définissant ainsi un invariant de graphe. Pour certaines familles de graphes, les nombres booléens satisfont des propriétés combinatoires intriguantes. Ce travail est une collaboration entre Claesson, Kitaev, et Ragnarsson.

scholarly journals Lebesgue measure of escaping sets of entire functions

2018 ◽  
Vol 40 (1) ◽  
pp. 89-116 ◽  
Author(s):  
WEIWEI CUI
Keyword(s):  
Entire Function ◽  
Recent Work ◽  
Lebesgue Measure ◽  
Finite Order ◽  
Infinite Order ◽  
Entire Functions ◽  
Transcendental Entire Function ◽  
Positive Lebesgue Measure

For a transcendental entire function $f$ of finite order in the Eremenko–Lyubich class ${\mathcal{B}}$, we give conditions under which the Lebesgue measure of the escaping set ${\mathcal{I}}(f)$ of $f$ is zero. This complements the recent work of Aspenberg and Bergweiler [Math. Ann. 352(1) (2012), 27–54], in which they give conditions on entire functions in the same class with escaping sets of positive Lebesgue measure. We will construct an entire function in the Eremenko–Lyubich class to show that the condition given by Aspenberg and Bergweiler is essentially sharp. Furthermore, we adapt our idea of proof to certain infinite-order entire functions. Under some restrictions to the growth of these entire functions, we show that the escaping sets have zero Lebesgue measure. This generalizes a result of Eremenko and Lyubich.

scholarly journals The Many Faces of Alternating-Sign Matrices

2001 ◽  
Vol DMTCS Proceedings vol. AA,... (Proceedings) ◽  
Author(s):  
James Propp
Keyword(s):  
Height Function ◽  
Original Form ◽  
Alternating Sign Matrices ◽  
International Audience ◽  
Order Ideals ◽  
The Many

International audience I give a survey of different combinatorial forms of alternating-sign matrices, starting with the original form introduced by Mills, Robbins and Rumsey as well as corner-sum matrices, height-function matrices, three-colorings, monotone triangles, tetrahedral order ideals, square ice, gasket-and-basket tilings and full packings of loops.

scholarly journals $Star^1$-convex functions on tropical linear spaces of complete graphs

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Complete Graph ◽  
Piecewise Linear ◽  
Negative Answer ◽  
Linear Functions ◽  
One Dimension ◽  
Complete Graphs ◽  
Piecewise Linear Functions ◽  
Linear Spaces ◽  
International Audience ◽  
Tropical Linear Space

International audience Given a fan $\Delta$ and a cone $\sigma \in \Delta$ let $star^1(\sigma )$ be the set of cones that contain $\sigma$ and are one dimension bigger than $\sigma$ . In this paper we study two cones of piecewise linear functions defined on $\delta$ : the cone of functions which are convex on $star^1(σ\sigma)$ for all cones, and the cone of functions which are convex on $star^1(σ\sigma)$ for all cones of codimension 1. We give nice combinatorial descriptions for these two cones given two different fan structures on the tropical linear space of complete graphs. For the complete graph $K_5$, we prove that with the finer fan subdivision the two cones are not equal, but with the coarser subdivision they are the same. This gives a negative answer to a question of Gibney-Maclagan that for the finer subdivision the two cones are the same. Soit $\Delta$ un fan, pour $\sigma \in \Delta$ nous définissons $star^1(\sigma )$ comme l'ensemble de cônes qui contiennent $\sigma$ dont la dimension est un de plus que la dimension de $\sigma$ . Nous étudions deux cônes d'applications linéaires par morceaux définis sur $\Delta$ : le cône de fonctions convexes sur$star^1(\sigma )$, où $\sigma \in \Delta$ est un cône quelconque, et le cône de fonctions convexes sur $star^1(σ\sigma)$ où σ est un cône de codimension 1. étant donnés deux structures sur l'espace tropical linéaire de graphes complets, nous donnons de beaux descriptions combinatoires des cônes décrits en haut. Pour le graphe complet $K_5$, on démontre que avec la subdivision en fans plus fine, les deux cônes sont différentes, mais avec la subdivision plus gros ils sont cônes sont les mêmes. Ce résultant réponde négativement une question de Gibney-Maclagan.

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