R-systems

Galashin, Pavel; Pylyavskyy, Pavlo

doi:10.1007/s00029-019-0470-2

Cited by 10 publications

(8 citation statements)

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“…For the rectangle, birational rowmotion enjoys remarkable integrability properties related to cluster algebra Y-system dynamics (a.k.a., Zamolodchikov periodicity) [92]. Birational rowmotion was also the principal example motivating the recent definition of R-systems due to Galashin and Pylyavskyy [26].…”

Section: · · ·mentioning

confidence: 99%

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Hopkins

2020

Experimental Mathematics

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We relate Reiner, Tenner, and Yong's coincidental down-degree expectations (CDE) property of posets to the minuscule doppelgänger pairs studied by Hamaker, Patrias, Pechenik, and Williams. Via this relation, we put forward a series of conjectures which suggest that the minuscule doppelgänger pairs behave "as if" they had isomorphic comparability graphs, even though they do not. We further explore the idea of minuscule doppelgänger pairs pretending to have isomorphic comparability graphs by considering the rowmotion operator on order ideals. We conjecture that the members of a minuscule doppelgänger pair behave the same way under rowmotion, as they would if they had isomorphic comparability graphs. Moreover, we conjecture that these pairs continue to behave the same way under the piecewise-linear and birational liftings of rowmotion introduced by Einstein and Propp. This conjecture motivates us to study the homomesies (in the sense of Propp and Roby) exhibited by birational rowmotion. We establish the birational analog of the antichain cardinality homomesy for the major examples of posets known or conjectured to have finite birational rowmotion order (namely: minuscule posets and root posets of coincidental type).

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Section: · · ·mentioning

confidence: 99%

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Hopkins

2020

Experimental Mathematics

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“…Birational rowmotion was studied by Grinberg and Roby ; Glick later showed that one of the Grinberg–Roby results was equivalent to an instance of Zamolodchikov periodicity. Most recently, Galashin and Pylyavskyy generalized the notion of birational rowmotion to strongly connected directed graphs by introducing

R

‐systems .…”

Section: History and Examplesmentioning

confidence: 99%

Rowmotion in slow motion

Thomas

Williams

2019

Proc. London Math. Soc.

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Rowmotion is a simple cyclic action on the distributive lattice of order ideals of a poset: it sends the order ideal x to the order ideal generated by the minimal elements not in x. It can also be computed in ‘slow motion’ as a sequence of local moves. We use the setting of trim lattices to generalize both definitions of rowmotion, proving many structural results along the way. We introduce a flag simplicial complex (similar to the canonical join complex of a semidistributive lattice) and relate our results to recent work of Barnard by proving that extremal semidistributive lattices are trim. As a corollary, we prove that if A is a representation finite algebra and modA has no cycles, then the torsion classes of A ordered by inclusion form a trim lattice.

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“…Rowmotion is an action originally defined on hypergraphs by P. Duchet [13] and generalized to order ideals of an arbitrary finite poset by A. Brouwer and A. Schrijver [7]. Given I ∈ J(P ), rowmotion on I, denoted Row(I), is the order ideal generated by the minimal elements of P \ I. Rowmotion has recently generated significant interest as a prototypical action in the emerging subfield of dynamical algebraic combinatorics; see [38] for a detailed history and [3,6,10,11,12,14,15,17,18,19,21,22,26,27,28,29,35,37,36,41] for more recent developments.…”

Section: 22mentioning

confidence: 99%

Rowmotion and increasing labeling promotion

Dilks¹,

Striker²,

Vorland³

2019

Journal of Combinatorial Theory, Series A

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In 2012, N. Williams and the second author showed that on order ideals of ranked partially ordered sets (posets), rowmotion is conjugate to (and thus has the same orbit structure as) a different toggle group action, which in special cases is equivalent to promotion on linear extensions of posets constructed from two chains. In 2015, O. Pechenik and the first and second authors extended these results to show that increasing tableaux under K-promotion naturally corresponds to order ideals in a product of three chains under a toggle group action conjugate to rowmotion they called hyperplane promotion.In this paper, we generalize these results to the setting of arbitrary increasing labelings of any finite poset with given restrictions on the labels. We define a generalization of K-promotion in this setting and show it corresponds to a toggle group action we call toggle-promotion on order ideals of an associated poset. When the restrictions on labels are particularly nice (for example, specifying a global bound on all labels used), we show that toggle-promotion is conjugate to rowmotion. Additionally, we show that any poset that can be nicely embedded into a Cartesian product has a natural toggle-promotion action conjugate to rowmotion. This paper builds upon work of N. Williams and the second author [38] regarding promotion and rowmotion on posets with a two-dimensional lattice projection and subsequent work of O. Pechenik with the first and second authors [12] which n-dimensionalized this result, relating K-promotion on increasing tableaux to rowmotion on the product of three chains poset.We describe our main results in Subsection 1.1 and then give a brief history of promotion and rowmotion in Subsection 1.2, with a focus on the motivating results from these two papers.1.1. Main Results. Throughout this paper, let P be a finite partially ordered set (poset).Definition 1.1. We say that a function f : P → Z is an increasing labeling if p 1 < p 2 in P implies that f (p 1 ) < f (p 2 ) (with the usual total ordering on the integers). We will be interested in sets of increasing labelings on P given a restriction function R : P → P(Z) indicating which labels each poset element is allowed to attain (where P(Z) is the power set of Z). We require R(p) to be nonempty and finite for each p ∈ P . Call the set of such increasing labelings Inc R (P ).See Figures 1 and 2 for examples.Remark 1.2. Up to conventions of increasing versus decreasing, increasing labelings can also be thought of as strict P -partitions with restricted parts. We use the terminology of increasing labelings rather than P -partitions since we are generalizing from increasing tableaux rather than from integer partitions. See Remarks 1.4, 2.30, and 2.31 for more on the connection to P -partitions.We consider a natural partial order on Inc R (P ), where f ≤ g if and only if f (p) ≤ g(p) for all p ∈ P . This is a distributive lattice, so we may apply Birkhoff's Representation Theorem to obtain a representation in terms of order ideals in a poset. A subset I of P ...

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R-systems

Cited by 10 publications

References 56 publications

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

Rowmotion in slow motion

Rowmotion and increasing labeling promotion

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