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 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 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 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.

FINITE POSETS AND THEIR REPRESENTATION ALGEBRAS

2010 ◽  
Vol 20 (01) ◽  
pp. 27-38 ◽  
Author(s):  
MURRAY GERSTENHABER ◽  
MARY SCHAPS
Keyword(s):  
Valuation Ring ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Dimensional Algebra ◽  
Finite Poset ◽  
Finite Dimensional ◽  
Triangular Representation ◽  
Complete Discrete Valuation Ring ◽  
Characteristic 2 ◽  
Finite Posets

A finite poset (P ≼ S) determines a finite dimensional algebra TP over the field 𝔽 of two elements, with an upper triangular representation. We determine the structure of the radical of the representation algebra A of the monoid (TP,·) over a field of characteristic different from 2. We also consider degenerations of A over a complete discrete valuation ring with residue field of characteristic 2.

scholarly journals ON MAZUR'S CONJECTURE FOR TWISTED L-FUNCTIONS OF ELLIPTIC CURVES OVER TOTALLY REAL OR CM FIELDS

2010 ◽  
Vol 53 (1) ◽  
pp. 207-210
Author(s):  
CRISTIAN VIRDOL
Keyword(s):  
Functional Equation ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Finite Order ◽  
Number Field ◽  
Meromorphic Continuation ◽  
Minimal Order ◽  
Finite Set ◽  
Entire Complex ◽  
Totally Real

Let E be an elliptic curve defined over a number field F, and let Σ be a finite set of finite places of F. Let L(s, E, ψ) be the L-function of E twisted by a finite-order Hecke character ψ of F. It is conjectured that L(s, E, ψ) has a meromorphic continuation to the entire complex plane and satisfies a functional equation s ↔ 2 − s. Then one can define the so called minimal order of vanishing ats = 1 of L(s, E, ψ), denoted by m(E, ψ) (see Section 2 for the definition).

scholarly journals Birational Rowmotion and Coxeter-Motion on Minuscule Posets

10.37236/9557 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Soichi Okada
Keyword(s):  
Dynamical System ◽  
Discrete Dynamical System ◽  
Order Ideal ◽  
Positive Real ◽  
Finite Poset ◽  
Coxeter Number ◽  
Order Ideals ◽  
Minuscule Poset

Birational rowmotion is a discrete dynamical system on the set of all positive real-valued functions on a finite poset, which is a birational lift of combinatorial rowmotion on order ideals. It is known that combinatorial rowmotion for a minuscule poset has order equal to the Coxeter number, and exhibits the file homomesy phenomenon for refined order ideal cardinality statistics. In this paper we generalize these results to the birational setting. Moreover, as a generalization of birational promotion on a product of two chains, we introduce birational Coxeter-motion on minuscule posets, and prove that it enjoys periodicity and file homomesy.

scholarly journals Computing all border bases for ideals of points

2019 ◽  
Vol 18 (06) ◽  
pp. 1950102 ◽  
Author(s):  
Amir Hashemi ◽  
Martin Kreuzer ◽  
Samira Pourkhajouei
Keyword(s):  
Las Vegas ◽  
Error Correcting Codes ◽  
Multivariate Polynomials ◽  
Vanishing Ideal ◽  
Algebraic Algorithms ◽  
Finite Set ◽  
Order Ideals ◽  
Set Of Points ◽  
Term Ordering ◽  
Vanishing Ideals

In this paper, we consider the problem of computing all possible order ideals and also sets connected to 1, and the corresponding border bases, for the vanishing ideal of a given finite set of points. In this context, two different approaches are discussed: based on the Buchberger–Möller Algorithm [H. M. Möller and B. Buchberger, The construction of multivariate polynomials with preassigned zeros, EUROCAM ’82 Conf., Computer Algebra, Marseille/France 1982, Lect. Notes Comput. Sci. 144, (1982), pp. 24–31], we first propose a new algorithm to compute all possible order ideals and the corresponding border bases for an ideal of points. The second approach involves adapting the Farr–Gao Algorithm [J. B. Farr and S. Gao, Computing Gröbner bases for vanishing ideals of finite sets of points, in 16th Int. Symp. Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC-16, Las Vegas, NV, USA (Springer, Berlin, 2006), pp. 118–127] for finding all sets connected to 1, as well as the corresponding border bases, for an ideal of points. It should be noted that our algorithms are term ordering free. Therefore, they can compute successfully all border bases for an ideal of points. Both proposed algorithms have been implemented and their efficiency is discussed via a set of benchmarks.

scholarly journals Formalization of some central theorems in combinatorics of finite sets

10.29007/r7fg ◽  
2018 ◽  
Author(s):  
Abhishek Kr Singh
Keyword(s):  
Ordered Set ◽  
Decomposition Theorem ◽  
Proof Assistant ◽  
Combinatorial Objects ◽  
Finite Poset ◽  
The Coq Proof Assistant ◽  
Coq Proof Assistant ◽  
Finite Partially Ordered Set ◽  
Dilworth’S Theorem ◽  
Finite Posets

We present fully formalized proofs of some central theorems from combinatorics. These are Dilworth's decomposition theorem, Mirsky's theorem, Hall's marriage theorem and the Erdős-Szekeres theorem. Dilworth's decomposition theorem is the key result among these. It states that in any finite partially ordered set (poset), the size of a smallest chain cover and a largest antichain are the same. Mirsky's theorem is a dual of Dilworth's decomposition theorem, which states that in any finite poset, the size of a smallest antichain cover and a largest chain are the same. We use Dilworth's theorem in the proofs of Hall's Marriage theorem and the Erdős-Szekeres theorem. The combinatorial objects involved in these theorems are sets and sequences. All the proofs are formalized in the Coq proof assistant. We develop a library of definitions and facts that can be used as a framework for formalizing other theorems on finite posets.

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 The radial oscillation of solutions to ode's in the complex domain

1996 ◽  
Vol 39 (3) ◽  
pp. 473-483 ◽  
Author(s):  
John Rossi ◽  
Shupei Wang
Keyword(s):  
Entire Function ◽  
Finite Order ◽  
Upper Bound ◽  
Complex Domain ◽  
Radial Oscillation ◽  
Real Zeros ◽  
The Third ◽  
Finite Set ◽  
Oscillation Of Solutions

We prove three results concerning the oscillation near a ray of solutions to (*)w″ + Aw = 0, where A is an entire function. The first result assumes that A is a polynomial and gives an upper bound on the number of its real zeros if (*) admits a solution with only real zeros and infinitely many. The second result proves that for A of finite order a solution w to (*) has “few” zeros “near” a ray if and only if the same is true for w′. The third result involves the density of the zeros of a solution to (*) “away” from a finite set of rays.

Sign in / Sign up

Export Citation Format

Share Document