• Corpus ID: 119302820

Iterative properties of birational rowmotion

@article{Grinberg2014IterativePO,
  title={Iterative properties of birational rowmotion},
  author={Darij Grinberg and Tom Roby},
  journal={arXiv: Combinatorics},
  year={2014}
}
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… 
Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets
TLDR
It is proved that a birational map associated to any finite poset $P has finite order for a certain class of posets which the authors call "skeletal", and a parallel analysis of classical rowmotion on this kind of poset is made, proving that the order equals the order of birational rowmotion.
Paths to Understanding Birational Rowmotion on Products of Two Chains
Birational rowmotion is an action on the space of assignments of rational functions to the elements of a finite partially-ordered set (poset). It is lifted from the well-studied rowmotion map on
Rowmotion and generalized toggle groups
TLDR
The notion of the toggle group is generalized from the set of order ideals of a poset to any family of subsets of a finite set, and structure theorems for certain finite generalized toggle groups are proved, similar to the theorem of Cameron and Fon-der-Flaass in the case ofOrder ideals.
Antichain Toggling and Rowmotion
TLDR
This paper examines the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups (for a finite poset) and describes a piecewise-linear analogue of toggling to the Stanley’s chain polytope.
Piecewise-linear and birational toggling
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
Studies on quasisymmetric functions
In 1983, Ira Gessel introduced the ring of quasisymmetric functions (QSym), an extension of the ring of symmetric functions and nowadays one of the standard examples of a combinatorial Hopf algebra.
Toggling Involutions and Homomesies for Maps on Finite Sets, Noncrossing Partitions, and Independent Sets of Path Graphs
This paper explores the orbit structure and homomesy properties of various actions on finite sets. The homomesy phenomenon, meaning constant averages over orbits, was proposed by Propp and Roby in
Zamolodchikov integrability via rings of invariants
Zamolodchikov periodicity is periodicity of certein recursions associated with box products $X \square Y$ of two finite type Dynkin diagrams. We suggest an affine analog of Zamolodchikov periodicity,
NOTES FROM THE AIM WORKSHOP ON DYNAMICAL ALGEBRAIC COMBINATORICS
These are notes from the “Dynamical algebraic combinatorics” workshop held March 23rd–27th, 2015 at the American Institute of Mathematics in San Jose, California. The organizers were James Propp, Tom
Affine type A geometric crystal structure on the Grassmannian
We construct a type A(1) n−1 affine geometric crystal structure on the Grassmannian Gr(k, n). The tropicalization of this structure recovers the combinatorics of crystal operators on semistandard

References

SHOWING 1-10 OF 57 REFERENCES
The order of birational rowmotion
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
Combinatorial, piecewise-linear, and birational homomesy for products of two chains
The purpose of this article is to illustrate the dynamical concept of {\em homomesy} in three kinds of dynamical systems -- combinatorial, piecewise-linear, and birational -- and to show the
Piecewise-linear and birational toggling
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
A uniform bijection between nonnesting and noncrossing partitions
In 2007, D.I. Panyushev defined a remarkable map on the set of nonnesting partitions (antichains in the root poset of a finite Weyl group). In this paper we use Panyushev's map, together with the
Promotion and rowmotion
On orbits of order ideals of minuscule posets
An action on order ideals of posets considered by Fon-Der-Flaass is analyzed in the case of posets arising from minuscule representations of complex simple Lie algebras. For these minuscule posets,
Combinatorial Markov chains on linear extensions
TLDR
This work considers generalizations of Schützenberger’s promotion operator on the set $\mathcal{L}$ of linear extensions of a finite poset of size n and provides explicit eigenvalues of the transition matrix in general when the poset is a rooted forest.
Geometric RSK correspondence, Whittaker functions and symmetrized random polymers
We show that the geometric lifting of the RSK correspondence introduced by A.N. Kirillov (Physics and Combinatorics. Proc. Nagoya 2000 2nd Internat Workshop, pp. 82–150, 2001) is volume preserving
GROUPS GENERATED BY INVOLUTIONS GELFAND TSETLIN PATTERNS AND COMBINATORICS OF YOUNG TABLEAUX
We construct families of piecewise linear representations cpl representations of the symmetric group Sn and the a ne Weyl group e Sn of type A n acting on the space of triangles Xn We nd a nontrivial
...
1
2
3
4
5
...