# 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…

## 10 Citations

Iterative Properties of Birational Rowmotion I: Generalities and Skeletal Posets

- MathematicsElectron. J. Comb.
- 2016

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

- MathematicsAlgebraic Combinatorics
- 2019

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

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2018

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

- MathematicsElectron. J. Comb.
- 2019

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

- Mathematics
- 2014

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

- Mathematics
- 2016

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

- Mathematics
- 2017

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

- Mathematics
- 2015

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

- Mathematics
- 2015

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

- Mathematics
- 2016

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…

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