Iterative properties of birational rowmotion. II: Rectangles and triangles.

Darij Grinberg¹ ; Tom Roby²

¹Massachusetts Institute of Technology
²University of Connecticut

The electronic journal of combinatorics, Tome 22 (2015) no. 3

Résumé

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.

Détail

DOI : 10.37236/4335

Classification : 06A07, 05E99
Mots-clés : rowmotion, finite posets, order ideals, Zamolodchikov periodicity, root systems, promotion, graded posets, Grassmannians, tropicalization