Iterative properties of birational rowmotion. II: Rectangles and triangles. (English) Zbl 1339.06001

Grinberg, Darij; Roby, Tom

Electron. J. Comb. 22, No. 3, Research Paper P3.40, 49 p. (2015).

Summary: 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 [for part I see ibid. 23, No. 1, Research Paper P1.33 (2016; Zbl 1338.06003)], we prove that it has order \(p+q\) on the \((p, q)\)-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 A. Yu. Volkov [Commun. Math. Phys. 276, No. 2, 509-517 (2007; Zbl 1136.82011)] 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.

MSC:

06A07	Combinatorics of partially ordered sets
05E99	Algebraic combinatorics

Keywords:

rowmotion; finite posets; order ideals; Zamolodchikov periodicity; root systems; promotion; graded posets; Grassmannians; tropicalization