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Iterative properties of birational rowmotion
arXiv (Cornell University)
25 Feb 2014
Abstract
Electronic Journal of Combinatorics, Volume 23, Issue 1 (2016),
Paper #P1.33 (part 1, abridged) and Electronic Journal of Combinatorics,
Volume 22, Issue 3 (2015), Paper #P3.40 (part 2, abridged) 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 prove that
birational rowmotion 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 furthermore compute
its orders on some triangle-shaped posets and on a class of posets which we
call "skeletal" (this class includes all graded forests). In all cases
mentioned, birational rowmotion turns out to have a 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. We
also make a digression to study classical rowmotion on skeletal posets, since
this case has seemingly been overlooked so far.
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Details
- Title
- Iterative properties of birational rowmotion
- Creators
- Darij GrinbergTom Roby
- Publication Details
- arXiv (Cornell University)
- Resource Type
- Preprint
- Language
- English
- Academic Unit
- Mathematics
- Other Identifier
- 991021862367604721