91 research outputs found
Orbits of Plane Partitions of Exceptional Lie Type
For each minuscule flag variety , there is a corresponding minuscule
poset, describing its Schubert decomposition. We study an action on plane
partitions over such posets, introduced by P. Cameron and D. Fon-der-Flaass
(1995). For plane partitions of height at most , D. Rush and X. Shi (2013)
proved an instance of the cyclic sieving phenomenon, completely describing the
orbit structure of this action. They noted their result does not extend to
greater heights in general; however, when is one of the two minuscule flag
varieties of exceptional Lie type , they conjectured explicit instances of
cyclic sieving for all heights.
We prove their conjecture in the case that is the Cayley-Moufang plane of
type . For the other exceptional minuscule flag variety, the Freudenthal
variety of type , we establish their conjecture for heights at most ,
but show that it fails generally. We further give a new proof of an unpublished
cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height
in the case is an even-dimensional quadric hypersurface. Our argument uses
ideas of K. Dilks, O. Pechenik, and J. Striker (2017) to relate the action on
plane partitions to combinatorics derived from -theoretic Schubert calculus.Comment: 25 pages, 7 figures, 3 tables. Section 5 rewritten and simplifie
Resonance in orbits of plane partitions
International audienceWe introduce a new concept of resonance on discrete dynamical systems. Our main result is an equivariant bijection between plane partitions in a box under rowmotion and increasing tableaux under K-promotion, using a generalization of the equivariance of promotion and rowmotion [J. Striker–N. Williams '12] to higher dimensional lattices. This theorem implies new results for K-promotion and new proofs of previous results on plane partitions
Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass
Let be a graded poset of rank and let be a -element
chain. For an order ideal of , its rowmotion
is the smallest ideal containing the minimal elements of the complementary
filter of . The map defines invertible dynamics on the set of ideals.
We say that that has NRP ("not relatively prime") rowmotion if no
-orbit has cardinality relatively prime to .
In work with R. Patrias (2020), we proved a 1995 conjecture of P. Cameron and
D. Fon-Der-Flaass by establishing NRP rowmotion for the product of two chains, the poset whose order ideals correspond to
the Schubert varieties of a Grassmann variety
under containment. Here, we initiate the general study of posets with NRP
rowmotion.
Our first main result establishes NRP rowmotion for all minuscule posets ,
posets whose order ideals reflect the Schubert stratification of minuscule flag
varieties. Our second main result is that NRP promotion depends only on the
isomorphism class of the comparability graph of .Comment: 15 pages, 5 figure
- …