Orbits of plane partitions of exceptional Lie type

Abstract

For each minuscule flag variety X, 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 2, 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 X is one of the two minuscule flag varieties of exceptional Lie type E, they conjectured explicit instances of cyclic sieving for all heights.We prove their conjecture in the case that X is the Cayley-Moufang plane of type E 6 . For the other exceptional minuscule flag variety, the Freudenthal variety of type E 7 , we establish their conjecture for heights at most 4, 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 We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case X is an even-dimensional quadric hypersurface.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 K-theoretic Schubert calculus.

Minuscule Doppelgängers, The Coincidental down-Degree Expectations Property, and Rowmotion

“…Observe that the size of this orbit is 4, and that the average of ddeg along this orbit is ) for a minuscule poset P was studied by Rush and Shi [70] and later by Mandel and Pechenik [48] in the context of cyclic sieving, but the results they obtained were somewhat sporadic. We believe rowmotion acting on PP ℓ (P ) behaves better than rowmotion acting on J(P × [ℓ]): for example, below we conjecture a uniform cyclic sieving result for rowmotion acting on PP ℓ (P ) when P is a minuscule poset.…”

“…For example, for the rectangle poset P = [a] × [b], the order of row acting on PP ℓ (P ) is a + b for all ℓ ≥ 1 (see Theorem 4.20 below); while Cameron and Fon-Der-Flaass [10] showed that row acting on J(P × [2]) has order a + b + 1. Order ideal rowmotion acting on J(P × [ℓ]) for a minuscule poset P was studied by Rush and Shi [70] and later by Mandel and Pechenik [48] in the context of cyclic sieving, but the results they obtained were somewhat sporadic. We believe rowmotion acting on PP ℓ (P ) behaves better than rowmotion acting on J(P × [ℓ]): for example, below we conjecture a uniform cyclic sieving result for rowmotion acting on PP ℓ (P ) when P is a minuscule poset.…”

Minuscule analogues of the plane partition periodicity conjecture of Cameron and Fon-Der-Flaass

“…In this section, we recall the definition and classification of minuscule posets. For additional background, see, e.g., [Pro84,Ste94,TY09a,RS13,MP18,HPPW20,Oka21].…”

“…Unfortunately, our proof of Theorem 1.2 is not type-uniform, but relies on the classification of minuscule posets, with corresponding case-by-case analysis. In the exceptional types, it relies on the author's explicit computer calculations reported in [MP18]. It would be very interesting to have a uniform proof of Theorem 1.2.…”

The genomic Schur function is fundamental-positive

“…Increasing tableaux were perhaps first studied in their own right in [TY09], although they appeared earlier in various contexts (e.g., [EG87,JPS98]). As in [MP18], we say an increasing tableau T is gapless if the set of numbers appearing in T is an initial segment of Z ą0 . We write Inc gl pλq for the set of all gapless increasing tableaux of shape λ.…”