Portal de Periódicos da CAPES

Orbits of plane partitions of exceptional Lie type

2018; Elsevier BV; Volume: 74; Linguagem: Inglês

10.1016/j.ejc.2018.07.007

1095-9971

Holly Mandel, Oliver Pechenik,

Advanced Mathematical Identities

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 Cameron and Fon-der-Flaass (1995). For plane partitions of height at most 2, Rush and 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 E6. For the other exceptional minuscule flag variety, the Freudenthal variety of type E7, 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 We further give a new proof of an unpublished cyclic sieving of Rush and Shi (2011) for plane partitions of any height in the case X is an even-dimensional quadric hypersurface. Our argument uses ideas of Dilks et al. (2017) to relate the action on plane partitions to combinatorics derived from K-theoretic Schubert calculus.