Orbits of Plane Partitions of Exceptional Lie Type

TL;DR

This paper proves conjectures related to cyclic sieving phenomena for plane partitions over minuscule posets of exceptional Lie types, specifically confirming the case for type E6 and partial results for E7, using K-theoretic Schubert calculus.

Contribution

It confirms the cyclic sieving conjecture for the Cayley-Moufang plane of type E6 and provides partial results for the Freudenthal variety of type E7, introducing new combinatorial proofs.

Findings

Proved cyclic sieving for E6 minuscule flag variety.

Established partial cyclic sieving results for E7 for heights up to 4.

Provided a new proof for cyclic sieving in even-dimensional quadrics.

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 We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case $X$ is an even-dimensional quadric hypersurface.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.