Orbits of Plane Partitions of Exceptional Lie Type
Holly Mandel Oliver Pechenik
math.CO 05A15, 05E18, 06A07, 17B25










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theoremTheorem[section]
lemma[theorem]Lemma
proposition[theorem]Proposition
corollary[theorem]Corollary
conjecture[theorem]Conjecture

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definition[theorem]Definition
examplex[theorem]Example

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Orbits of plane partitions]Orbits of plane partitions of exceptional Lie type  


H. Mandel]Holly Mandel
[HM]Department of Mathematics, University of California, Berkeley, Berkeley, CA 94720
holly.mandel@berkeley.edu


O. Pechenik]Oliver Pechenik
[OP]Department of Mathematics, University of Michigan, Ann Arbor, MI 48109
pechenik@umich.edu



[2010]Primary 05A15; Secondary 05E18, 06A07, 17B25









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 We further give a new proof of an unpublished cyclic sieving of D. Rush and X. Shi (2011) for plane partitions of any height in the case X is an even-dimensional quadric hypersurface.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


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    December 30, 2023
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§ INTRODUCTION





The minuscule posets are a remarkable collection of partially-ordered sets that arise naturally from the representation theory of Lie algebras or alternatively from the Schubert calculus of generalized flag varieties. We study dynamical enumerative properties of plane partitions over these posets.

A special case is the set of ordinary plane partitions that fit inside a fixed rectangular box. In this context, P. Cameron and D. Fon-der-Flaass <cit.> initiated the study of a combinatorially-natural operator Ψ. This operator is now generally known as rowmotion and has become a subject of intense study (cf., e.g., <cit.>).
We will describe minuscule posets and the operation of rowmotion in Sections <ref> and <ref>, respectively.

For any poset P, let ^k(P) denote the set of plane partitions of height at most k over P, or equivalently, the set of order ideals in the product P × k of P with a chain poset of k elements. Let f_P^k denote the generating function that enumerates the elements of ^k(P)  by cardinality, so 
f_P^k(q) ∑_ℐ∈^k(P) q^|ℐ|. In the special case k ≤ 2 and P minuscule, D. Rush and X. Shi <cit.> showed that f_P^k also encodes the orbit structure of rowmotion via an instance of the cyclic sieving phenomenon (introduced by V. Reiner, D. Stanton, and D. White <cit.>). Thus for k ≤ 2, the number of minuscule plane partitions fixed by the d-fold application of rowmotion Ψ^∘ d is the evaluation of the polynomial f_P^k at ζ^d, where ζ is any primitive nth root of unity and n is the period of Ψ on ^k(P). (Since some of our other operators have superscripts in their names, we denote the d-fold composition of an operator τ by τ^∘ d.)

It was noted in <cit.> that this instance of cyclic sieving does not extend to the case k≥ 3 for general minuscule posets.
However, D. Rush and X. Shi conjectured the following. (The posets in question are illustrated in Figure <ref>.)

Let P be one of the two minuscule posets associated to an exceptional Lie algebra of type E and let k ∈ℤ_≥ 0. Then f_P^k is a cyclic sieving polynomial for the action of Ψ on ^k(P).




Our main result is to completely resolve Conjecture <ref>. 

Conjecture <ref> holds for the E_6 minuscule poset P_CM (cf. Figure <ref>A) and all k, but holds for the E_7 minuscule poset P_F (cf. Figure <ref>B) only when k ≤ 4. 
  
Verification of Conjecture <ref> in the the cases k ≤ 4 for P_CM and k ≤ 3 for P_F was previously reported in <cit.>. Our new results are therefore:


  * cyclic sieving for P_CM when k > 4,

  * cyclic sieving for P_F when k = 4, and

  * failure of the conjectured cyclic sieving for P_F when k > 4.


Our approach to proving Theorem <ref> is to use the ideas of K. Dilks, O. Pechenik, J. Striker and C. Vorland <cit.> to relate the action of Ψ to the action of K-promotion on increasing tableaux. K-promotion was first studied in <cit.>, building on combinatorial tools for K-theoretic Schubert calculus due to H. Thomas and A. Yong <cit.>. We show that the action of K-promotion is controlled by its behavior on a finite subset of increasing tableaux. By understanding the orbit structure of this subset, we are able to determine the complete orbit structure, thereby establishing Theorem <ref>.

Having developed these methods, it becomes straightforward to prove the following additional result.

Let P be a minuscule poset associated to an even-dimensional quadric of type D_p+1 (cf. Figure <ref>C) and let k ∈ℤ_≥ 0. Then f_P^k is a cyclic sieving polynomial for the action of Ψ on ^k(P).

Theorem <ref> was previously announced by D. Rush and X. Shi <cit.>; however, they omitted their proof <cit.> from the published paper. We believe that our alternative proof of Theorem <ref> via K-theoretic combinatorics provides different insight.


Often instances of cyclic sieving can be proven using representation-theoretic techniques <cit.>, and when cyclic sieving is established in a more direct fashion, as we do here, it may be a clue toward new underlying algebra (cf. <cit.>). The results in Theorems <ref> and <ref> perhaps suggest the existence of new symmetric group module structures on the sets ^k(P). Developing such representations would be an interesting direction for future work. In particular, we do not have a good understanding of why the posets P_CM and P_F behave so differently in Theorem <ref>, even though the associated algebra and geometry seems very similar; a representation-theoretic construction might shed light on this mystery. It is also possible that the difference between P_CM and P_F in Theorem <ref> could be explained via monodromy in real Schubert calculus by vastly extending the geometric constructions of <cit.>; from conversations with J. Levinson and K. Purbhoo, it seems that there are many obstacles, however, to developing the necessary geometry to give such a geometric explanation of the results here.


This paper is organized as follows. In Section <ref>, we define the minuscule posets, recalling their classification and other properties we will use. Section <ref> gives the precise definition of rowmotion and, following <cit.>, notes the close relation between rowmotion and K-promotion. We then develop new tools for understanding the orbit structure of K-promotion in Sections <ref>, <ref>, and <ref>. Specifically, Section <ref> introduces the operations of inflation and deflation, retracting the set of increasing tableaux onto the finite subset of gapless tableaux. 
In Section <ref>, we recall the precise definition of K-promotion on increasing tableaux and show how K-promotion is governed by its restriction to gapless tableaux. Section <ref> uses this information to determine the period of K-promotion on general increasing tableaux.
Finally, Section <ref> combines these ideas to prove Theorems <ref> and <ref>.



§ MINUSCULE POSETS


Let G be a complex connected reductive Lie group with maximal torus T. Denote by W the Weyl group N_ G( T)/ T. The root system Φ of G may be partitioned Φ^+ ⊔Φ^- into positive and negative roots according to a choice Δ of simple roots. There is a natural poset structure on Φ^+ obtained as the transitive closure of the covering relation α⋖β if and only if β - α∈Δ. The choice of bipartition of Φ into positive and negative roots further specifies a choice of a Borel subgroup B_+ ⊂ G and an opposite Borel subgroup B_- ⊂ G with B_+ ∩ B_- =  T.

We say δ∈Δ is minuscule if for every α∈Φ^+, δ^∨ appears with multiplicity at most 1 in the simple coroot expansion of α^∨. The classification of minuscule roots is well known and is illustrated in Figure <ref> in terms of Dynkin diagrams.

