Cyclic Sieving for Plane Partitions and Symmetry
Cyclic Sieving for Plane Partitions and Symmetry
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DOI：
10.3842/sigma.2020.130
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发表时间：
2019-07
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影响因子：
--
通讯作者：
S. Hopkins
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文献类型：
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作者：
S. Hopkins
赖纳、斯坦顿和白色的循环筛选现象表明，我们通常可以通过将单位根插入与组合集相关的多项式中来计算作用于该组合集的循环群的元素的不动点。循环筛选现象最令人印象深刻的例子之一是罗兹的一个定理，该定理断言在推广作用下矩形盒中的平面划分集具有循环筛选。在Rhoades的结果中，筛分多项式是这些平面划分的尺寸生成函数，由于MacMahon，它具有众所周知的乘积公式。我们还考虑平面分区的对称性：具体地说，互补和换位扩展罗兹的结果。这里的相关多项式是对称平面划分的尺寸生成函数，其乘积公式由MacMahon给出，并由Andrews和Macdonald证明。最后，我们解释了这些对称性的结果也适用于rowmotion运营商的平面分区，这是密切相关的促进。
The cyclic sieving phenomenon of Reiner, Stanton, and White says that we can often count the fixed points of elements of a cyclic group acting on a combinatorial set by plugging roots of unity into a polynomial related to this set. One of the most impressive instances of the cyclic sieving phenomenon is a theorem of Rhoades asserting that the set of plane partitions in a rectangular box under the action of promotion exhibits cyclic sieving. In Rhoades's result the sieving polynomial is the size generating function for these plane partitions, which has a well-known product formula due to MacMahon. We extend Rhoades's result by also considering symmetries of plane partitions: specifically, complementation and transposition. The relevant polynomial here is the size generating function for symmetric plane partitions, whose product formula was conjectured by MacMahon and proved by Andrews and Macdonald. Finally, we explain how these symmetry results also apply to the rowmotion operator on plane partitions, which is closely related to promotion.