Refined Cyclic Sieving
Refined Cyclic Sieving
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精制循环筛分
DOI：
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发表时间：
2017
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影响因子：
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通讯作者：
Joshua P. Swanson
中科院分区：
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作者：
Connor Ahlbach;Joshua P. Swanson
Reiner-Stanton-White(2004)定义了与有限循环群作用和多项式有关的循环筛选现象(CSP)。一个关键的例子来自有限Coxeter群的抛物商的最小长度陪集表示的长度母函数。在类型A中，这一结果可以用对固定内容的词的自然循环作用来表述。对于许多CSP来说，有一个自然的精化概念。我们通过跟踪一个词的循环下降类型以及它的内容来制定并证明上述CSP的精化。所提出的论点与Reiner-Stanton-White的表征理论方法完全不同。它是组合性的，虽然不是完全的，但在很大程度上是双射的。我们论证的一个基础涉及对移位子集和的循环筛选，这也出现在Reiner-Stanton-White中。我们通过推广Wagon-Wilf(1994)的一些思想，给出了这一结果的改进的另一种主要是双射的证明。
Reiner-Stanton-White (2004) defined the cyclic sieving phenomenon (CSP) associated to a finite cyclic group action and polynomial. A key example arises from the length generating function for minimal length coset representatives of a parabolic quotient of a finite Coxeter group. In type A, this result can be phrased in terms of the natural cyclic action on words of fixed content. There is a natural notion of refinement for many CSP’s. We formulate and prove a refinement of the aforementioned CSP arising from tracking the cyclic descent type of a word in addition to its content. The argument presented is completely different from Reiner-Stanton-White’s representation-theoretic approach. It is combinatorial and largely, though not entirely, bijective. A building block of our argument involves cyclic sieving for shifted subset sums, which also appeared in Reiner-Stanton-White. We give an alternate, largely bijective proof of a refinement of this result by extending some ideas of Wagon-Wilf (1994).