Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey
Surveys in Combinatorics 2011: The cyclic sieving phenomenon: a survey
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组合学调查 2011：循环筛选现象：一项调查
DOI：
10.1017/cbo9781139004114.006
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发表时间：
2010
期刊：
影响因子：
--
通讯作者：
B. Sagan
中科院分区：
文献类型：
--
作者：
B. Sagan
循环筛分现象是由Reiner， Stanton和White在2004年的一篇论文中定义的。设X是一个有限集合，C是作用于X的有限循环群，f(q)是q中的一个系数为非负整数的多项式。如果对于C中的所有g，我们有# X^g = f(w)，其中#表示基数，X^g是g的不动点集，w是选择与g具有相同阶次的单位根，那么三元组（X，C,f(q)）表现出循环筛分现象。将单位根替换为具有整数系数的多项式似乎不太可能具有枚举意义。但是现在已经发现了许多循环筛分现象的实例。此外，这种现象的证明往往涉及表征理论中有趣的，有时甚至是深刻的结果。我们将调查循环筛分的当前文献，提供必要的背景表示，考克斯特群，和其他代数方面的需要。
The cyclic sieving phenomenon was defined by Reiner, Stanton, and White in a 2004 paper. Let X be a finite set, C be a finite cyclic group acting on X, and f(q) be a polynomial in q with nonnegative integer coefficients. Then the triple (X,C,f(q)) exhibits the cyclic sieving phenomenon if, for all g in C, we have # X^g = f(w) where # denotes cardinality, X^g is the fixed point set of g, and w is a root of unity chosen to have the same order as g. It might seem improbable that substituting a root of unity into a polynomial with integer coefficients would have an enumerative meaning. But many instances of the cyclic sieving phenomenon have now been found. Furthermore, the proofs that this phenomenon hold often involve interesting and sometimes deep results from representation theory. We will survey the current literature on cyclic sieving, providing the necessary background about representations, Coxeter groups, and other algebraic aspects as needed.