Characters of symmetric groups: sharp bounds and applications
Characters of symmetric groups: sharp bounds and applications
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对称群的特征：锐界及其应用
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
A. Shalev
中科院分区：
文献类型：
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作者：
M. Larsen;A. Shalev
我们给出了对称群的特征标值的新估计，这些特征标值对所有特征标成立，并且在某种意义上是最好可能的。从我们的一般界可以得出结论：如果一个置换σ∈Sn至多没有长度为<；m的(1)个圈，则对Sn的所有不可约特征标χ(σ)|≤χ(1)1/m+o(1)。这是Fomin和Lulov的一个结果的广泛推广.然后，我们利用我们的各种特征标界来解决关于随机游动的混合时间、被共轭类的幂覆盖以及字映射的概率和组合性质的广泛的公开问题.特别地，我们证明了Rudvalis和Vishne关于覆盖数的猜想以及Lulov和Pak关于Sn上某些随机游动的混合时间的猜想.我们的特征标论方法也给出了交错群An的Waring型问题的最佳可能解，证明了如果w是非平凡的词，且n≫0，则An的每个元素都是w的两个值的乘积。
We provide new estimates on character values of symmetric groups which hold for all characters and which are in some sense best possible. It follows from our general bound that if a permutation σ∈Sn has at most no(1) cycles of length <m, then |χ(σ)|≤χ(1)1/m+o(1) for all irreducible characters χ of Sn. This is a far reaching generalization of a result of Fomin and Lulov.We then use our various character bounds to solve a wide range of open problems regarding mixing times of random walks, covering by powers of conjugacy classes, as well as probabilistic and combinatorial properties of word maps.In particular we prove a conjecture of Rudvalis and of Vishne on covering numbers, and a conjecture of Lulov and Pak on mixing times of certain random walks on Sn.Our character-theoretic methods also yield best possible solutions to Waring type problems for alternating groups An, showing that if w is a non-trivial word, and n≫0, then every element of An is a product of two values of w.