对称群上的超压缩性
Hypercontractivity on the symmetric group
摘要：
超压缩不等式是分析中的一个基本结果，在整个离散数学、理论计算机科学、组合数学等领域有许多应用。到目前为止，这个不等式的变体主要是在乘积空间中被证明的，这就提出了一个问题，即类似的结果是否适用于非乘积域。本文考虑对称群$S_n$，最基本的非积域之一，建立了其上的超压缩不等式，这些不等式对$S_n$上的一类\emph{global functions}是最有效的，这类函数在限制输入的$O(1)$坐标时其$2$ -范数保持很小，并断言对于$q>2$，低次整体函数具有很小的$q$ -范数.作为应用，我们表明：1。超立方体上的水平- $d$不等式的一个类似，断言低次上的整体函数的质量非常小。我们还展示了如何使用这个不等式来约束交替群$A_n$中的全局无积集的大小。2.整体函数的$S_n$的转置Cayley图上的等周不等式，类似于KKL定理和布尔超立方体中的小集展开性质. 3.多重切片上的超压缩不等式，以及Kruskal-Katona定理在某些参数下的稳定性。
英文摘要：
The hypercontractive inequality is a fundamental result in analysis, with many applications throughout discrete mathematics, theoretical computer science, combinatorics and more. So far, variants of this inequality have been proved mainly for product spaces, which raises the question of whether analogous results hold over non-product domains. We consider the symmetric group, $S_n$, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of \emph{global functions} on $S_n$, which are functions whose $2$-norm remains small when restricting $O(1)$ coordinates of the input, and assert that low-degree, global functions have small $q$-norms, for $q>2$. As applications, we show: 1. An analog of the level-$d$ inequality on the hypercube, asserting that the mass of a global function on low-degrees is very small. We also show how to use this inequality to bound the size of global, product-free sets in the alternating group $A_n$. 2. Isoperimetric inequalities on the transposition Cayley graph of $S_n$ for global functions, that are analogous to the KKL theorem and to the small-set expansion property in the Boolean hypercube. 3. Hypercontractive inequalities on the multi-slice, and stability versions of the Kruskal--Katona Theorem in some regimes of parameters.
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