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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对称群上的超收缩性

超收缩不等式是分析中的一个基本结果，在离散数学、理论计算机科学、组合学等领域有许多应用。到目前为止，这种不等式的变体主要在乘积空间中得到证明，这就提出了类似结果是否适用于非乘积域的问题。我们考虑对称群$S_n$，这是最基本的非乘积域之一，并在其上建立超收缩不等式。我们的不等式对于 $S_n$ 上的 \emph{全局函数} 类最有效，这些函数在限制输入的 $O(1)$ 坐标时 $2$-范数仍然很小，并断言低度，全局函数有小的 $q$-norms，因为 $q>2$。作为应用，我们展示： 1. 超立方体上的 level-$d$ 不等式的模拟，断言低度上的全局函数的质量非常小。我们还展示了如何使用这个不等式来限制交替组 $A_n$ 中全局无乘积集的大小。2. 全局函数的 $S_n$ 转置凯莱图上的等周不等式，类似于 KKL 定理和布尔超立方体中的小集展开性质。3. Kruskal--Katona 定理在某些参数范围内的多重切片和稳定性版本的超收缩不等式。类似于 KKL 定理和布尔超立方体中的小集扩展属性。3. Kruskal--Katona 定理在某些参数范围内的多重切片和稳定性版本的超收缩不等式。类似于 KKL 定理和布尔超立方体中的小集扩展属性。3. Kruskal--Katona 定理在某些参数范围内的多重切片和稳定性版本的超收缩不等式。