Hypercontractivity on the symmetric group

TL;DR

This paper establishes hypercontractive inequalities on the symmetric group for global functions, extending fundamental analysis tools to non-product domains with applications in combinatorics and theoretical computer science.

Contribution

It introduces hypercontractive inequalities for the symmetric group, particularly for global functions, and applies these results to isoperimetric inequalities, level-d inequalities, and stability results.

Findings

Proved hypercontractive inequalities for global functions on $S_n$.

Derived isoperimetric inequalities analogous to KKL and small-set expansion.

Established stability versions of the Kruskal--Katona Theorem.

Abstract

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.