TY - JOUR

T1 - Hypercontractivity on the symmetric group

AU - Filmus, Yuval

AU - Kindler, Guy

AU - Lifshitz, Noam

AU - Minzer, Dor

N1 - Publisher Copyright:
© 2024 The Author(s). Published by Cambridge University Press.

PY - 2024/1/8

Y1 - 2024/1/8

N2 - 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, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on, which are functions whose -norm remains small when restricting coordinates of the input, and assert that low-degree, global functions have small q-norms, for 2. As applications, we show the following: 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. 2. Isoperimetric inequalities on the transposition Cayley graph of 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.

AB - 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, one of the most basic non-product domains, and establish hypercontractive inequalities on it. Our inequalities are most effective for the class of global functions on, which are functions whose -norm remains small when restricting coordinates of the input, and assert that low-degree, global functions have small q-norms, for 2. As applications, we show the following: 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. 2. Isoperimetric inequalities on the transposition Cayley graph of 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.

UR - http://www.scopus.com/inward/record.url?scp=85183297593&partnerID=8YFLogxK

U2 - 10.1017/fms.2023.118

DO - 10.1017/fms.2023.118

M3 - Article

AN - SCOPUS:85183297593

SN - 2050-5094

VL - 12

JO - Forum of Mathematics, Sigma

JF - Forum of Mathematics, Sigma

M1 - e6

ER -