Hypercontractivity on the symmetric group

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 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.

On the Largest Product-free Subsets of the Alternating Groups

“…Then there exists some t-umvirate with t ≤ 4r in which A has density at least n t/4 µ(A). This 1% stability result will be deduced from a combination of the trace method, a recent level-d inequality due to Filmus, Kindler, Lifshitz and Minzer [5], and novel upper bounds on eigenvalues of Cayley graphs over the symmetric group.…”

Hypercontractivity on High Dimensional Expanders: a Local-to-Global Approach for Higher Moments

“…What's more, hypercontractivity (and its resulting applications) actually extend beyond the hypercube. After KKL's seminal work, many authors studied extensions and applications of hypercontractivity [BKK + 92, Tal94, FK96, Fri98], but it wasn't until recently that tight analogs of Equation (1) were developed for general product spaces [KLLM19] as well as for other structured domains such as the symmetric group [FKLM20] and Grassmannian [KMS18]. These extended domains differ from the hypercube in that they are only hypercontractive for special classes of pseudorandom functions, but are nevertheless responsible for an impressive set of applications including analogs of classical results, a variety of new sharp threshold theorems [KLLM19, LM19, KLLM21], and perhaps most famously the proof of the 2-2 Games Conjecture [KMS17, DKK + 18b, DKK + 18a, BKS18, KMMS18, KMS18].…”

“…Another line of work has examined hypercontractivity on what are often called "exotic" domains: specific objects beyond products such as the slice [KMMS18], multislice [FOW18], Grassmannian [KMS18] (or similarly the degree-two short code [BKS18]), and symmetric group [FKLM20]. Like KLLM's improved result for product distributions, most of these examples are only hypercontractive for pseudorandom functions (with the multislice being the only exception).…”

Hypercontractivity on High Dimensional Expanders: Approximate Efron-Stein Decompositions for $\varepsilon$-Product Spaces

“…Filmus et al [23] used their hypercontractivity theorem to prove a stability result for the Kruskal-Katona theorem. We prove a similar stability result for ǫ-HDX.…”

“…In this work, we focus on analysis of Boolean functions on high dimensional expanders, whose systematic study was recently initiated by Dikstein et al [9]. This continues a long line of investigation of Fourier analysis of Boolean functions on extended domains beyond the Boolean hypercube, such as the Boolean slice [44,20,25,24], the Grassmann scheme [15,39,18], the symmetric group [23,21,8], the p-biased cube [17,40,22], and the multi-slice [26,6]. The foregoing extended domains arise naturally throughout theoretical computer science, and indeed, the study of analysis of Boolean functions on extended domains has recently led to a breakthrough regarding the unique games conjecture [38,16,15,39].…”