Hypercontractivity is the secret sauce in Boolean function analysis. Yet it many domains, hypercontractivity is useless for general functions. In recent work, Keevash, Lifshitz, Long and Mintzer showed that in such cases, hypercontractivity does hold for global functions, which are functions whose expectation doesn’t change significantly when restricting to subdomains of small codimension.
In this work, we extend this theory to functions on the symmetric group. As applications, we bound the size of global, product-free sets in the alternative group (via a level $k$ inequality), and prove a robust version of Kruskal–Katona.