  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.
