  Finding upper bounds for character ratios is a fundamental problem in
asymptotic group theory. Previous bounds in the symmetric group have led to
remarkable applications in unexpected domains. The existing approaches
predominantly relied on algebraic methods, whereas our approach combines
analytic and algebraic tools. Specifically, we make use of a tool called
`hypercontractivity for global functions' from the theory of Boolean functions.
By establishing sharp upper bounds on the $L^p$-norms of characters of the
symmetric group, we improve existing results on character ratios from the work
of Larsen and Shalev [Larsen M., Shalev A. Characters of the symmetric group:
sharp bounds and applications. Invent. math. 174 645-687 (2008)]. We use our
norm bounds to bound Kronecker coefficients, Fourier coefficients of class
functions, product mixing of normal sets, and mixing time of normal Cayley
graphs. Our approach bypasses the need for the $S_n$-specific
Murnaghan--Nakayama rule. Instead we leverage more flexible representation
theoretic tools, such as Young's branching rule, which potentially extend the
applicability of our method to groups beyond $S_n$.
