Bounds for Characters of the Symmetric Group: A Hypercontractive Approach

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 symmetric groups: sharp bounds and applications. Invent. math. 174, 645-687 (2008)]. We use our norm bounds to bound Fourier coefficients of class functions, product mixing of normal sets, mixing time of normal Cayley graphs, and Kronecker coefficients. 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 $S_n$.