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 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 $S_n$

On the sum of the L1 influences of bounded functions

Let $f\colon \{-1,1\}^n \to [-1,1]$ have degree $d$ as a multilinear polynomial. It is well-known that the total influence of $f$ is at most $d$. Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also be bounded as a function of $d$. Ba\v{c}kurs and Bavarian answered this question in the affirmative, providing a bound of $O(d^3)$ for general functions and $O(d^2)$ for homogeneous functions. We improve on their results by providing a bound of $d^2$ for general functions and $O(d\log d)$ for homogeneous functions. In addition, we prove a bound of $d/(2 \pi)+o(d)$ for monotone functions, and provide a matching example.Comment: 16 pages; accepted for publication in the Israel Journal of Mathematic

Sharp Hypercontractivity for Global Functions

For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion $f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality for the noise operator allows bounding norms of $T_\rho f=\sum_S\rho^{|S|}\hat f(S)\chi_S$ in terms of norms of $f$. If $f$ is Boolean-valued, the level-$d$ inequality allows bounding the norm of $f^{=d}=\sum_{|S|=d}\hat f(S)\chi_S$ in terms of $E[f]$. These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube is endowed with the uniform measure, they do not hold for more general discrete product spaces. Finding a `natural' generalization was a long-standing open problem. In [P. Keevash et al., Global hypercontractivity and its applications, J. Amer. Math. Soc., to appear], a hypercontractive inequality for this setting was presented, that holds for functions which are `global' -- namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group $S_n$ and to other settings where sharpness of the bound is crucial. Also, no level-$d$ inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level-$d$ inequality for this setting. Our inequalities open the way for diverse applications to extremal set theory and to group theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications to the study of functions on $S_n$ -- including hypercontractivity and level-$d$ inequalities, character bounds, variants of Roth's theorem and of Bogolyubov's lemma, and diameter bounds, that were obtained using our techniques