Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups

Abstract

For any finite group $G$, any transitive $G$-set $X$ and any field $\mathbb{F}$, we consider the vector space ${\mathbb{F}}^X$ of all functions from $X$ to $\mathbb{F}$, which is a $G$-space isomorphic to the permutation $\mathbb{F}G$-module $\mathbb{F}X$. When the group algebra $\mathbb{F}G$ is semisimple and split, we find a specific basis $\hat{X}$ of ${\mathbb{F}}^X$ and, for $f\in {\mathbb{F}}^X$, construct the Fourier transform $\hat{f}\in {\mathbb{F}}^{\hat{X}}$. We define the rank support $\mbox{rk-supp}(\widehat f)$ and prove that $\mbox{rk-supp}(\widehat f)=\dim {\Bbb F} G f$, where $\mathbb{F}Gf$ is the submodule of $\mathbb{F}X$ generated by the element $f={\sum }_{x\in X}f(x)x$. Next, we extend and strengthen the sharpened uncertainty principle for finite abelian groups, established by Feng, Hollmann, and Xiang in 2019, to a broader framework and a sharp version. For $0\ne f\in {\mathbb{F}}^X$, we construct a block $X_{\mathrm{supp}(f)}$ of $X$ and a subset ${\mathcal{S}}^{\prime -1}$ of $G$ determined by the support $\mathrm{supp}(f)$ of $f$, and show that $\dim \mathbb{F}Gf-\dim \mathbb{F}{\mathcal{S}}^{\prime -1}f\ge 1$ and $$|\mathrm{supp}(f)|\cdot \dim \mathbb{F}Gf\ge |X|+(\dim \mathbb{F}Gf-\dim \mathbb{F}{\mathcal{S}}^{\prime -1}f)\cdot |\mathrm{supp}(f)|-|X_{\mathrm{supp}(f)}|,$$ where $\mathbb{F}{\mathcal{S}}^{\prime -1}f$ denotes the subspace of $\mathbb{F}X$ spanned by the subset ${\mathcal{S}}^{\prime -1}f=\{\hat{I}\pm f|\hat{I}\pm \in {\mathcal{S}}^{\prime -1}\}\subseteq \mathbb{F}X$. We provide necessary and sufficient conditions for the above inequality to achieve equality. As corollaries, we derive many sharpened or classical versions of the finite-dimensional uncertainty principle, address an open question posed by Feng, Hollmann, and Xiang. When $|G|$ is a prime and $X=G$, we give a lower bound on $\dim \mathbb{F}Gf$ that recovers Tao's 2005 strong uncertainty principle, along with a precise characterization of the equality case.