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

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 $\widehat{X}$ of ${\mathbb{F}}^X$ and, for $f\in {\mathbb{F}}^X$, construct the Fourier transform $\widehat{f}\in {\mathbb{F}}^{\widehat{X}}$. We define the rank support $rk-supp(\widehat{f})$ and prove that $rk-supp(\widehat{f})=\dim \mathbb{F}Gf$, 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 $0\mathrm{\ne }f\in {\mathbb{F}}^X$, we construct a block $X_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)}$ of $X$ and a subset ${\mathcal{S}}^{\mathrm{\prime }-\text{\hspace{0.17em}⁣}1}$ of $G$ determined by the support $\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)$ of $f$, and show that $\dim \mathbb{F}Gf-\dim \mathbb{F}{\mathcal{S}}^{\mathrm{\prime }-\text{\hspace{0.17em}⁣}1}\text{\hspace{0.17em}⁣}f\ge 1$ and $$\mathrm{\mid }\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)\mathrm{\mid }\cdot \dim \mathbb{F}Gf\ge \mathrm{\mid }X\mathrm{\mid }+(\text{\hspace{0.17em}⁣}\dim \mathbb{F}Gf-\dim \mathbb{F}{\mathcal{S}}^{\mathrm{\prime }-1}f)\cdot \mathrm{\mid }\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)\mathrm{\mid }-\mathrm{\mid }{X}_{\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(f)}\mathrm{\mid },$$ where $\mathbb{F}{\mathcal{S}}^{\mathrm{\prime }-1}f$ denotes the subspace of $\mathbb{F}X$ spanned by the subset ${\mathcal{S}}^{\mathrm{\prime }-1}f=\{\alpha f\mathrm{\mid }\alpha \in {\mathcal{S}}^{\mathrm{\prime }-1}\}\subseteq \mathbb{F}X$. 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 $\mathrm{\mid }G\mathrm{\mid }$ 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.