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 ${\Bbb F}$, we consider the vector space ${\Bbb F}^X$ of all functions from $X$ to ${\Bbb F}$, which is a $G$-space isomorphic to the permutation ${\Bbb F} G$-module ${\Bbb F} X$. When the group algebra ${\Bbb F} G$ is semisimple and split, we find a specific basis $\widehat X$ of ${\Bbb F}^X$ and, for $f\in{\Bbb F}^X$, construct the Fourier transform $\widehat f\in{\Bbb F}^{\widehat X}$. We define the rank support $\mbox{rk-supp}(\widehat f)$ and prove that $\mbox{rk-supp}(\widehat f)=\dim {\Bbb F} G f$, where ${\Bbb F} G f$ is the submodule of ${\Bbb F} X$ 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 $0\ne f\in{\Bbb F}^X$, we construct a block $X_{{\rm supp}(f)}$ of $X$ and a subset ${\mathscr S}'^{-\!1}$ of $G$ determined by the support ${\rm supp}(f)$ of $f$, and show that $\dim{\Bbb F} Gf-\dim{\Bbb F}{\mathscr S}'^{-\!1}\!f\ge 1$ and $$ |{\rm supp}(f)|\cdot \dim{\Bbb F} Gf \ge |X|+ (\!\dim{\Bbb F} Gf-\dim{\Bbb F}{\mathscr S}'^{-1}f) \cdot|{\rm supp}(f)| -|X_{{\rm supp}(f)}|, $$ where ${\Bbb F}{\mathscr S}'^{-1}f$ denotes the subspace of ${\Bbb F}X$ spanned by the subset ${\mathscr S}'^{-1}f=\{αf\,|\,α\in{\mathscr S}'^{-1}\}\subseteq{\Bbb F} X$. 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 $\dim {\Bbb F}Gf$ that recovers Tao's 2005 strong uncertainty principle, along with a precise characterization of the equality case.