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

TL;DR

This paper extends the uncertainty principle to transitive G-sets over arbitrary fields, providing a sharp inequality relating support size and module dimension, with conditions for equality and applications to finite groups.

Contribution

It generalizes and sharpens the uncertainty principle for finite groups and sets, introducing new bounds and characterizations in a broad algebraic framework.

Findings

Established a new uncertainty inequality for transitive G-sets.

Provided conditions for equality in the uncertainty bounds.

Recovered Tao's strong uncertainty principle for prime order groups.

Abstract

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 ${\mathscr S}'^{-1}f=\{\alpha f\,|\,\alpha\in{\mathscr S}'^{-1}\}\subseteq{\Bbb F} X$. 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.