Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , |
|---|---|
| Format: | Preprint |
| Publié: |
2022
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866917081534955520 |
|---|---|
| author | Chen, Bocong Fan, Yun Luo, Gaojun |
| author_facet | Chen, Bocong Fan, Yun Luo, Gaojun |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2211_11204 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups Chen, Bocong Fan, Yun Luo, Gaojun Group Theory Discrete Mathematics 05E18, 20B05, 20C15, 43A30, 94B60 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. |
| title | Sharp Uncertainty Principle for Transitive $G$-Sets over Arbitrary Fields and Finite Groups |
| topic | Group Theory Discrete Mathematics 05E18, 20B05, 20C15, 43A30, 94B60 |
| url | https://arxiv.org/abs/2211.11204 |