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An uncertainty inequality for groups of order $pq$. (English) Zbl 0790.43007
Let $G$ be a finite group. For a function $f: G\to \bbfC$ and a representation $\rho : G \to GL(V)$ of $G$ the endomorphism $\widehat{f}(\rho) = \sum\sb{x \in G}f(x)\rho(x)$ of $V$ is called the Fourier transform of $f$ at $\rho$. If $\rho\sb 1,\rho\sb 2,\dots,\rho\sb t$ denote the complex irreducible representations of $G$, where $\rho\sb i: G\to GL(V\sb i)$, and $\text{deg }\rho\sb i = \dim V\sb i = n\sb i$, then define $\mu(f) = \sum\sp t\sb{i = 1}\dim V\sb i\cdot \text{rank }\widehat{f}(\rho\sb i)$. The main result of this paper is theorem 1: Let $0 \ne f : G \to \bbfC$. Then (a) $\vert \text{Supp }f\vert \mu(f) \geq \vert G\vert$ (b) Suppose $f(1) = 1$. Then $\vert \text{Supp}(f)\vert \mu(f) = \vert G\vert$ if and only if $H = \text{Supp }f$ is a subgroup of $G$, and $f(x) = 1\sb H(x)\chi(x)$ where $\chi$ is a 1-dimensional character of $G$, and $1\sb H(x)$ is the indicator function of $H\subseteq G$. As an application of this theorem the author obtains an uncertainty-type inequality for direct products of non-abelian groups of order $pq$ where $p$ and $q$ are prime numbers and $p\mid q - 1$.

43A30Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
20C15Ordinary representations and characters of groups
20D60Arithmetic and combinatorial problems on finite groups
Full Text: DOI
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