Non-vanishing elements and complex group algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929543438139392 |
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| author | Ebrahimi, Mahdi |
| author_facet | Ebrahimi, Mahdi |
| contents | Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $χ$ in $\mathrm{Irr}(G)$, we have $χ(x)=0$. Also the element $x$ is called rational if $x$ is conjugate to $x^i$ for every integer $i$ co-prime to the order of $x$. We define the weight of $G$ as $ω(G):=(\sum_{χ\in \mathrm{Irr}(G)}χ(1))^2/|G|$. In this paper, we show that for every rational non-vanishing element $x\in G$, the order of $C_G(x)$ is at least $ω(G)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11313 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-vanishing elements and complex group algebras Ebrahimi, Mahdi Group Theory 20C15, 05C50, 05C92 Let $G$ be a finite group, and let $\mathrm{Irr}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $χ$ in $\mathrm{Irr}(G)$, we have $χ(x)=0$. Also the element $x$ is called rational if $x$ is conjugate to $x^i$ for every integer $i$ co-prime to the order of $x$. We define the weight of $G$ as $ω(G):=(\sum_{χ\in \mathrm{Irr}(G)}χ(1))^2/|G|$. In this paper, we show that for every rational non-vanishing element $x\in G$, the order of $C_G(x)$ is at least $ω(G)$. |
| title | Non-vanishing elements and complex group algebras |
| topic | Group Theory 20C15, 05C50, 05C92 |
| url | https://arxiv.org/abs/2410.11313 |