On common zeros of characters of finite groups
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912041050046464 |
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| author | Lewis, Mark L. Morotti, Lucia Pacifici, Emanuele Sanus, Lucia Tong-Viet, Hung P. |
| author_facet | Lewis, Mark L. Morotti, Lucia Pacifici, Emanuele Sanus, Lucia Tong-Viet, Hung P. |
| contents | Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $χ\in\text{Irr}(G)$ such that $χ(g)=0$ (i.e., $g$ is a zero of $χ$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $χ\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14811 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On common zeros of characters of finite groups Lewis, Mark L. Morotti, Lucia Pacifici, Emanuele Sanus, Lucia Tong-Viet, Hung P. Group Theory Representation Theory Let $G$ be a finite group, and let $\text{Irr}(G)$ denote the set of the irreducible complex characters of $G$. An element $g\in G$ is called a vanishing element of $G$ if there exists $χ\in\text{Irr}(G)$ such that $χ(g)=0$ (i.e., $g$ is a zero of $χ$) and, in this case, the conjugacy class $g^G$ of $g$ in $G$ is called a vanishing conjugacy class. In this paper we consider several problems concerning vanishing elements and vanishing conjugacy classes; in particular, we consider the problem of determining the least number of conjugacy classes of a finite group $G$ such that every non-linear $χ\in\text{Irr}(G)$ vanishes on one of them. We also consider the related problem of determining the minimum number of non-linear irreducible characters of a group such that two of them have a common zero. |
| title | On common zeros of characters of finite groups |
| topic | Group Theory Representation Theory |
| url | https://arxiv.org/abs/2409.14811 |