Mark L. Lewis, Lucia Morotti, Emanuele Pacifici, Lucia Sanus, Hung P. Tong-Viet
{"title":"关于有限群的公零","authors":"Mark L. Lewis, Lucia Morotti, Emanuele Pacifici, Lucia Sanus, Hung P. Tong-Viet","doi":"10.1007/s10468-025-10320-1","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a finite group, and let <span>\\(\\textrm{Irr}(G)\\)</span> denote the set of the irreducible complex characters of <i>G</i>. An element <span>\\(g\\in G\\)</span> is called a <i>vanishing element</i> of <i>G</i> if there exists <span>\\(\\chi \\in \\textrm{Irr}(G)\\)</span> such that <span>\\(\\chi (g)=0\\)</span> (i.e., <i>g</i> is a <i>zero</i> of <span>\\(\\chi \\)</span>) and, in this case, the conjugacy class <span>\\(g^G\\)</span> of <i>g</i> in <i>G</i> is called a <i>vanishing conjugacy class</i>. 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 <i>G</i> such that every non-linear <span>\\(\\chi \\in \\textrm{Irr}(G)\\)</span> 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.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"28 2","pages":"395 - 406"},"PeriodicalIF":0.6000,"publicationDate":"2025-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10468-025-10320-1.pdf","citationCount":"0","resultStr":"{\"title\":\"On Common Zeros of Characters of Finite Groups\",\"authors\":\"Mark L. Lewis, Lucia Morotti, Emanuele Pacifici, Lucia Sanus, Hung P. Tong-Viet\",\"doi\":\"10.1007/s10468-025-10320-1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>G</i> be a finite group, and let <span>\\\\(\\\\textrm{Irr}(G)\\\\)</span> denote the set of the irreducible complex characters of <i>G</i>. An element <span>\\\\(g\\\\in G\\\\)</span> is called a <i>vanishing element</i> of <i>G</i> if there exists <span>\\\\(\\\\chi \\\\in \\\\textrm{Irr}(G)\\\\)</span> such that <span>\\\\(\\\\chi (g)=0\\\\)</span> (i.e., <i>g</i> is a <i>zero</i> of <span>\\\\(\\\\chi \\\\)</span>) and, in this case, the conjugacy class <span>\\\\(g^G\\\\)</span> of <i>g</i> in <i>G</i> is called a <i>vanishing conjugacy class</i>. 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 <i>G</i> such that every non-linear <span>\\\\(\\\\chi \\\\in \\\\textrm{Irr}(G)\\\\)</span> 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.</p></div>\",\"PeriodicalId\":50825,\"journal\":{\"name\":\"Algebras and Representation Theory\",\"volume\":\"28 2\",\"pages\":\"395 - 406\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2025-02-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://link.springer.com/content/pdf/10.1007/s10468-025-10320-1.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebras and Representation Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10468-025-10320-1\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebras and Representation Theory","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10468-025-10320-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Let G be a finite group, and let \(\textrm{Irr}(G)\) 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 \(\chi \in \textrm{Irr}(G)\) such that \(\chi (g)=0\) (i.e., g is a zero of \(\chi \)) 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 \(\chi \in \textrm{Irr}(G)\) 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.
期刊介绍:
Algebras and Representation Theory features carefully refereed papers relating, in its broadest sense, to the structure and representation theory of algebras, including Lie algebras and superalgebras, rings of differential operators, group rings and algebras, C*-algebras and Hopf algebras, with particular emphasis on quantum groups.
The journal contains high level, significant and original research papers, as well as expository survey papers written by specialists who present the state-of-the-art of well-defined subjects or subdomains. Occasionally, special issues on specific subjects are published as well, the latter allowing specialists and non-specialists to quickly get acquainted with new developments and topics within the field of rings, algebras and their applications.