{"title":"巴特勒方法应用于 $$\\mathbb {Z}_p[C_p\\times C_p]$$ -Permutation 模块","authors":"John W. MacQuarrie, Marlon Estanislau","doi":"10.1007/s10468-024-10277-7","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>G</i> be a finite <i>p</i>-group with normal subgroup <span>\\(\\varvec{N}\\)</span> of order <span>\\(\\varvec{p}\\)</span>. The first author and Zalesskii have previously shown that a <span>\\(\\mathbb {Z}_p\\)</span> <span>\\(\\varvec{G}\\)</span>-lattice is a permutation module if, and only if, its <span>\\(\\varvec{N}\\)</span>-invariants, its <span>\\(\\varvec{N}\\)</span>-coinvariants, and a third module are all <i>G</i>/<i>N</i> permutation modules over <span>\\(\\mathbb {Z}_p, \\mathbb {Z}_p\\)</span> and <span>\\(\\mathbb {Z}_p\\)</span> respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a <span>\\(\\mathbb {Z}_p\\)</span> <span>\\(\\varvec{G}\\)</span>-lattice for an abelian <span>\\(\\varvec{p}\\)</span>-group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the <span>\\(\\varvec{N}\\)</span>-invariants and the <span>\\(\\varvec{N}\\)</span>-coinvariants of a given lattice <span>\\(\\varvec{U}\\)</span> are permutation modules, then so is <span>\\(\\varvec{U}\\)</span>. Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice.</p></div>","PeriodicalId":50825,"journal":{"name":"Algebras and Representation Theory","volume":"27 4","pages":"1671 - 1680"},"PeriodicalIF":0.5000,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Butler’s Method Applied to \\\\(\\\\mathbb {Z}_p[C_p\\\\times C_p]\\\\)-Permutation Modules\",\"authors\":\"John W. MacQuarrie, Marlon Estanislau\",\"doi\":\"10.1007/s10468-024-10277-7\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Let <i>G</i> be a finite <i>p</i>-group with normal subgroup <span>\\\\(\\\\varvec{N}\\\\)</span> of order <span>\\\\(\\\\varvec{p}\\\\)</span>. The first author and Zalesskii have previously shown that a <span>\\\\(\\\\mathbb {Z}_p\\\\)</span> <span>\\\\(\\\\varvec{G}\\\\)</span>-lattice is a permutation module if, and only if, its <span>\\\\(\\\\varvec{N}\\\\)</span>-invariants, its <span>\\\\(\\\\varvec{N}\\\\)</span>-coinvariants, and a third module are all <i>G</i>/<i>N</i> permutation modules over <span>\\\\(\\\\mathbb {Z}_p, \\\\mathbb {Z}_p\\\\)</span> and <span>\\\\(\\\\mathbb {Z}_p\\\\)</span> respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a <span>\\\\(\\\\mathbb {Z}_p\\\\)</span> <span>\\\\(\\\\varvec{G}\\\\)</span>-lattice for an abelian <span>\\\\(\\\\varvec{p}\\\\)</span>-group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the <span>\\\\(\\\\varvec{N}\\\\)</span>-invariants and the <span>\\\\(\\\\varvec{N}\\\\)</span>-coinvariants of a given lattice <span>\\\\(\\\\varvec{U}\\\\)</span> are permutation modules, then so is <span>\\\\(\\\\varvec{U}\\\\)</span>. Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice.</p></div>\",\"PeriodicalId\":50825,\"journal\":{\"name\":\"Algebras and Representation Theory\",\"volume\":\"27 4\",\"pages\":\"1671 - 1680\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-07-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Algebras and Representation Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s10468-024-10277-7\",\"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/s10468-024-10277-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Butler’s Method Applied to \(\mathbb {Z}_p[C_p\times C_p]\)-Permutation Modules
Let G be a finite p-group with normal subgroup \(\varvec{N}\) of order \(\varvec{p}\). The first author and Zalesskii have previously shown that a \(\mathbb {Z}_p\)\(\varvec{G}\)-lattice is a permutation module if, and only if, its \(\varvec{N}\)-invariants, its \(\varvec{N}\)-coinvariants, and a third module are all G/N permutation modules over \(\mathbb {Z}_p, \mathbb {Z}_p\) and \(\mathbb {Z}_p\) respectively. The necessity of the first two conditions is easily shown but the necessity of the third was not known. We apply a correspondence due to Butler, which associates to a \(\mathbb {Z}_p\)\(\varvec{G}\)-lattice for an abelian \(\varvec{p}\)-group a set of simple combinatorial data, to demonstrate the necessity of the conditions, using the correspondence to construct highly non-trivial counterexamples to the claim that if both the \(\varvec{N}\)-invariants and the \(\varvec{N}\)-coinvariants of a given lattice \(\varvec{U}\) are permutation modules, then so is \(\varvec{U}\). Our approach, which is new, is to translate the desired properties to the combinatorial side, find the counterexample there, and translate it back to a lattice.
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.