巴特勒方法应用于 $$\mathbb {Z}_p[C_p\times C_p]$$ -Permutation 模块

IF 0.5 4区 数学 Q3 MATHEMATICS
John W. MacQuarrie, Marlon Estanislau
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引用次数: 0

摘要

让 G 是一个有限 p 群,具有阶为 \(\varvec{p}\) 的正常子群 \(\varvec{N}\)。第一作者和 Zalesskii 之前已经证明,当且仅当其\(\varvec{N}\)-不变式是一个置换模块时,一个\(\mathbb {Z}_p\) \(\varvec{G}\)-晶格才是一个置换模块、和第三个模块都是分别覆盖 \(\mathbb {Z}_p, \mathbb {Z}_p\) 和 \(\mathbb {Z}_p\)的 G/N 置换模块。前两个条件的必要性很容易证明,但第三个条件的必要性却不为人所知。我们应用了巴特勒提出的一种对应关系,它将一组简单的组合数据关联到一个无性\(\varvec{p}\)-群的\(\mathbb {Z}_p\) \(\varvec{G}\)-晶格,从而证明了这些条件的必要性、利用对应关系来构造高度非难的反例,即如果给定网格 \(\varvec{U}) 的 \(\varvec{N}\)-invariants 和 \(\varvec{N}\)-coinvariants 都是置换模块,那么 \(\varvec{U}\) 也是置换模块。我们的新方法是把所需的性质转换到组合方面,在那里找到反例,然后再把它转换回晶格。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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.

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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
61
审稿时长
6-12 weeks
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.
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