{"title":"半线性张量分解","authors":"K.K. Mahavadi , A.J.E. Ryba","doi":"10.1016/j.jaca.2024.100013","DOIUrl":null,"url":null,"abstract":"<div><p>We prove that a <em>kG</em>-module has a <em>semilinear tensor decomposition</em> if and only if its endomorphism algebra has a pair of mutually centralizing, unital, <em>G</em>-invariant subalgebras that are not commutative and are isomorphic to complete matrix algebras over an extension field <em>K</em> of <em>k</em>. We give an algorithm that constructs a semilinear tensor decomposition for any module whose endomorphism algebra contains appropriate invariant subalgebras.</p></div>","PeriodicalId":100767,"journal":{"name":"Journal of Computational Algebra","volume":"9 ","pages":"Article 100013"},"PeriodicalIF":0.0000,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S2772827724000032/pdfft?md5=3558c14d36b31fbd7274f355c1412fd1&pid=1-s2.0-S2772827724000032-main.pdf","citationCount":"0","resultStr":"{\"title\":\"Semilinear tensor decompositions\",\"authors\":\"K.K. Mahavadi , A.J.E. Ryba\",\"doi\":\"10.1016/j.jaca.2024.100013\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove that a <em>kG</em>-module has a <em>semilinear tensor decomposition</em> if and only if its endomorphism algebra has a pair of mutually centralizing, unital, <em>G</em>-invariant subalgebras that are not commutative and are isomorphic to complete matrix algebras over an extension field <em>K</em> of <em>k</em>. We give an algorithm that constructs a semilinear tensor decomposition for any module whose endomorphism algebra contains appropriate invariant subalgebras.</p></div>\",\"PeriodicalId\":100767,\"journal\":{\"name\":\"Journal of Computational Algebra\",\"volume\":\"9 \",\"pages\":\"Article 100013\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-03-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S2772827724000032/pdfft?md5=3558c14d36b31fbd7274f355c1412fd1&pid=1-s2.0-S2772827724000032-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S2772827724000032\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational Algebra","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S2772827724000032","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
我们证明，kG 模块具有半线性张量分解，当且仅当它的内象代数具有一对互为中心化、单存在、G 不变的子代数，这些子代数不交换，并且与 k 的扩展域 K 上的完整矩阵代数同构。我们给出了一种算法，可以为任何内象代数包含适当不变子代数的模块构造半线性张量分解。
We prove that a kG-module has a semilinear tensor decomposition if and only if its endomorphism algebra has a pair of mutually centralizing, unital, G-invariant subalgebras that are not commutative and are isomorphic to complete matrix algebras over an extension field K of k. We give an algorithm that constructs a semilinear tensor decomposition for any module whose endomorphism algebra contains appropriate invariant subalgebras.
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