{"title":"面向格拉斯曼流形 G͠ 2 t,3 的模 2 同调中的格氏基","authors":"Uroš A. Colović, Branislav I. Prvulović","doi":"10.1515/ms-2024-0015","DOIUrl":null,"url":null,"abstract":"For <jats:italic>n</jats:italic> a power of two, we give a complete description of the cohomology algebra <jats:italic>H</jats:italic> <jats:sup>*</jats:sup>(<jats:italic>G͠</jats:italic> <jats:sub> <jats:italic>n</jats:italic>,3</jats:sub>; ℤ<jats:sub>2</jats:sub>) of the Grassmann manifold <jats:italic>G͠</jats:italic> <jats:sub> <jats:italic>n</jats:italic>,3</jats:sub> of oriented 3-planes in ℝ<jats:sup> <jats:italic>n</jats:italic> </jats:sup>. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for <jats:italic>H</jats:italic> <jats:sup>*</jats:sup>(<jats:italic>G͠</jats:italic> <jats:sub> <jats:italic>n</jats:italic>,3</jats:sub>; ℤ<jats:sub>2</jats:sub>).","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"16 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Gröbner bases in the mod 2 cohomology of oriented Grassmann manifolds G͠ 2 t,3\",\"authors\":\"Uroš A. Colović, Branislav I. Prvulović\",\"doi\":\"10.1515/ms-2024-0015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For <jats:italic>n</jats:italic> a power of two, we give a complete description of the cohomology algebra <jats:italic>H</jats:italic> <jats:sup>*</jats:sup>(<jats:italic>G͠</jats:italic> <jats:sub> <jats:italic>n</jats:italic>,3</jats:sub>; ℤ<jats:sub>2</jats:sub>) of the Grassmann manifold <jats:italic>G͠</jats:italic> <jats:sub> <jats:italic>n</jats:italic>,3</jats:sub> of oriented 3-planes in ℝ<jats:sup> <jats:italic>n</jats:italic> </jats:sup>. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for <jats:italic>H</jats:italic> <jats:sup>*</jats:sup>(<jats:italic>G͠</jats:italic> <jats:sub> <jats:italic>n</jats:italic>,3</jats:sub>; ℤ<jats:sub>2</jats:sub>).\",\"PeriodicalId\":18282,\"journal\":{\"name\":\"Mathematica Slovaca\",\"volume\":\"16 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-05-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematica Slovaca\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1515/ms-2024-0015\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ms-2024-0015","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
对于 n 的 2 次幂，我们给出了对ℝ n 中定向 3 平面的格拉斯曼流形 G͠ n,3 的同调代数 H *(G͠ n,3; ℤ2)的完整描述。为此，我们要为与这个同调代数密切相关的理想找到一个还原的格氏基。利用这个格氏基，我们还提出了 H *(G͠ n,3; ℤ2)的加法基。
Gröbner bases in the mod 2 cohomology of oriented Grassmann manifolds G͠ 2 t,3
For n a power of two, we give a complete description of the cohomology algebra H*(G͠n,3; ℤ2) of the Grassmann manifold G͠n,3 of oriented 3-planes in ℝn. We do this by finding a reduced Gröbner basis for an ideal closely related to this cohomology algebra. Using this Gröbner basis we also present an additive basis for H*(G͠n,3; ℤ2).
期刊介绍：
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca. It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc. The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.
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