{"title":"Some remarks on spin-orbits of unit vectors","authors":"Tariq Syed","doi":"10.1016/j.jpaa.2024.107802","DOIUrl":null,"url":null,"abstract":"<div><p>For <span><math><mi>n</mi><mo>∈</mo><mi>N</mi></math></span> and a commutative ring <em>R</em> with <span><math><mn>2</mn><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>×</mo></mrow></msup></math></span>, the group <span><math><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> acts on the set <span><math><mi>U</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of unimodular vectors of length <em>n</em> and <span><math><mi>S</mi><mi>p</mi><mi>i</mi><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> acts on the set of unit vectors <span><math><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. We give an example of a ring for which the comparison map <span><math><mi>U</mi><msub><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>/</mo><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>→</mo><msub><mrow><mi>U</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo><mo>/</mo><mi>S</mi><mi>p</mi><mi>i</mi><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> fails to be bijective.</p></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924001993/pdfft?md5=92a252c768ddb3132dc1cf4aa6995e84&pid=1-s2.0-S0022404924001993-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Pure and Applied Algebra","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022404924001993","RegionNum":2,"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
Abstract
For and a commutative ring R with , the group acts on the set of unimodular vectors of length n and acts on the set of unit vectors . We give an example of a ring for which the comparison map fails to be bijective.
期刊介绍:
The Journal of Pure and Applied Algebra concentrates on that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications.