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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Classification of skew symmetric matrices


Author: Berndt Brenken
Journal: Proc. Amer. Math. Soc. 108 (1990), 163-169
MSC: Primary 15A72; Secondary 15A21
MathSciNet review: 986646
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Abstract: The group $ {\text{GL(}}d,\mathbb{Z}{\text{) = Aut(}}{\mathbb{Z}^d}{\text{)}}$ acts on the $ \mathbb{Z}$-module $ \operatorname{Hom} {\text{(}}{\Lambda ^2}{\mathbb{Z}^d},\mathbb{Z}/a\mathbb{Z}... ...alpha {\text{)}}\quad {\text{(}}\alpha \in {\text{Aut}}{\mathbb{Z}^d}{\text{)}}$. Associated with each $ \varphi $ in $ \operatorname{Hom} {\text{(}}{\Lambda ^2}{\mathbb{Z}^d},\mathbb{Z}/a\mathbb{Z})$ is a finite set of invariants completely describing the orbit of $ \varphi $ under this action. The result holds with $ \mathbb{Z}$ replaced by an arbitrary commutative principal ideal domain.


References [Enhancements On Off] (What's this?)

  • [1] Berndt Brenken, A classification of some noncommutative tori, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 389–397. MR 1065837, 10.1216/rmjm/1181073114
  • [2] F. G. Frobenius, Theorie der linearen Formen mit ganzen coefficienten, J. Reine Angew. Math. 86 (1880), 96-116.
  • [3] Morris Newman, Integral matrices, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR 0340283

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1990-0986646-4
Keywords: Module, principal ideal domain, pfaffian, skew symmetric matrix, automorphism
Article copyright: © Copyright 1990 American Mathematical Society