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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] B. Brenken, A classification of some noncommutative tori (presented at the Great Plains Operator Theory Seminar, May, 1987 in Lawrence, Kansas, (to appear). MR 1065837 (91e:46099)
  • [2] F. G. Frobenius, Theorie der linearen Formen mit ganzen coefficienten, J. Reine Angew. Math. 86 (1880), 96-116.
  • [3] M. Newman, Integral matrices, Academic Press, 1972. MR 0340283 (49:5038)

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Keywords: Module, principal ideal domain, pfaffian, skew symmetric matrix, automorphism
Article copyright: © Copyright 1990 American Mathematical Society

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