Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 15A72, 15A21

Retrieve articles in all journals with MSC: 15A72, 15A21

Additional Information

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