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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Classification of skew symmetric matrices
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by Berndt Brenken PDF
Proc. Amer. Math. Soc. 108 (1990), 163-169 Request permission

Abstract:

The group ${\text {GL(}}d,\mathbb {Z}{\text {) = Aut(}}{\mathbb {Z}^d}{\text {)}}$ acts on the $\mathbb {Z}$-module $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z}){\text {by}}\varphi \to \varphi {\text {(}}\alpha \Lambda \alpha {\text {)}}\quad {\text {(}}\alpha \in {\text {Aut}}{\mathbb {Z}^d}{\text {)}}$. 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
  • 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, DOI 10.1216/rmjm/1181073114
  • F. G. Frobenius, Theorie der linearen Formen mit ganzen coefficienten, J. Reine Angew. Math. 86 (1880), 96-116.
  • Morris Newman, Integral matrices, Pure and Applied Mathematics, Vol. 45, Academic Press, New York-London, 1972. MR 0340283
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 108 (1990), 163-169
  • MSC: Primary 15A72; Secondary 15A21
  • DOI: https://doi.org/10.1090/S0002-9939-1990-0986646-4
  • MathSciNet review: 986646