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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

$ m$-symplectic matrices


Author: Edward Spence
Journal: Trans. Amer. Math. Soc. 170 (1972), 447-457
MSC: Primary 15A21
MathSciNet review: 0311684
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Abstract: The symplectic modular group $ \mathfrak{M}$ is the set of all $ 2n \times 2n$ matrices $ M$ with rational integral entries, which satisfy $ MJM' = J,J = \left[ {\begin{array}{*{20}{c}} 0 & I \\ I & 0 \\ \end{array} } \right]$, $ I$ being the identity $ n \times n$ matrix. Let $ m$ be a positive integer. Then the $ 2n \times 2n$ matrix $ N$ is said to be $ m$-symplectic if it has rational integral entries and if it satisfies $ NJN' = mJ$. In this paper we consider canonical forms for $ m$-symplectic matrices under left-multiplication by symplectic modular matrices (corresponding to Hermite's normal form) and under both left- and right-multiplication by symplectic modular matrices (corresponding to Smith's normal form). The number of canonical forms in each case is determined explicitly in terms of the prime divisors of $ m$. Finally, corresponding results are stated, without proof, for 0-symplectic matrices; these are $ 2n \times 2n$ matrices $ M$ with rational integral entries and which satisfy $ MJM' = M'JM = 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0311684-8
PII: S 0002-9947(1972)0311684-8
Keywords: Symplectic modular group, unimodular matrices, canonical forms, elementary divisor theory, multiplicative function
Article copyright: © Copyright 1972 American Mathematical Society