Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-1972-0311684-8
MathSciNet review: 0311684
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $MJM’ = M’JM = 0$.


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


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 15A21

Retrieve articles in all journals with MSC: 15A21


Additional Information

Keywords: Symplectic modular group, unimodular matrices, canonical forms, elementary divisor theory, multiplicative function
Article copyright: © Copyright 1972 American Mathematical Society