# Transactions of the American Mathematical Society

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

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## $m$-symplectic matricesHTML articles powered by AMS MathViewer

by Edward Spence
Trans. Amer. Math. Soc. 170 (1972), 447-457 Request permission

## 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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