Skip to Main Content

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.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

$m$-symplectic matrices
HTML articles powered by AMS MathViewer

by Edward Spence PDF
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 $MJM’ = M’JM = 0$.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 15A21
  • Retrieve articles in all journals with MSC: 15A21
Additional Information
  • © Copyright 1972 American Mathematical Society
  • 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