$m$-symplectic matrices
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- 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
-
C. C. MacDuffee, The theory of matrices, Chelsea, New York, 1946.
- Hans Maass, Die Primzahlen in der Theorie der Siegelschen Modulfunktionen, Math. Ann. 124 (1951), 87–122 (German). MR 47075, DOI 10.1007/BF01343553
- Masao Sugawara, On the transformation theory of Siegel’s modular group of the $n$-th. degree, Proc. Imp. Acad. Tokyo 13 (1937), no. 9, 335–338. MR 1568477
- Edward Spence, Matrix divisors of $mI$, Acta Arith. 20 (1972), 189–201. MR 320032, DOI 10.4064/aa-20-2-189-201
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