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- W2026752669 abstract "The symplectic modular group $mathfrak {M}$ is the set of all $2n times 2n$ matrices $M$ with rational integral entries, which satisfy $MJMâ = J,J = left [ {begin {array}{*{20}{c}} 0 & I I & 0 end {array} } right ]$, $I$ 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 $NJNâ = mJ$. In this paper we consider canonical forms for $m$-symplectic matrices under left-multiplication by symplectic modular matrices (corresponding to Hermiteâs normal form) and under both left- and right-multiplication by symplectic modular matrices (corresponding to Smithâs 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 $MJMâ = MâJM = 0$." @default.
- W2026752669 created "2016-06-24" @default.
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- W2026752669 date "1972-01-01" @default.
- W2026752669 modified "2023-09-26" @default.
- W2026752669 title "$m$-symplectic matrices" @default.
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- W2026752669 doi "https://doi.org/10.1090/s0002-9947-1972-0311684-8" @default.
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