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In Masoud Kamgarpour's paper "Weil Representations" he uses a set of generators for the symplectic group, referring to a book by R. Steinberg which I do not have access to. If it matters at all, I am working in characteristic zero.

After choosing a symplectic basis, the generators can be written \begin{equation} \left( \begin{array}{cc} A & 0 \newline 0 & (A^t)^{-1} \end{array} \right), \ \left( \begin{array}{cc} I & B \newline 0 & I \end{array} \right), \ \text{and} \ \left( \begin{array}{cc} 0 & I \newline -I & 0 \end{array} \right), \end{equation} where $A$ ranges through invertible matrices and $B$ ranges through symmetric matrices. Does anyone know of a reference or an explanation for this, especially a coordinate-free conceptual and/or geometric one?

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Dieudonné's book on the geometry of classical groups argues that for most fields (and surely for all fields of characteristic zero) the symplectic group is generated by symplectic transvections (and that there is a simple bound on the number of these needed to write an element) It references [Dieudonné, Jean Sur les générateurs des groupes classiques. (French) Summa Brasil. Math. 3 (1955), 149–149.] –  Mariano Suárez-Alvarez May 9 '11 at 6:33

2 Answers 2

I think this is essentially a bloc version of LU decompostion (called Bruhat decomposition) : any symplectic bloc upper triangular matrix can be written as a product of the first two (types of matrices), conjugacy by the third gives you bloc lower triangulars. The point being that your symplectic bloc-diagonal matrix are of the of the first matrix you describe. So you can prove this formally by checking Bruhat in GL(2), which is both obvious and geometric (look up flag).

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I don't know if this precisely answers your question, but a study of generators by symplectic transvections for fields of characteristic $\ne 2$ was carried out by methods using graphs in

R. Brown and S.P. Humphries, ``Orbits under symplectic transvections I'', Proc. London Math. Soc. (3) 52 (1986) 517-531.

The main result is: for a symplectic space $V$ with symplectic form $\cdot$ and a subset $S$ of $V$, define a graph $G(S)$ with vertex set $S$ and an edge between $a$ and $b$ if and only if $a\cdot b \ne 0$. Then the transvections corresponding to the elements of $S$ generate the symplectic group of $(V, \cdot)$ if and only if $S$ spans $V$ and $G(S)$ is connected.

(The immediately following sequel did the more complicated case of characteristic $2$.)

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