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I would like to know the properties of orthogonal matrices and symplectic matrices in terms of the forms they preserve. Could someone please add and/or correct, maybe give some refs/examples?

AFAIK, given a quadratic form q on a vector space V over a field F, there is an associated orthogonal group O(2n) ,a subgroup of GL(n,F), which preserve q; if F is the reals O(2n) preserves q= inner-product and norm (since in R, the norm is induced by the inner-product). Symplectic matrices only preserve symplectic forms, i.e., bilinear,antisymmetric,non-degenerate forms.

Are there relations between these groups; do they overlap, intersect, etc?

I am interested mostly in the case where the field is Z/2.

Thanks

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    Thanks a lot to both, ver$y$ helpful; unfortunately I don't have the status necessary to give you points.2011-04-14

1 Answers 1

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Over the reals the intersection of the orthogonal and symplectic groups in even dimension is isomorphic to the unitary group in the half dimension.

$ U(n) = O(2n, \mathbf{R}) \cap Sp(2n, \mathbf{R})$

This is the 2 out of 3 property expressing the compatibility the symplectic structure with the symmetric bilinear form of the orthogonal group.

The orthogonal group over $Z_2$ is a subgroup of the symplectic group because a symmetric bilinear form is also alternating (since $-1 = +1$).

The full symplectic group $Sp(2n, Z_2)$ can be realized from the action of the $2^n$ dimensional Clifford group on the bits of the binary representation of the basis vectors (up to a phase) as explained in Daniel Gottesmann's paper.

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    Thanks, David, that's what I was looking for.2011-06-22