For a positive even integer $d$ the **unitary symplectic group** $\mathrm{USp}(d)$ is
the group of unitary transformations of a $d$-dimensional $\C$-vector space equipped with a symplectic form $\Omega$. In other words, the subgroup of $\GL_d(\mathbb C)$ whose elements $A$ satisfy:

- $A^{-1} = \bar A^{\intercal}$ (unitary);
- $A^\intercal \Omega A=\Omega$ (symplectic).

It is a compact real Lie group that can also be viewed as the intersection of $\mathrm{U}(d)$ and $\mathrm{Sp}(d,\C)$ in $\GL_d(\C)$.

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- Review status: beta
- Last edited by Andrew Sutherland on 2021-01-16 14:28:11

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