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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 $\GL_d(\C)$.

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