Overgroups of elementary block-diagonal subgroups in the classical symplectic group over an arbitrary commutative ring

Abstract

In this paper we prove a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $R$ that contain the elementary block-diagonal (or subsystem) subgroup $\mathrm{Ep}(\nu ,R)$ corresponding to a unitary equivalence realation $\nu$ such that all self-conjugate equivalence classes of $\nu$ are of size at least 4 and all not-self-conjugate classes of $\nu$ are of size at least 5. Namely, given a subgroup $H$ of $\mathrm{Sp}(2n,R)$ such that $\mathrm{Ep}(\nu ,R)\le H$ we show that there exists a unique exact major form net of ideals $(\sigma ,\Gamma )$ over $R$ such that $\mathrm{Ep}(\sigma ,\Gamma )\le H\le N_{\mathrm{Sp}(2n,R)}(\mathrm{Sp}(\sigma ,\Gamma ))$. Further, we describe the normalizer $N_{\mathrm{Sp}(2n,R)}(\mathrm{Sp}(\sigma ,\Gamma ))$ in terms of congruences.