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 $\operatorname{Ep}(\nu, R)$ 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 $\operatorname{Sp}(2n, R)$ such that $\operatorname{Ep}(\nu, R) \le H$ we show that there exists a unique exact major form net of ideals $(\sigma, \Gamma)$ over $R$ such that $\operatorname{Ep}(\sigma, \Gamma) \le H \le N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(\sigma, \Gamma))$. Further, we describe the normalizer $N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(\sigma, \Gamma))$ in terms of congruences.