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

TL;DR

This paper establishes a classification of subgroups in the symplectic group over a commutative ring that contain certain elementary block-diagonal subgroups, using form nets of ideals and describing their normalizers.

Contribution

It proves a sandwich classification theorem for subgroups containing elementary block-diagonal subgroups in symplectic groups over arbitrary rings, introducing the concept of form nets of ideals.

Findings

Existence of a unique form net of ideals associated with such subgroups.

Description of the normalizer in terms of congruences.

Extension of classification results to arbitrary commutative rings.

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.