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

Shchegolev, A.

doi:10.1090/spmj/1580

Overgroups of elementary block-diagonal subgroups in the classical symplectic group over an arbitrary commutative ring
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by A. V. Shchegolev
Translated by: the author

St. Petersburg Math. J. 30 (2019), 1007-1041

DOI: https://doi.org/10.1090/spmj/1580

Published electronically: September 16, 2019

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Abstract:

The paper provides 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 relation $\nu$ such that all selfconjugate equivalence classes of $\nu$ are of size at least 4 and all nonselfconjugate classes of $\nu$ are of size at least 5. Namely, given a subgroup $H \ge \mathrm {Ep}(\nu , R)$ of $\mathrm {Sp}(2n, R)$, it is shown that there exists a unique exact major form net of ideals $(\sigma , \Gamma )$ over $R$ such that $\mathrm {Ep}(\sigma , \Gamma ) \le H \le \mathrm {N}_{\operatorname {Sp}(2n,R)}(\mathrm {Sp}(\sigma , \Gamma ))$. Next, the normalizer $\mathrm {N}_{\mathrm {Sp}(2n,R)}(\mathrm {Sp}(\sigma , \Gamma ))$ is described in terms of congruences.

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