Symplectic spaces over commutative rings and normality of subspace posets
Jun Guo, Fenyan Liu, Yanfang Qi
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Source: Crossref
Published: Oct 8, 2026
DOI: 10.1142/s1793830926501065
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Let [Formula: see text] be the [Formula: see text]-dimensional symplectic space and [Formula: see text] be the corresponding symplectic group of degree [Formula: see text] over a finite commutative ring [Formula: see text]. By using a matrix method, we describe the orbits of subspaces of [Formula: see text] under the action of [Formula: see text], and determine the lengths of these orbits, which is a generalization of similar results of the symplectic space over a finite field. All totally isotropic subspaces of [Formula: see text] form a poset ordered by inclusion, denoted by [Formula: see text]. For a fixed [Formula: see text]-totally isotropic subspace [Formula: see text] of [Formula: see text], each set [Formula: see text] is a subposet of [Formula: see text], where [Formula: see text]. We show that [Formula: see text] has the normalized matching (NM) property, which implies that [Formula: see text] has the strong Sperner property and the LYM property.
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