Abstract
Let $\Pi$ be a polar space of type $\textsf{D}_{n}$. Denote by ${\mathcal G}_{\delta}(\Pi)$, $\delta\in \{+,-\}$ the associated half-spin Grassmannians and write $\Gamma_{\delta}(\Pi)$ for the corresponding half-spin Grassmann graphs. In the case when $n\ge 4$ is even, the apartments of ${\mathcal G}_{\delta}(\Pi)$ will be characterized as the images of isometric embeddings of the half-cube graph $\frac{1}{2}H_n$ in $\Gamma_{\delta}(\Pi)$. As an application, we describe all isometric embeddings of $\Gamma_{\delta}(\Pi)$ in the half-spin Grassmann graphs associated to a polar space of type $\textsf{D}_{n'}$ under the assumption that $n\ge 6$ is even.
Highlights
In the present paper we continue to discuss the problem of metric characterization of apartments in building Grassmannians [11, 12]
All apartments are isomorphic to a certain Coxeter complex, i.e. the simplicial complex associated to a Coxeter system, which defines the type of the building
The vertex set of ∆ can be labeled by the nodes of the diagram corresponding to the associated Coxeter system
Summary
In the present paper we continue to discuss the problem of metric characterization of apartments in building Grassmannians [11, 12]. The polar Grassmannian whose elements are maximal singular subspaces is called the dual polar space and the associated Grassmann graph is known as the dual polar graph. By [12], the apartments in the dual polar space can be characterized as the images of isometric embeddings of the n-dimensional hypercube graph Hn in the dual polar graph. As an application of the main result, we describe all isometric embeddings of the half-spin Grassmann graphs of a polar space of type Dn, where n 6 is even, in the half-spin Grassmann graphs associated to a polar the electronic journal of combinatorics 21(4) (2014), #P4.4 space of type Dn (Theorem 4). Note that in [3] apartments in Grassmannians of finite-dimensional vector spaces, dual polar spaces and half-spin Grassmannians were characterized in terms of independent subsets in the associated partial linear spaces. See [4] for a survey
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