\documentclass[12pt]{article} \usepackage{e-jc} \specs{P4.4}{21(4)}{2014} \usepackage{amsthm,amsmath,amssymb} \usepackage{graphicx} \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{rem}{Remark} \newtheorem{exmp}{Example} \newtheorem{cor}{Corollary} \newtheorem{fact}{Fact} \newtheorem*{prob}{Problem} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} \title{\bf Isometric embeddings of half-cube graphs\\ in half-spin Grassmannians} % input author, affilliation, address and support information as follows; % the address should include the country, and does not have to include % the street address \author{Mark Pankov\\ \small Department of Mathematics and Computer Science\\[-0.8ex] \small University of Warmia and Mazury\\[-0.8ex] \small Olsztyn, Poland\\ \small\tt pankov@matman.uwm.edu.pl\\} \date{\dateline{Feb 12, 2014}{Sep 22, 2014}{Oct 2, 2014}\\ \small Mathematics Subject Classifications: 51A50, 51E24} \begin{document} \maketitle \begin{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. \bigskip\noindent \textbf{Keywords:} half-cube graph; half-spin Grassmann graph; isometric embedding. \end{abstract} \section{Introduction} In the present paper we continue to discuss the problem of metric characterization of apartments in building Grassmannians \cite{Pankov2,Pankov3}. This problem is connected to the results obtained in \cite{CKS,Coop-sur,Kasikova1,Kasikova2,Pankov4}. By \cite{Tits}, a {\it building} is a simplicial complex $\Delta$ containing a family of subcomplexes called {\it apartments} and characterized by some properties. 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. We suppose that our building is {\it spherical}, i.e. the associated Coxeter system is finite. Maximal simplices of $\Delta$ are said to be {\it chambers}. They have the same finite cardinality $n$ called the {\it rank} of $\Delta$. %Two chambers are {\it adjacent} if their intersection consists of $n-1$ vertices. The vertex set of $\Delta$ can be labeled by the nodes of the diagram corresponding to the associated Coxeter system (such labeling is unique up to a permutation on the set of nodes). Every set consisting of all vertices labeled by the same node is said to be a {\it Grassmannian} (more general objects are investigated in \cite{Pasini}). The intersection of every chamber and every Grassmannian is a single vertex. Thus the vertex set of $\Delta$ is decomposed in precisely $n$ distinct Grassmannians. The intersections of apartments with one of the building Grassmannians are called {\it apartments in this Grassmannian}. Let ${\mathcal G}$ be one of the building Grassmannians. We say that $a,b\in {\mathcal G}$ are {\it adjacent} if there is a simplex $P$ such that $P\cup\{a\}$ and $P\cup\{b\}$ are chambers. Consider the associated {\it Grassmann graph} $\Gamma$, i.e. the graph whose vertex set is ${\mathcal G}$ and whose edges are pairs of adjacent vertices. Suppose that ${\mathcal A}$ is an apartment of ${\mathcal G}$ and denote by $\Gamma_{\mathcal A}$ the restriction of the graph $\Gamma$ to ${\mathcal A}$. We want to distinguish all cases such that the image of every isometric embedding of $\Gamma_{\mathcal A}$ in $\Gamma$ is an apartment of ${\mathcal G}$. Note that in some cases this does not hold \cite{CS,Pankov2}. By \cite{Tits}, there are the following seven types of {\it irreducible thick spherical buildings} of rank $\ge 3$: three classical types ${\textsf A}_{n}, {\textsf B}_{n}={\textsf C}_{n}, {\textsf D}_{n}$ and four exceptional types ${\textsf F}_{4}, {\textsf E}_{i},i=6,7,8$. Every building of type ${\textsf A}_{n-1}$ is the flag complex of a certain $n$-dimensional vector space $V$ (over a division ring). The associated Grassmannians are ${\mathcal G}_{k}(V)$, where $k\in\{1,\dots,n-1\}$. Each ${\mathcal G}_{k}(V)$ consists of all $k$-dimensional subspaces of $V$ and two elements of ${\mathcal G}_{k}(V)$ are adjacent if their intersection is $(k-1)$-dimensional. The corresponding Grassmann graph is denoted by $\Gamma_{k}(V)$. The case when $k=1,n-1$ is trivial --- any two distinct vertices of $\Gamma_{k}(V)$ are adjacent. Every apartment of ${\mathcal G}_{k}(V)$ is defined by a certain base of $V$: it consists of all $k$-dimensional subspaces spanned by subsets of this base. All apartments of ${\mathcal G}_{k}(V)$ are the images of isometric embeddings of the Johnson graph $J(n,k)$ in $\Gamma_{k}(V)$. However, the image of every isometric embedding of $J(n,k)$ in $\Gamma_{k}(V)$ is an apartment if and only if $n=2k$ \cite{Pankov2}. All buildings of types $\textsf{C}_{n}$ and $\textsf{D}_{n}$ are defined by polar spaces. Every building of type $\textsf{C}_{n}$ is the flag complex formed by singular subspaces of a rank $n$ polar space. The associated Grassmannians are the {\it polar Grassmannians} consisting of singular subspaces of the same dimension. The polar Grassmannian whose elements are maximal singular subspaces is called the {\it dual polar space} and the associated Grassmann graph is known as the {\it dual polar graph}. By \cite{Pankov3}, the apartments in the dual polar space can be characterized as the images of isometric embeddings of the $n$-dimensional hypercube graph $H_{n}$ in the dual polar graph. Every building of type $\textsf{D}_{n}$ can be obtained from a polar space of type $\textsf{D}_{n}$. This construction is known as the {\it oriflamme complex}. The Grassmannians of this building are some of the polar Grassmannians and so-called {\it half-spin Grassmannians}. The Grassmann graphs associated to the half-spin Grassmannians are called the {\it half-spin Grassmann graphs}. In this paper we show that the apartments of the half-spin Grassmannians can be characterized as the images of isometric embeddings of the half-cube graph $\frac{1}{2}H_n$ in the half-spin Grassmann graphs if $n\ge 4$ is even (Theorem \ref{theorem-apart-halfspin}). In the case when $n$ is odd, we conjecture the existence of isometric embeddings of $\frac{1}{2}H_n$ in the half-spin Grassmann graphs whose images are not apartments (Section 6). As an application of the main result, we describe all isometric embeddings of the half-spin Grassmann graphs of a polar space of type $\textsf{D}_{n}$, where $n\ge 6$ is even, in the half-spin Grassmann graphs associated to a polar space of type $\textsf{D}_{n'}$ (Theorem \ref{theorem-half-spin-emb}). Note that in \cite{CKS} 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. Some more general results can be found in \cite{Kasikova1,Kasikova2}. See \cite{Coop-sur} for a survey. \section{Basic definitions and constructions} \subsection{Graphs} The {\it distance} between two vertices in a connected graph is the smallest number $i$ such that there is a path of length $i$ (a path consisting of $i$ edges) connecting these vertices. A path connecting two vertices is said to be a {\it geodesic} if the number of edges in this path is equal to the distance between the vertices. The maximum of all distances between vertices in a graph is called the {\it diameter} of the graph. In the case when the diameter is finite, two vertices are said to be {\it opposite} if the distance between them is equal to the diameter. A subset in the vertex set of a graph is called a {\it clique} if any two distinct elements of this subset are adjacent vertices. Maximal cliques exist and every clique is contained in a certain maximal clique. For finite graph this is trivial and we use Zorn lemma for infinite graphs. An injective mapping of the vertex set of a graph $\Gamma$ to the vertex set of a graph $\Gamma'$ is an {\it embedding} of $\Gamma$ in $\Gamma'$ if it transfers adjacent vertices to adjacent vertices and non-adjacent vertices to non-adjacent vertices. Every surjective embedding is an isomorphism between the graphs. An embedding is said to be {\it isometric} if it preserves the distance between vertices. \subsection{Hypercube and half-cube graphs} The vertex set of the $n$-dimensional {\it hypercube graph} $H_{n}$ is formed by all sequences $(a_{1},\dots,a_{n})$, where each $a_{i}$ is equal to $0$ or $1$. For any two such sequences $a=(a_{1},\dots,a_{n})$ and $b=(b_{1},\dots,b_{n})$ we define the distance $$d(a,b):=|a_{1}-b_{1}|+\dots+|a_{n}-b_{n}|.$$ The sequences $a$ and $b$ are adjacent vertices of $H_{n}$ if $d(a,b)=1$. The graph $H_{n}$ is connected, $d$ is the distance on $H_{n}$ and the diameter of $H_{n}$ is equal to $n$. \begin{lemma}\label{lemma-cube} For every vertex of $H_{n}$ there is unique vertex of $H_{n}$ opposite to this vertex. If $v$ and $w$ are opposite vertices of $H_{n}$ then every vertex of $H_{n}$ belongs to a geodesic connecting $v$ and $w$. \end{lemma} \begin{proof} Easy verification. \end{proof} \begin{rem}\label{rem-cube}{\rm Consider the $(2n)$-element set $J=\{\pm 1,\dots,\pm n\}$. A subset $I\subset J$ is said to be {\it singular} if for every $i\in I$ we have $-i\not\in I$. Every maximal singular subset consists of $n$ elements. The hypercube graph $H_{n}$ can be defined as the graph whose vertex set is formed by all maximal singular subsets. Two such subsets are adjacent vertices of $H_{n}$ if their intersection consists of $n-1$ elements. }\end{rem} The vertex set of $H_{n}$ can be uniquely decomposed in the sum of two disjoint subsets $A_{+}$ and $A_{-}$ such that the distance between any two elements of $A_{\delta}$, $\delta\in \{+,-\}$ is even and the distance between every element of $A_{+}$ and every element of $A_{-}$ is odd. Consider the graph $H_{\delta}$, $\delta\in \{+,-\}$ whose vertex set is $A_{\delta}$ and two vertices of $H_{\delta}$ are adjacent if the distance between them in $H_{n}$ is equal to $2$. The graphs $H_{+}$ and $H_{-}$ are isomorphic. Each of these graphs is called the $n$-dimensional {\it half-cube graph} and denoted by $\frac{1}{2}H_{n}$. The graph $\frac{1}{2}H_{n}$ is connected. If $v$ and $u$ are vertices of $\frac{1}{2}H_{n}$ then the distance between them is equal to $\frac{d(v,u)}{2}$. The diameter of $\frac{1}{2}H_{n}$ is equal to $\frac{n}{2}$ or $\frac{n-1}{2}$ if $n$ is even or odd, respectively. There is the following analogue of Lemma \ref{lemma-cube}. \begin{lemma}\label{lemma-half-cube-even} Suppose that $n$ is even. For every vertex of $\frac{1}{2}H_{n}$ there is a unique vertex of $\frac{1}{2}H_{n}$ opposite to this vertex. If $v$ and $w$ are opposite vertices of $\frac{1}{2}H_{n}$ then every vertex of $\frac{1}{2}H_{n}$ belongs to a geodesic connecting $v$ and $w$. \end{lemma} \begin{proof} Easy verification. \end{proof} If $n$ is odd then the above statement fails and for every vertex of $\frac{1}{2}H_{n}$ there are precisely $n$ vertices of $\frac{1}{2}H_{n}$ opposite to this vertex. \subsection{Partial linear spaces} Consider a non-empty set $P$ whose elements are called {\it points} and a family ${\mathcal L}$ of proper subsets of $P$ called {\it lines}. We say that two or more points are {\it collinear} if there is a line containing all of them. Suppose that the pair $\Pi=(P,{\mathcal L})$ is a {\it partial linear space}, i.e. the following axioms hold: \begin{enumerate} \item[$\bullet$] every line contains at least two points and every point belongs to a line, \item[$\bullet$] for any distinct collinear points $p,q\in P$ there is precisely one line $pq$ containing them. \end{enumerate} A subset $S\subset P$ is a {\it subspace} of $\Pi$ if for any distinct collinear points $p,q\in S$ the line $pq$ is contained in $S$. A subspace is said to be {\it singular} if any two distinct points of the subspace are collinear (by the definition, the empty set and a single point are singular subspaces). Using Zorn lemma, we establish the existence of maximal singular subspaces and the fact that every singular subspace is contained in a certain maximal singular subspace. For every subset $X\subset P$ the minimal subspace containing $X$, i.e. the intersection of all subspaces containing $X$, is called {\it spanned} by $X$ and denoted by $\langle X\rangle$. We say that $X$ is {\it independent} if the subspace $\langle X\rangle$ cannot be spanned by a proper subset of $X$. Let $S$ be a subspace of $\Pi$ (possible $S = P$). An independent subset $X\subset S$ is a {\it base} of $S$ if $\langle X\rangle=S$. The {\it dimension} of $S$ is the smallest cardinality $\alpha$ such that $S$ has a base of cardinality $\alpha+1$. The dimension of the empty set and a single point is equal to $-1$ and $0$ (respectively), lines are $1$-dimensional subspaces. Every $2$-dimensional singular subspace will be called a {\it plane}. Two partial linear spaces $\Pi=(P,{\mathcal L})$ and $\Pi'=(P',{\mathcal L}')$ are {\it isomorphic} if there is a bijection $f:P\to P'$ satisfying $f({\mathcal L})={\mathcal L}'$. Every such bijection is called a {\it collineation} of $\Pi$ to $\Pi'$. \subsection{Polar spaces} By \cite{BuekenhoutCohen-book,Pankov1,Shult-book,Ueberberg}, a {\it polar space} is a partial linear space $\Pi=(P,{\mathcal L})$ satisfying the following axioms: \begin{enumerate} \item[$\bullet$] every line contains at least three points, \item[$\bullet$] there is no point collinear to all points, \item[$\bullet$] if $p\in P$ and $L\in {\mathcal L}$ then $p$ is collinear to one or all points of the line $L$, \item[$\bullet$] any sequence of distinct mutually incident singular subspaces is finite. \end{enumerate} If our polar space contains a singular subspace whose dimension is not less than $2$ then all maximal singular subspaces of $\Pi$ are projective spaces of the same dimension $n\ge 2$ and the number $n+1$ is called the {\it rank} of $\Pi$. The collinearity relation of $\Pi$ will be denoted by $\perp$. For points $p,q\in P$ we write $p\perp q$ if $p$ is collinear to $q$ and $p\not\perp q$ otherwise. Moreover, if $X,Y\subset P$ then $X\perp Y$ means that every point of $X$ is collinear to all points of $Y$. If $X\perp X$ then the subspace $\langle X\rangle$ is singular. For every polar space of rank $n$ one of the following possibilities is realized: \begin{enumerate} \item[$\bullet$] type ${\mathsf C}_{n}$ --- every $(n-2)$-dimensional singular subspace is contained in at least three maximal singular subspaces, \item[$\bullet$] type ${\mathsf D}_{n}$ --- every $(n-2)$-dimensional singular subspace is contained in precisely two maximal singular subspaces. \end{enumerate} All polar spaces of rank $\ge 3$ were described by J. Tits \cite{Tits}. We will focus our attention on polar spaces of type ${\mathsf D}_{n}$, $n\ge 4$. \begin{exmp}\label{exmp-polar-D}{\rm Let $V$ be a $(2n)$-dimensional vector space over a field. Suppose that the characteristic of this field is not equal to $2$ and consider a non-degenerate symmetric bilinear form on $V$ such that maximal totally isotropic subspaces are $n$-dimensional. The associated polar space whose points are $1$-dimensional isotropic subspaces and whose lines are defined by $2$-dimensional totally isotropic subspaces (the line corresponding to a $2$-dimensional totally isotropic subspace $S$ consists of all $1$-dimensional subspaces of $S$) is of type ${\mathsf D}_{n}$. In the case when the characteristic of the field is equal to $2$, we consider a non-defect quadratic form on $V$ such that maximal singular subspaces are $n$-dimensional. The associated polar space (the points are $1$-dimensional singular subspaces and the lines are defined by $2$-dimensional singular subspaces) is also of type ${\mathsf D}_{n}$. }\end{exmp} It follows from Tits's description of polar spaces that every polar space of type ${\mathsf D}_{n}$, $n\ge 4$ is isomorphic to one of the polar spaces presented in Example \ref{exmp-polar-D}. \subsection{Polar Grassmannians} Let $\Pi=(P,{\mathcal L})$ be a polar space of rank $n$. For every $k\in\{0,1,\dots,n-1\}$ we denote by ${\mathcal G}_{k}(\Pi)$ the polar Grassmannian consisting of all $k$-dimensional singular subspaces of $\Pi$; in particular, ${\mathcal G}_{n-1}(\Pi)$ is formed by maximal singular subspaces. The {\it dual polar graph} $\Gamma_{n-1}(\Pi)$ is the graph whose vertex set is ${\mathcal G}_{n-1}(\Pi)$ and two distinct elements of ${\mathcal G}_{n-1}(\Pi)$ are adjacent vertices of $\Gamma_{n-1}(\Pi)$ if their intersection is an $(n-2)$-dimensional singular subspace. The graph $\Gamma_{n-1}(\Pi)$ is connected and the distance $d(S,U)$ between $S,U\in {\mathcal G}_{n-1}(\Pi)$ is equal to $$n-1-\dim(S\cap U).$$ The diameter of $\Gamma_{n-1}(\Pi)$ is equal to $n$ and two vertices of $\Gamma_{n-1}(\Pi)$ are opposite if and only if they are disjoint singular subspaces. Let $S$ and $U$ be singular subspaces of $\Pi$ such that $S\subset U$. If $$\dim Sm$ we denote by $[M\rangle_{k}$ the set of all elements of ${\mathcal G}_{k}(\Pi)$ containing $M$. Now suppose that $m4$. If $n=4$ and there is a star whose image is a special subset then Lemma \ref{lemma4-2} implies that all stars go to special subsets and all special subsets go to stars. In the next section we will use the following fact: if $h$ transfers stars to stars and special subsets to special subsets then for every $S\in {\mathcal G}_{k}(\Pi)$, $k\in \{0,\dots,n-2\}$ spanned by a subset of ${\mathcal F}$ there is $S'\in {\mathcal G}_{k}(\Pi)$ spanned by a subset of the frame associated to ${\mathcal A}'$ such that $$h({\mathcal A}\cap [S\rangle_{\delta})={\mathcal A}'\cap [S'\rangle_{\delta}.$$ \section{Proof of Theorem \ref{theorem-apart-halfspin}} \subsection{Reduction} We show that the general case can be reduced to the case when $m=n$. Let $\Pi=(P,{\mathcal L})$ be a polar space of rank $n$. \begin{lemma}[Lemma 2 in \cite{Pankov3}]\label{lemma5-1red} If $X_{0},\dots, X_{i}$ is a geodesic in $\Gamma_{n-1}(\Pi)$ then $$X_{0}\cap X_{i} \subset X_{j}$$ for every $j\in\{1,\dots,i-1\}$. \end{lemma} Now suppose that $\Pi$ is a polar space of type $\textsf{D}_{n}$ and $\delta \in \{+,-\}$. \begin{lemma}\label{lemma5-2red} If $Y_{0},\dots, Y_{i}$ is a geodesic in $\Gamma_{\delta}(\Pi)$ then $$Y_{0}\cap Y_{i} \subset Y_{j}$$ for every $j\in\{1,\dots,i-1\}$. \end{lemma} \begin{proof} There exists a geodesic $X_{0},\dots, X_{2i}$ in $\Gamma_{n-1}(\Pi)$ such that $X_{2j}=Y_{j}$ for every $j\in\{0,\dots,i\}$. Lemma \ref{lemma5-1red} gives the claim. \end{proof} \begin{lemma}\label{lemma5-3red} If $m$ is an even integer not greater than $n$ then for every isometric embedding $f$ of $\frac{1}{2}H_{m}$ in $\Gamma_{\delta}(\Pi)$ there is an $(n-m-1)$-dimensional singular subspace $M$ such that the image of $f$ is contained in $[M\rangle_{\delta}$, in other words, $f$ can be considered as an isometric embedding of $\frac{1}{2}H_{m}$ in $\Gamma_{\delta}(\Pi_{M})$, where $\Pi_{M}$ is the associated polar space of type {\rm $\textsf{D}_{m}$}. \end{lemma} \begin{proof} Let $v$ and $w$ be opposite vertices of $\frac{1}{2}H_{m}$. Then $$m=2d_{\delta}((f(v),f(w)))=d(f(v),f(w))=n-1-\dim(f(v)\cap f(w))$$ which implies that $$M:=f(v)\cap f(w)$$ is an $(n-m-1)$-dimensional singular subspace. Our statement is a simple consequence of Lemmas \ref{lemma-half-cube-even} and \ref{lemma5-2red}. \end{proof} By Lemma \ref{lemma5-3red}, it is sufficient to prove Theorem \ref{theorem-apart-halfspin} only in the case when $m=n$. \begin{rem}{\rm In the case when $m$ is odd, the graph $\frac{1}{2}H_{m}$ is more complicated (Subsection 2.2) and we cannot prove the latter statement. }\end{rem} \subsection{Proof of Theorem \ref{theorem-apart-halfspin} for the case $m=n$} Let $\Pi=(P,{\mathcal L})$ be a polar space of type $\textsf{D}_{n}$ and let ${\mathcal F}=\{p_{1},\dots,p_{2n}\}$ be a frame of $\Pi$. For every integer $k\in \{1,\dots, n-1\}$ we denote by ${\mathcal A}_{k}$ the associated apartment of ${\mathcal G}_{k}(\Pi)$, i.e. the set formed by all $k$-dimensional singular subspaces of type $$ \langle p_{i_{1}},\dots, p_{i_{k+1}}\rangle. $$ Note that points $p_{i_{1}},\dots, p_{i_{k}}$ span a $(k-1)$-dimensional singular subspace if and only if $$\{i_{1},\dots,i_{k}\}\cap \{\sigma(i_{1}),\dots,\sigma(i_{k})\}=\emptyset,$$ or equivalently, these points are mutually collinear. In this case, we will say that $i_{1},\dots,i_{k}$ form a {\it singular} set. Let $\delta\in \{+,-\}$. Then $${\mathcal A}:={\mathcal A}_{n-1}\cap {\mathcal G}_{\delta}(\Pi)$$ is the apartment of ${\mathcal G}_{\delta}(\Pi)$ associated to ${\mathcal F}$. For every singular set $\{i_{1},\dots,i_{k}\}$ we denote by ${\mathcal A}(i_{1},\dots,i_{k})$ the set of all elements of ${\mathcal A}$ containing $p_{i_{1}},\dots, p_{i_{k}}$, i.e. $${\mathcal A}(i_{1},\dots,i_{k})={\mathcal A}\cap [S\rangle_{\delta},$$ where $S=\langle p_{i_{1}},\dots, p_{i_{k}}\rangle$. It was noted in Subsection 4.2 that the restriction of $\Gamma_{\delta}(\Pi)$ to ${\mathcal A}$ is isomorphic to $\frac{1}{2}H_{n}$. Let $f:{\mathcal A}\to {\mathcal G}_{\delta}(\Pi)$ be an isometric embedding of $\frac{1}{2}H_{n}$ in $\Gamma_{\delta}(\Pi)$. Then $$\dim (X\cap Y)=\dim (f(X)\cap f(Y))$$ for all $X,Y\in {\mathcal A}$. Using induction we show that ${\mathcal X}:=f({\mathcal A})$ is an apartment of ${\mathcal G}_{\delta}(\Pi)$ if $n$ is even. For $n=4$ this is true, see Example \ref{exmp3-1}. Suppose that the statement holds for $n=2l\ge 4$ and consider the case when $n=2(l+1)\ge 6$. For every singular set $\{i_{1},\dots,i_{k}\}$ we define $${\mathcal X}(i_{1},\dots,i_{k}):=f({\mathcal A}(i_{1},\dots,i_{k})).$$ If $i,j$ is a singular pair then the restriction of $\Gamma_{\delta}(\Pi)$ to ${\mathcal A}(i,j)$ is isomorphic to $\frac{1}{2}H_{n-2}$ and Lemma \ref{lemma5-3red} implies the existence of a line $L_{ij}$ such that $${\mathcal X}(i,j)\subset [L_{ij}\rangle_{\delta}.$$ \begin{lemma}\label{lemma5-4} If $i,j$ and $s,t$ are singular pairs satisfying $\{i,j\}\cap \{s,t\}=\emptyset$ then $$L_{ij}\cap L_{st}=\emptyset.$$ \end{lemma} \begin{proof} We choose $X\in {\mathcal A}(i,j)$ and $Y\in {\mathcal A}(s,t)$ such that $X\cap Y=\emptyset$. Then $$f(X)\cap f(Y)=\emptyset.$$ The statement follows from the fact that $f(X)\in {\mathcal X}(i,j)$ and $f(Y)\in {\mathcal X}(s,t)$ contain $L_{ij}$ and $L_{st}$, respectively. \end{proof} Let $i,j$ be a singular pair. Then $\Pi_{p_{i}p_{j}}$ is a polar space of type $\textsf{D}_{n-2}$ and ${\mathcal A}(i,j)$ is an apartment of ${\mathcal G}_{\delta}(\Pi_{p_{i}p_{j}})$. The inductive hypothesis implies that ${\mathcal X}(i,j)$ is an apartment of ${\mathcal G}_{\delta}(\Pi_{L_{ij}})$. Our embedding induces an isomorphism between the restrictions of $\Gamma_{\delta}(\Pi)$ to ${\mathcal A}(i,j)$ and ${\mathcal X}(i,j)$ (they are isomorphic to $\frac{1}{2}H_{n-2}$) and, by Subsection 4.2, we have the following possibilities: \begin{enumerate} \item[(1)] $f$ transfers every star of ${\mathcal A}(i,j)$ to a star of ${\mathcal X}(i,j)$, \item[(2)] $n=6$ and $f$ sends every star of ${\mathcal A}(i,j)$ to a special subset of ${\mathcal X}(i,j)$. \end{enumerate} \begin{lemma}\label{lemma5-5} The second possibility is not realized. \end{lemma} \begin{proof} Suppose that $n=6$ and $f$ transfers every star of ${\mathcal A}(i,j)$ to a special subset of ${\mathcal X}(i,j)$. We take any $t$ such that $i,j,t$ form a singular set. Then ${\mathcal A}(i,j,t)$ is a star of ${\mathcal A}(i,j)$ and, by our assumption, ${\mathcal X}(i,j,t)$ is a special subset of ${\mathcal X}(i,j)$. The latter implies that the intersection of all elements from ${\mathcal X}(i,j,t)$ coincides with $L_{ij}$. On the other hand, ${\mathcal A}(i,j,t)$ is a star in ${\mathcal A}(i,t)$ and ${\mathcal A}(j,t)$. Hence ${\mathcal X}(i,j,t)$ is a special subset in ${\mathcal X}(i,t)$ and ${\mathcal X}(j,t)$. By the above arguments, the lines $L_{it}$ and $L_{jt}$ both coincide with $L_{ij}$. Now we take any $s\ne t$ such that $i,j,s$ form a singular set. Then ${\mathcal A}(i,j,s)$ is a star of ${\mathcal A}(i,j)$ and ${\mathcal X}(i,j,s)$ is a special subset of ${\mathcal X}(i,j)$. As above, we establish that $L_{is}$ coincides with $L_{ij}$. Thus $L_{is}=L_{jt}$ which contradicts Lemma \ref{lemma5-4}. \end{proof} %As above, we suppose that $i,j$ is an admissible pair. Lemma \ref{lemma5-5} shows that $f$ transfers every star of ${\mathcal A}(i,j)$ to a star of ${\mathcal X}(i,j)$. Then, by Subsection 4.2, for every $t$ such that $\{i,j,t\}$ is a singular set there is a plane $S_{ij}(t)$ containing $L_{ij}$ and such that ${\mathcal X}(i,j,t)$ consists of all elements of ${\mathcal X}(i,j)$ containing $S_{ij}(t)$. The intersection of all elements from ${\mathcal X}(i,j,t)$ coincides with $S_{ij}(t)$. Since $${\mathcal A}(i,j,t)= {\mathcal A}(i,j)\cap {\mathcal A}(i,t)\cap {\mathcal A}(j,t),$$ we have $${\mathcal X}(i,j,t)={\mathcal X}(i,j)\cap{\mathcal X}(i,t)\cap{\mathcal X}(j,t)$$ and $$S_{ij}(t)=S_{it}(j)=S_{jt}(i).$$ In what follows this plane will be denoted by $S_{ijt}$. The following fact is obvious. \begin{lemma}\label{lemma5-6} If $i,j$ is a singular pair then all $S_{ijt}$ with $t\not\in \{i,j,\sigma(i),\sigma(j)\}$ form a frame of the polar space $\Pi_{L_{ij}}$ and ${\mathcal X}(i,j)$ is the apartment of ${\mathcal G}_{\delta}(\Pi_{L_{ij}})$ associated to this frame. \end{lemma} The plane $S_{ijt}$ contains the lines $L_{ij},L_{it},L_{jt}$. Hence for any pair of these lines one of the following possibilities is realized: the lines are coincident or they are distinct and span $S_{ijt}$. \begin{lemma}\label{lemma5-7} If $i,j,t$ form a singular set then the lines $L_{ij},L_{it},L_{jt}$ are mutually distinct, i.e. any two of these lines span the plane $S_{ijt}$. \end{lemma} \begin{proof} Suppose that $L_{ij}=L_{it}$. Consider $X\in {\mathcal A}(i,j)$ and $Y\in {\mathcal A}(i,t)$ such that $$X\cap Y=p_{i}p_{s}\;\mbox{ and }\;s\ne j,t.$$ Then $X,Y\in {\mathcal A}(i,s)$ and $f(X),f(Y)\in {\mathcal X}(i,s)$. Since $f(X)\cap f(Y)$ is a line, we have $$ f(X)\cap f(Y)=L_{is}. $$ On the other hand, $f(X)\in {\mathcal X}(i,j)$ contains $L_{ij}$ and $f(Y)\in {\mathcal X}(i,t)$ contains $L_{it}$. This means that the line $L_{ij}=L_{it}$ is contained in both $f(X)$ and $f(Y)$, i.e. it coincides with $L_{is}$. Therefore, the lines $L_{is}$ and $L_{jt}$ have a non-empty intersection ($L_{is}$ coincides with $L_{ij}$ and the latter line has a non-empty intersection with $L_{jt}$) which is impossible by Lemma \ref{lemma5-4}. \end{proof} Consider the graph $\Gamma_{1}(\Pi)$ whose vertex set is ${\mathcal G}_{1}(\Pi)$ and two lines are adjacent vertices of $\Gamma_{1}(\Pi)$ if they span a plane. By \cite[Proposition 4.16]{Pankov1}, maximal cliques of this graph are of the following two types: \begin{enumerate} \item[$\bullet$] the {\it star} $[p, U]_{1}$, where $p$ is a point of $U\in {\mathcal G}_{n-1}(\Pi)$; \item[$\bullet$] the {\it top} $\langle T]_{1}$, where $T\in {\mathcal G}_{2}(\Pi)$. \end{enumerate} An easy verification shows that the restriction of the graph $\Gamma_{1}(\Pi)$ to ${\mathcal A}_{1}$ has precisely the following two types of maximal cliques: \begin{enumerate} \item[$\bullet$] the {\it star} ${\mathcal A}_{1}\cap[p_{i}, U]_{1}$, where $U\in {\mathcal A}_{n-1}$ contains $p_{i}$; \item[$\bullet$] the {\it top} ${\mathcal A}_{1}\cap \langle T]_{1}$, where $T\in {\mathcal A}_{2}$. \end{enumerate} Denote by $g$ the mapping of ${\mathcal A}_{1}$ to ${\mathcal G}_{1}(\Pi)$ sending every line $p_{i}p_{j}$ to the line $L_{ij}$. It follows from Lemma \ref{lemma5-7} that $g$ transfers adjacent vertices of $\Gamma_{1}(\Pi)$ to adjacent vertices of $\Gamma_{1}(\Pi)$. Then $g$ sends every star of ${\mathcal A}_{1}$ to a subset in a star or a top of ${\mathcal G}_{1}(\Pi)$. We cannot state that $g$ is injective, but its restriction to a star or a top of ${\mathcal A}_{1}$ is injective. \begin{lemma}\label{lemma5-8} Let $U\in {\mathcal A}$ and $p_{i}\in U$. Then \begin{equation}\label{eq5-1} g({\mathcal A}_{1}\cap [p_{i},U]_{1}) \end{equation} is contained in a star of ${\mathcal G}_{1}(\Pi)$. \end{lemma} \begin{proof} Let $J$ be the set of all $j\ne i$ such that $p_{j}\in U$. Then ${\mathcal A}_{1}\cap [p_{i},U]_{1}$ consists of all lines $p_{i}p_{j}$ with $j\in J$ and \eqref{eq5-1} is formed by all $L_{ij}$ with $j\in J$. Now we fix $j\in J$ and consider $f(U)$ as an element of ${\mathcal X}(i,j)$. Since $U$ belongs to ${\mathcal A}(i,j,t)$ for every $t\in J\setminus \{j\}$, the planes $S_{ijt}$, $t\in J\setminus \{j\}$ are contained in $f(U)$ and Lemma \ref{lemma5-6} implies that $f(U)$ is spanned by these planes. Then Lemma \ref{lemma5-7} guarantees that $f(U)$ is spanned by all lines belonging to \eqref{eq5-1}. This means that \eqref{eq5-1} cannot be contained in a top and we get the claim. \end{proof} Let $U$ and $T$ be distinct elements of ${\mathcal A}$ containing $p_{i}$. Lemma \ref{lemma5-8} implies the existence of points $q_{i}(U),q_{i}(T)$ and maximal singular subspaces $U',T'$ such that $$g({\mathcal A}_{1}\cap [p_{i},U]_{1})\subset [q_{i}(U),U']_{1}$$ and $$g({\mathcal A}_{1}\cap [p_{i},T]_{1})\subset [q_{i}(T),T']_{1}.$$ If $U$ and $T$ are adjacent vertices of the graph $\Gamma_{\delta}(\Pi)$ then $$\dim(U\cap T)=n-3\ge 3$$ and $$({\mathcal A}_{1}\cap [p_{i},U]_{1})\cap({\mathcal A}_{1}\cap [p_{i},T]_{1})$$ contains more than one element. The same holds for $$[q_{i}(U),U']_{1}\cap [q_{i}(T),T']_{1}$$ (since the restriction of $g$ to every star is injective). This means that $q_{i}(U)=q_{i}(T)$ (if $q_{i}(U)$ and $q_{i}(T)$ are distinct then the latter intersection contains at most one element). The graph $\Gamma_{\delta}(\Pi)$ is connected and the equality $q_{i}(U)=q_{i}(T)$ holds for any $U,T\in {\mathcal A}$ containing $p_i$. Since every line $p_{i}p_{j}$ is contained in a certain $U\in {\mathcal A}$, we get the following: for every point $p_{i}$ there is a point $q_{i}$ such that $$g({\mathcal A}_{1}\cap [p_{i}\rangle_{1})\subset [q_{i}\rangle_{1}.$$ The point $q_{i}$ is the intersection of all lines $L_{ij}$. We show that $q_{i}\ne q_{j}$ if $i\ne j$. The lines $L_{is}$ and $L_{jt}$ contain $q_{i}$ and $q_{j}$ (respectively) and, by Lemma \ref{lemma5-4}, these lines are disjoint if $\{i,s\}\cap \{j,t\}=\emptyset$. Thus $L_{ij}$ is spanned by $q_{i}$ and $q_{j}$. Then for every $U\in {\mathcal A}$ the subspace $f(U)$ is spanned by all $q_{i}$ such that $p_{i}\in U$ (see the proof of Lemma \ref{lemma5-8}). So, the required statement is a simple consequence of the following. \begin{lemma}\label{lemma5-9} The points $q_{1},\dots,q_{2n}$ form a frame of $\Pi$. \end{lemma} To prove Lemma \ref{lemma5-9} we need some elementary properties of collinearity relation in polar spaces. \begin{lemma}\label{lemma5-10} The following assertions are fulfilled: \begin{enumerate} \item[(1)] if $p\in P$ and $X\subset P$ then $p\perp X$ implies that $p\perp \langle X\rangle$, \item[(2)] if $p\in P$ and $S$ is a maximal singular subspace then $p\perp S$ implies that $p\in S$. \end{enumerate} \end{lemma} \begin{proof} See, for example, \cite[Section 4.1]{Pankov1}. \end{proof} \begin{proof}[Proof of Lemma \ref{lemma5-9}] We have $q_{i}\perp q_{j}$ if $j\ne \sigma(i)$ (since these points span the line $L_{ij}$). Suppose that $q_{i}\perp q_{\sigma(i)}$ for a certain $i$. Then $q_{\sigma(i)}$ is collinear to all points $q_{j}$. Every element of ${\mathcal X}$ is spanned by some $q_{j}$ and the first statement of Lemma \ref{lemma5-10} guarantees that $q_{\sigma(i)}\perp S$ for every $S\in {\mathcal X}$. By the second statement of Lemma \ref{lemma5-10}, every element of ${\mathcal X}$ contains $q_{\sigma(i)}$ and any two elements of ${\mathcal X}$ have a non-empty intersection. The latter contradicts the fact that ${\mathcal X}$ is the image of an isometric embedding of $\frac{1}{2}H_{n}$ in $\Gamma_{\delta}(\Pi)$. \end{proof} \section{Conjecture} In this section we will suppose that $\Pi=(P,{\mathcal L})$ is a polar space of type $\textsf{D}_n$, where $n$ is an odd integer greater than $4$. Let ${\mathcal A}$ be the apartment of ${\mathcal G}_{\delta}(\Pi)$, $\delta\in \{+,-\}$ defined by a frame ${\mathcal F}=\{p_{1},\dots,p_{2n}\}$. Let also $f:{\mathcal A}\to {\mathcal G}_{\delta}(\Pi)$ be an isometric embedding of $\frac{1}{2}H_{n}$ in $\Gamma_{\delta}(\Pi)$. For every $i\in \{1,\dots,2n\}$ the restriction of $\Gamma_{\delta}(\Pi)$ to ${\mathcal A}\cap [p_{i}\rangle_{\delta}$ is isomorphic to $\frac{1}{2}H_{n-1}$ and Lemma \ref{lemma5-3red} implies the existence of a point $q_{i}$ such that $$f({\mathcal A}\cap [p_{i}\rangle_{\delta})\subset [q_{i}\rangle_{\delta}.$$ We can prove that there are precisely the following possibilities: \begin{enumerate} \item[(1)] all $q_{i}$ are coincident, \item[(2)] all $q_{i}$ are mutually distinct. \end{enumerate} In the second case, an easy verification shows that all $q_{i}$ form a frame and $f({\mathcal A})$ is the apartment of ${\mathcal G}_{\delta}(\Pi)$ associated to this frame. We conjecture that the first possibility is realized, i.e. there exists an isometric embedding of $\frac{1}{2}H_{n}$ in $\Gamma_{\delta}(\Pi)$ whose image is not an apartment of ${\mathcal G}_{\delta}(\Pi)$. In the first case, $f$ can be considered as an isometric embedding of $\frac{1}{2}H_{n}$ in $\Gamma_{\delta}(\Pi_{q})$, where $q=q_{1}=\dots=q_{2n}$. The rank of the polar space $\Pi_{q}$ is equal to $n-1$. Thus the graphs $\frac{1}{2}H_{n}$ and $\Gamma_{\delta}(\Pi_{q})$ are of the same diameter $\frac{n-1}{2}$. This trivial observation supports our conjecture. \section{Isometric embeddings of half-spin Grassmann \\ graphs} \subsection{Application of Theorem \ref{theorem-apart-halfspin}} Let $\Pi=(P,{\mathcal L})$ and $\Pi'=(P',{\mathcal L}')$ be polar spaces of rank $n$ and $n'$, respectively. The existence of isometric embeddings of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n'-1}(\Pi')$ implies that the diameter of $\Gamma_{n-1}(\Pi)$ is not greater than the diameter of $\Gamma_{n'-1}(\Pi')$, i.e. $n\le n'$. First, we consider the case when $n=n'$. Let $f:P\to P'$ be a mapping which transfers every frame of $\Pi$ to a frame of $\Pi'$. By \cite[Subsection 4.9.6]{Pankov1}, $f$ sends lines of $\Pi$ to subsets contained in lines of $\Pi'$; moreover, if $S$ is a singular subspace of $\Pi$ then $f(S)$ spans a singular subspace of $\Pi'$ whose dimension is equal to the dimension of $S$. The mapping which transfers every $X\in {\mathcal G}_{n-1}(\Pi)$ to $\langle f(X)\rangle$ is an isometric embedding of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n-1}(\Pi')$. If $\Pi$ and $\Pi'$ both are polar spaces of type $\textsf{D}_{n}$ then the restriction of this mapping to ${\mathcal G}_{\delta}(\Pi)$, $\delta \in \{+,-\}$ is an isometric embedding of $\Gamma_{\delta}(\Pi)$ in $\Gamma_{\gamma}(\Pi')$, $\gamma \in \{+,-\}$. Suppose that $n\le n'$ and $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. Then $\Pi'_{M}$ is a polar space of rank $n$ (if $n=n'$ then $M=\emptyset$ and $\Pi'_{M}$ coincides with $\Pi'$). Every frame preserving mapping of $\Pi$ to $\Pi'_{M}$ induces an isometric embedding of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n-1}(\Pi'_{M})$. This is an isometric embedding of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n'-1}(\Pi')$, since $\Gamma_{n-1}(\Pi'_{M})$ can be naturally identified with the restriction of $\Gamma_{n'-1}(\Pi')$ to $[M\rangle_{n'-1}$. In the case when $\Pi$ and $\Pi'$ are polar spaces of type $\textsf{D}_{n}$ and $\textsf{D}_{n'}$ (respectively), we get an isometric embedding of $\Gamma_{\delta}(\Pi)$, $\delta \in \{+,-\}$ in $\Gamma_{\gamma}(\Pi')$, $\gamma \in \{+,-\}$. \begin{theorem}[\cite{Pankov3}]\label{theorem-dual-polar-emb} Every isometric embedding of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n'-1}(\Pi')$ is induced by a frame preserving mapping of $\Pi$ to $\Pi'_{M}$, where $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. \end{theorem} As an application of Theorem \ref{theorem-apart-halfspin} we prove the following. \begin{theorem}\label{theorem-half-spin-emb} Suppose that $\Pi$ and $\Pi'$ are polar spaces of type {\rm $\textsf{D}_{n}$} and {\rm $\textsf{D}_{n'}$}, respectively. If $n$ is an even integer not less than $6$ then every isometric embedding of $\Gamma_{\delta}(\Pi)$, $\delta \in \{+,-\}$ in $\Gamma_{\gamma}(\Pi')$, $\gamma \in \{+,-\}$ is induced by a frame preserving mapping of $\Pi$ in $\Pi'_{M}$, where $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. \end{theorem} \subsection{Proof of Theorem \ref{theorem-half-spin-emb}} Suppose that $f:{\mathcal G}_{\delta}(\Pi)\to {\mathcal G}_{\gamma}(\Pi')$ is an isometric embedding of $\Gamma_{\delta}(\Pi)$ in $\Gamma_{\gamma}(\Pi')$ and $n$ is even. Then $f$ transfers maximal cliques of $\Gamma_{\delta}(\Pi)$ (stars and special subspaces) to subsets in maximal cliques of $\Gamma_{\gamma}(\Pi')$. \begin{lemma}\label{lemma7-1} The following assertions are fulfilled: \begin{enumerate} \item[{\rm (1)}] distinct maximal cliques go to subsets of distinct maximal cliques, \item[{\rm (2)}] the image of every maximal clique of $\Gamma_{\delta}(\Pi)$ is contained in a unique maximal clique of $\Gamma_{\gamma}(\Pi')$. \end{enumerate} \end{lemma} \begin{proof} (1). If ${\mathcal X}$ and ${\mathcal Y}$ are distinct maximal cliques of $\Gamma_{\delta}(\Pi)$ whose images are contained in the same maximal clique of $\Gamma_{\gamma}(\Pi')$ then there exist non-adjacent vertices $X\in {\mathcal X}$ and $Y\in {\mathcal Y}$ such that $f(X)$ and $f(Y)$ are adjacent vertices of $\Gamma_{\gamma}(\Pi')$. This contradicts the fact that $f$ is an embedding. (2). Let ${\mathcal C}$ be a maximal clique of $\Gamma_{\delta}(\Pi)$ and let $S$ be the associated singular subspace of $\Pi$ (an element of ${\mathcal G}_{n-4}(\Pi)$ or ${\mathcal G}_{-\delta}(\Pi)$). By \cite[Proposition 4.7]{Pankov1}, there is a frame ${\mathcal F}$ such that $S$ is spanned by a subset of ${\mathcal F}$. Let ${\mathcal A}$ be the apartment of ${\mathcal G}_{\delta}(\Pi)$ defined by ${\mathcal F}$. Then ${\mathcal A}\cap {\mathcal C}$ is a base of the projective space ${\mathcal C}$ (see Subsection 4.2). Theorem \ref{theorem-apart-halfspin} states that $f({\mathcal A})$ is an apartment of ${\mathcal G}_{\delta}(\Pi'_{M})$, where $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. This guarantees that $f({\mathcal A}\cap {\mathcal C})$ is an independent subset in every maximal clique of $\Gamma_{\gamma}(\Pi')$ containing $f({\mathcal C})$ and the dimension of the subspace spanned by $f({\mathcal A}\cap {\mathcal C})$ is not less than $3$. Now suppose that $f({\mathcal C})$ is contained in two distinct maximal cliques of $\Gamma_{\gamma}(\Pi')$. If these cliques are of the same type then $f({\mathcal C})$ is contained in a line of ${\mathcal G}_{\gamma}(\Pi')$. If one of the cliques is a star and the other is a special subspace then $f({\mathcal C})$ spans a proper subspace in the star and the dimension of this subspace is not greater than $2$. Each of these possibilities contradicts the fact established above. \end{proof} \begin{lemma}\label{lemma7-2} One of the following possibilities is realized: \begin{enumerate} \item[{\rm (A)}] stars go to subsets of stars and special subspaces go to subsets of special subspaces, \item[{\rm (B)}] stars go to subsets of special subspaces and special subspaces go to subsets of stars. \end{enumerate} The second possibility can be realized only in the case when $n=4$. \end{lemma} \begin{proof} Let ${\mathcal C}$ and ${\mathcal C}'$ be maximal cliques of $\Gamma_{\delta}(\Pi)$ such that ${\mathcal C}\cap {\mathcal C}'$ is a plane. Then one of these cliques is a star $[M\rangle_{\delta}$, $M\in {\mathcal G}_{n-4}(\Pi)$ and the other is a special subspace $[U]_{\delta}$, $U\in {\mathcal G}_{-\delta}(\Pi)$. There is a frame ${\mathcal F}$ such that $M$ and $U$ are spanned by subsets of ${\mathcal F}$ \cite[Proposition 4.7]{Pankov1}. Let ${\mathcal A}$ be the apartment of ${\mathcal G}_{\delta}(\Pi)$ defined by ${\mathcal F}$. Then $${\mathcal A}\cap {\mathcal C}\cap {\mathcal C}'$$ is a base of the plane ${\mathcal C}\cap {\mathcal C}'$. By Theorem \ref{theorem-apart-halfspin}, $f({\mathcal A})$ is an apartment of ${\mathcal G}_{\delta}(\Pi'_{M})$, where $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. This means that $$f({\mathcal A}\cap {\mathcal C}\cap {\mathcal C}')$$ is an independent subset in the intersection of the maximal cliques of $\Gamma_{\gamma}(\Pi')$ containing $f({\mathcal C})$ and $f({\mathcal C}')$. Since this set contains three elements, the dimension of the intersection is not less than $2$. Therefore, the intersection of the maximal cliques of $\Gamma_{\gamma}(\Pi')$ containing $f({\mathcal C})$ and $f({\mathcal C}')$ is a plane. Lemma \ref{lemma7-1} implies that $f$ induces an injective mapping of the set of maximal cliques of $\Gamma_{\delta}(\Pi)$ to the set of maximal cliques of $\Gamma_{\gamma}(\Pi')$. Using Lemma \ref{lemma4-1}, we show that one of the possibilities (A) or (B) is realized. Now suppose that $n\ge 6$. Let $U\in {\mathcal G}_{-\delta}(\Pi)$. We take a frame ${\mathcal F}$ such that $U$ is spanned by a subset of ${\mathcal F}$ and denote by ${\mathcal A}$ the associated apartment of ${\mathcal G}_{\delta}(\Pi)$. Then ${\mathcal A}\cap [U]_{\delta}$ is a base of the projective space $[U]_{\delta}$. As above, we establish that $f({\mathcal A}\cap [U]_{\delta})$ is an independent subset in the maximal clique of $\Gamma_{\gamma}(\Pi')$ containing $f([U]_{\delta})$. Since $f({\mathcal A}\cap [U]_{\delta})$ consists of $n$ elements and $n>4$, the maximal clique containing $f([U]_{\delta})$ cannot be a star and the possibility (B) is not realized. \end{proof} From this moment we suppose that $f$ satisfies (A). Then for every $U\in {\mathcal G}_{-\delta}(\Pi)$ there is unique $U'\in {\mathcal G}_{-\gamma}(\Pi')$ such that $$f([U]_{\delta})\subset [U']_{\gamma}.$$ We set $f(U):=U'$ %for every $U\in {\mathcal G}_{-\delta}(\Pi)$ and extend $f$ to a mapping of ${\mathcal G}_{n-1}(\Pi)$ to ${\mathcal G}_{n'-1}(\Pi')$. The first part of Lemma \ref{lemma7-1} guarantees that the extension is injective. \begin{lemma}\label{lemma7-3} The following assertions are fulfilled: \begin{enumerate} \item[{\rm (1)}] $f$ transfers adjacent vertices of $\Gamma_{n-1}(\Pi)$ to adjacent vertices of $\Gamma_{n'-1}(\Pi')$, \item[{\rm (2)}] $d(X,Y)=d(f(X),f(Y))$ if $X,Y\in {\mathcal G}_{\delta}(\Pi)$. \end{enumerate} \end{lemma} \begin{proof} (1). If $X\in {\mathcal G}_{\delta}(\Pi)$ and $Y\in {\mathcal G}_{-\delta}(\Pi)$ are adjacent vertices of $\Gamma_{n-1}(\Pi)$ then $X$ belongs to $[Y]_{\delta}$ and $$f(X)\in [f(Y)]_{\gamma}$$ which implies that $f(X)$ and $f(Y)$ are adjacent vertices of $\Gamma_{n'-1}(\Pi')$. (2). This follows from the fact that the restriction of $f$ to ${\mathcal G}_{\delta}(\Pi)$ is an isometric embedding of $\Gamma_{\delta}(\Pi)$ in $\Gamma_{\gamma}(\Pi')$. \end{proof} By Theorem \ref{theorem-dual-polar-emb}, it is sufficient to show that $f$ is an isometric embedding of $\Gamma_{n-1}(\Pi)$ in $\Gamma_{n'-1}(\Pi')$. Since for any two distinct elements of ${\mathcal G}_{n-1}(\Pi)$ there is an apartment containing them \cite[Proposition 4.7]{Pankov1}, we need to show that the restriction of $f$ to every apartment of ${\mathcal G}_{n-1}(\Pi)$ is an isometric embedding of $H_{n}$ in $\Gamma_{n'-1}(\Pi')$. Let ${\mathcal A}$ be an apartment of ${\mathcal G}_{n-1}(\Pi)$. Then $${\mathcal A}_{\delta}:={\mathcal A}\cap {\mathcal G}_{\delta}(\Pi)$$ is an apartment of ${\mathcal G}_{\delta}(\Pi)$ and ${\mathcal A}\setminus {\mathcal A}_{\delta}$ is an apartment of ${\mathcal G}_{-\delta}(\Pi)$. Every special subset of ${\mathcal A}_{\delta}$ is defined by an element of ${\mathcal A}\setminus {\mathcal A}_{\delta}$, i.e. it is the intersection of ${\mathcal A}_{\delta}$ and $[U]_{\delta}$, where $U\in{\mathcal A}\setminus {\mathcal A}_{\delta}$. Theorem \ref{theorem-apart-halfspin} states that $f({\mathcal A}_{\delta})$ is an apartment of ${\mathcal G}_{\gamma}(\Pi'_{M})$, where $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. Let ${\mathcal A}'$ be the apartment of ${\mathcal G}_{n-1}(\Pi'_{M})$ containing $f({\mathcal A}_{\delta})$. Then ${\mathcal A}'\setminus f({\mathcal A}_{\delta})$ is an apartment of ${\mathcal G}_{-\gamma}(\Pi'_{M})$ and every special subset of $f({\mathcal A}_{\delta})$ is defined by an element of ${\mathcal A}'\setminus f({\mathcal A}_{\delta})$. Since $f$ transfers special subsets of ${\mathcal A}_{\delta}$ to special subsets of $f({\mathcal A}_{\delta})$, we have $$f({\mathcal A}\setminus {\mathcal A}_{\delta})={\mathcal A}'\setminus f({\mathcal A}_{\delta})$$ which implies that $f({\mathcal A})={\mathcal A}'$. Now we show that \begin{equation}\label{eq7-1} d(X,Y)=d(f(X),f(Y)) \end{equation} for all $X,Y\in {\mathcal A}$. By the second part of Lemma \ref{lemma7-3}, this is true if $X,Y\in {\mathcal A}_{\delta}$. In the case when $X\in {\mathcal A}\setminus{\mathcal A}_{\delta}$ and $Y\in {\mathcal A}_{\delta}$, we have $d(X,Y)\le n-1$ and there is a geodesic $Y_{0},Y_{1},\dots, Y_{i}$ in $\Gamma_{n-1}(\Pi)$ such that $$Y_{0}\in {\mathcal A}_{\delta},\;\;Y_{1}=X,\;\;Y_{i}=Y.$$ Lemma \ref{lemma7-3} shows that $$d(Y_{0},Y_{i})=d(f(Y_{0}),f(Y_{i}))$$ and $$f(Y_{0}),f(Y_{1}),\dots, f(Y_{i})$$ is a geodesic in $\Gamma_{n'-1}(\Pi')$ which implies \eqref{eq7-1}. If $X,Y\in{\mathcal A}\setminus{\mathcal A}_{\delta}$ and $d(X,Y)\le n-2$ then there is a geodesic $Y_{0},Y_{1},\dots, Y_{i}$ in $\Gamma_{n-1}(\Pi)$ such that $$Y_{0},Y_{i}\in {\mathcal A}_{\delta},\;\;Y_{1}=X\;\;Y_{i-1}=Y.$$ As above, the image of this geodesic is a geodesic in $\Gamma_{n'-1}(\Pi)$ and we get \eqref{eq7-1}. Consider the case when $X,Y\in{\mathcal A}\setminus{\mathcal A}_{\delta}$ are opposite vertices of $\Gamma_{n-1}(\Pi)$. It is clear that $${\mathcal A}\setminus\{Y\}=\{Z\in {\mathcal A}:d(X,Z)\le n-1\}$$ There is unique $Y'\in {\mathcal A}'$ satisfying $d(f(X),Y')=n$ and $${\mathcal A}'\setminus \{Y'\}=\{Z'\in {\mathcal A}':d(f(X),Z')\le n-1\}.$$ It follows from the above arguments that $$f({\mathcal A}\setminus\{Y\})={\mathcal A}'\setminus \{Y'\}.$$ Then $f(Y)=Y'$ (since $f({\mathcal A})={\mathcal A}'$) and \eqref{eq7-1} holds. So, we get the following. \begin{prop}\label{prop7-1} If $\Pi$ and $\Pi'$ are polar spaces of type {\rm $\textsf{D}_{n}$} and {\rm $\textsf{D}_{n'}$}{\rm (}respectively{\rm )} and $n$ is even then every isometric embedding of $\Gamma_{\delta}(\Pi)$, $\delta \in \{+,-\}$ in $\Gamma_{\gamma}(\Pi')$, $\gamma \in \{+,-\}$ satisfying {\rm (A)} is induced by a frame preserving mapping of $\Pi$ in $\Pi'_{M}$, where $M$ is an $(n'-n-1)$-dimensional singular subspace of $\Pi'$. \end{prop} Theorem \ref{theorem-half-spin-emb} is a direct consequence of Proposition \ref{prop7-1} and Lemma \ref{lemma7-2}. \subsection{Remarks on the case $n=4$} Suppose that $\Pi$ is a polar space of type $\textsf{D}_4$. Then the associated half-spin Grassmannians (together with the families of lines) also are polar spaces of type $\textsf{D}_4$. These polar spaces are denoted by $\Pi_{+}$ and $\Pi_{-}$ (Example \ref{exmp3-1}). Every maximal singular subspace of the polar space $\Pi_{\delta}$, $\delta\in \{+,-\}$ is a star (defined by a point of $\Pi$) or a special subspace (associated to an element of ${\mathcal G}_{-\delta}(\Pi)$). Using the intersection properties given in Section 4.1, we can show that one of the half-spin Grassmannians of $\Pi_{\delta}$ consists of all stars and the other is formed by all special subspaces. The corresponding polar spaces can be identified with $\Pi$ and $\Pi_{-\delta}$. By Remark \ref{rem-halfspin}, they are isomorphic. So, the polar spaces $\Pi$ and $\Pi_{\delta}$, $\delta\in \{+,-\}$ are isomorphic. Every collineation between $\Pi$ and $\Pi_{-\delta}$ induces a collineation of $\Pi_{\delta}$ to itself (an automorphism of $\Gamma_{\delta}(\Pi)$) which transfers stars to special subspaces and special subspaces to stars. Let $f$ be an isometric embedding of $\Gamma_{\delta}(\Pi)$, $\delta \in \{+,-\}$ in $\Gamma_{\gamma}(\Pi')$, $\gamma \in \{+,-\}$ satisfying {\rm (B)}. We take any automorphism $h$ of $\Gamma_{\delta}(\Pi)$ sending stars to special subspaces and special subspaces to stars. Then $fh$ is an isometric embedding of $\Gamma_{\delta}(\Pi)$ in $\Gamma_{\gamma}(\Pi')$ satisfying {\rm (A)} and we can apply Proposition \ref{prop7-1}. \subsection{Remarks on frame preserving mappings} In this subsection we suppose that $\Pi=(P,{\mathcal L})$ and $\Pi'=(P',{\mathcal L}')$ are polar spaces of an arbitrary type and embedable in the projective spaces $\Pi_{V}$ and $\Pi_{V'}$ associated to vector spaces $V$ and $V'$, respectively. It is well-known that every collineation of $\Pi$ to $\Pi'$ can be extended to a collineation of $\Pi_{V}$ to $\Pi_{V'}$ and, by the Fundamental Theorem of Projective Geometry, the latter collineation is induced by a semilinear isomorphism of $V$ to $V'$ \cite[Chapter III]{Die}. Let $f:P\to P'$ be a mapping which transfers frames of $\Pi$ to frames of $\Pi'$. It was noted above that this mapping sends lines of $\Pi$ to subsets in lines of $\Pi'$. We state that $f$ can be extended to a mapping of $\Pi_{V}$ to $\Pi_{V'}$ which transfers lines of $\Pi_{V}$ to subsets in lines of $\Pi_{V'}$ (the proof is a modification of the proof given in \cite[Chapter III]{Die}). It follows from a generalized version of the Fundamental Theorem of Projective Geometry \cite{FF,H} that the latter mapping is induced by a semilinear mapping $l:V\to V'$ (it must be pointed out that the associated homomorphism between the division rings is not necessarily surjective). Also, we state that the mapping $l:V\to V'$ is a strong semilinear embedding, i.e. it transfers any collection of linearly independent vectors to linearly independent vectors. In some special cases, for example, if our polar spaces are of type $\textsf{D}_{n}$, or for the symplectic polar spaces, this statement is obvious. In the general case, we need some algebraic technique to prove it. 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