Isometric embeddings of half-cube graphs in half-spin Grassmannians

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.


Introduction
In the present paper we continue to discussing the problem of metric characterization of apartments in building Grassmannians [11,12]. This problem is motivated by the well-known metric characterization of the intersections of apartments with the chamber sets of Tits buildings (see, for example, [2, p. 90]) and closely related with Cooperstein-Kasikova-Shult results [4].
By [13], a building is a simplicial complex ∆ containing a family of subcomplexes called apartments and satisfying some axioms. In particular, all apartments are isomorphic to a certain Coxeter complex (the simplicial complex associated with a Coxeter system) which defines the type of our building. Maximal simplices of ∆ are called chambers; they have the same cardinality n called the rank of ∆. Two chambers are said to be adjacent if their intersection consists of n − 1 vertices. The vertex set of ∆ can be labeled by the nodes of the diagram corresponding to the Coxeter system associated with ∆; such labeling is unique up to a permutation on the set of nodes. All vertices corresponding to the same node form a Grassmannian. So, the vertex set of ∆ is decomposed in precisely n distinct Grassmannians. The intersections of apartments of ∆ with a Grassmannian of ∆ are called apartments in this Grassmannian.
Let G be one of the Grassmannians associated with ∆. Two distinct vertices a, b ∈ G are adjacent if there exists a simplex P ∈ ∆ such that P ∪ {a} and P ∪ {b} both are chambers. Denote by Γ the associated Grassmann graph, i.e. the graph whose vertex set is G and whose edges are pairs of adjacent vertices. Let A be an apartment of G and let Γ A be the restriction of the graph Γ to A. We want to distinguish all cases such that the image of every isometric embedding of Γ A in Γ is an apartment of G (in some cases this does not hold [3,11]).
A building is spherical if the associated Coxeter system is finite. By [13], there are precisely the following seven types of irreducible thick spherical buildings of rank ≥ 3: three classical types A n , B n = C n , D n and four exceptional types F 4 , E i , i = 6, 7, 8.
Every building of type A n−1 is the flag complex of a certain n-dimensional vector space V (over a division ring). The Grassmannians of this building are the usual Grassmannians G k (V ), k ∈ {1, . . . , n − 1}. Two elements of G k (V ) are adjacent if their intersection is (k − 1)-dimensional. The associated Grassmann graph is denoted by Γ k (V ). The case when k = 1, n − 1 is trivial -any two distinct vertices of Γ k (V ) are adjacent. Every apartment of 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 G k (V ) are the images of some isometric embeddings of the Johnson graph J(n, k) in Γ k (V ). However, the image of every isometric embedding of J(n, k) in Γ k (V ) is an apartment of G k (V ) if and only if n = 2k. This follows from the classification of isometric embeddings of Johnson graphs J(l, m), 1 < m < l − 1 in the Grassmann graph Γ k (V ), 1 < k < n − 1 given in [11].
All buildings of types C n and D n are defined by polar spaces. Every C n -building is the flag complex formed by singular subspaces of a rank n polar space. The Grassmannians of this building are the associated polar Grassmannians and the Grassmannian consisting of maximal singular subspaces is called the dual polar space. By [12], 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 corresponding Grassmann graph.
Every D n -building can be obtained from a polar space of type D n (this construction is known as the oriflamme complex). The Grassmannians of such building are some of the polar Grassmannians and so-called half-spin Grassmannians. Apartments of the half-spin Grassmannians are the images of isometric embeddings of the half-cube graph 1 2 H n in the associated Grassmann graphs. In this paper we investigate isometric embeddings of 1 2 H m in the half-spin Grassmannians associated with a polar space of type D n with n ≥ m ≥ 4. Our main result states that if a such embedding can be extended to an embedding of H m in the dual polar space (the latter embedding is not assumed to be isometric) and m is even then the image is an apartment in a parabolic subspace and we get an apartment of the Grassmannian if n = m.
Note that in [4] apartments of usual Grassmannians, dual polar spaces and halfspin Grassmannians were characterized in terms of independent subsets of the corresponding Grassmann spaces. Some more general results can be found in [8,9].

Preliminaries
2.1. Graphs. We recall a few definitions from graph theory.
The distance between two vertices in a connected graph is the smallest number i such that there exists a path of length i (a path consisting of i edges) connecting these vertices. A path connecting two vertices is said to be a 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 diameter of the graph. In the case when the diameter is finite, two vertices are said to be opposite if the distance between them is equal to the diameter.
A subset in the vertex set of a graph is called a clique if any two distinct elements of this subset are adjacent vertices. Using Zorn lemma, we can show that maximal cliques exist and every clique is contained in a certain maximal clique.
An injective mapping of the vertex set of a graph Γ to the vertex set of a graph Γ ′ is an embedding of Γ in Γ ′ if two vertices of Γ are adjacent only in the case when their images are adjacent vertices of Γ ′ . We will use the following property of embeddings: if the distance between two vertices of Γ is equal to 2 then the distance between the images of these vertices also is equal to 2. An embedding is said to be isometric if it preserves the distance between vertices. We refer [5,Part III] for the general theory of isometric embeddings.
2.2. Partial linear spaces. Let P be a non-empty set whose elements will be called points. Let also L be a family of proper subsets of P . Elements of this family will be called lines. We say that two or more points are collinear if there is a line containing all of them. Now, assume that the pair Π = (P, L) is a partial linear space, i.e. the following axioms hold: • every line contains at least two points and every point belongs to a line; • for any distinct collinear points p, q ∈ P there is precisely one line containing them, this line will be denoted by p q. We say that S ⊂ P is a subspace of Π if for any distinct collinear points p, q ∈ S the line p q is contained in S. A subspace is said to be 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 ⊂ P the minimal subspace containing X, i.e. the intersection of all subspaces containing X, is called spanned by X and denoted by X . We say that X is independent if the subspace X can not be spanned by a proper subset of X.
Let S be a subspace of Π (possible S = P ). An independent subset X ⊂ S is a base of S if X = S. The dimension of S is the smallest cardinality α such that S has a base of cardinality α + 1. The dimension of the empty set and a single point is equal to −1 and 0 (respectively), lines are 1-dimensional subspaces. A 2-dimensional singular subspace is called a plane.
Two partial linear spaces Π = (P, L) and Π ′ = (P ′ , L ′ ) are isomorphic if there exists a bijection f : P → P ′ such that f (L) = L ′ . This bijection is called a collineation of Π to Π ′ .
2.3. Polar spaces. Following F. Buekenhout and E. E. Shult [1], we define a polar space as a partial linear space Π = (P, L) satisfying the following axioms: • every line contains at least three points, • there is no point collinear with all points, • if p ∈ P and L ∈ L then p is collinear with one or all points of the line L, • any flag formed by singular subspaces is finite. If our polar space Π contains a singular subspace whose dimension is not less than 2 then all maximal singular subspaces of Π are projective spaces of the same dimension n ≥ 2 and the number n + 1 is called the rank of Π.
The collinearity relation of Π will be denoted by ⊥. For points p, q ∈ P we write p ⊥ q if p is collinear with q and p ⊥ q otherwise. Moreover, if X, Y ⊂ P then X ⊥ Y means that every point of X is collinear with all points of Y . If X ⊥ X then the subspace X is singular. (1) if p ∈ P and X ⊂ P then p ⊥ X implies that p ⊥ X , (2) if p ∈ P and S is a maximal singular subspace of Π then p ⊥ S implies that p ∈ S.
For every polar space of rank n one of the following two possibilities is realized: • type C n -every (n − 2)-dimensional singular subspace is contained in at least three maximal singular subspaces, • type D n -every (n − 2)-dimensional singular subspace is contained in precisely two maximal singular subspaces.
2.4. Dual polar spaces. Let Π = (P, L) be a polar space of rank n. For every k ∈ {0, 1, . . . , n − 1} we denote by G k (Π) the polar Grassmannian consisting of all k-dimensional singular subspaces of Π; in particular, G n−1 (Π) is formed by maximal singular subspaces. The associated Grassmann graph is denoted by Γ k (Π). Two distinct elements of G n−1 (Π) (vertices of Γ n−1 (Π)) are adjacent if their intersection is (n − 2)-dimensional. The graph Γ n−1 (Π) is connected and we denote by d(S, U ) the distance between S, U ∈ G n−1 (Π); this distance is equal to The diameter of Γ n−1 (Π) is n and two vertices of Γ n−1 (Π) are opposite if and only if they are disjoint subspaces.
2.5. Half-spin Grassmannians. Let Π = (P, L) be a polar space of type D n . In this case the Grassmannian G n−1 (Π) can be uniquely decomposed in the sum of two disjoint subsets, we denote them by G + (Π) and G − (Π), such that the distance between any two elements of G δ (Π), δ ∈ {+, −} (in the Grassmann graph Γ n−1 (Π)) is even and the distance between any S ∈ G δ (Π) and U ∈ G −δ (Π) is odd (we write −δ for the complement of δ in the set {+, −}). These subsets are called the half-spin Grassmannians of the polar space Π. Two elements of the half-spin Grassmannian G δ (Π), δ ∈ {+, −} are adjacent if their intersection is (n−3)-dimensional, i.e. the distance between the corresponding vertices of Γ n−1 (Π) is equal to 2. The associated Grassmann graph Γ δ (Π) is the graph whose vertex set is G δ (Π) and whose edges are pairs of adjacent elements. This graph is connected and we denote by d δ (S, U ) the distance between S, U ∈ G δ (Π) in Γ δ (Π); it is equal to d(S, U )/2. If n is even then the diameter of Γ δ (Π) is equal to n 2 and two vertices are opposite if and only if they are disjoint subspaces. In the case when n is odd, the diameter is n−1 2 and two vertices are opposite if and only if their intersection is a single point. Let M be an m-dimensional singular subspace of Π and m ≤ n − 2. We define If m = n − 3 then this subset is said to be a line of G δ (Π). The half-spin Grassmannian G δ (Π) together with the set of all such lines is a partial linear space; it is called the half-spin Grassmann space associated with G δ (Π). Two distinct points of this space are collinear if and only if they are adjacent vertices of Γ δ (Π).
Let m < n − 3. Then [M δ is a subspace of the half-spin Grassmann space, this subspace is non-singular if m < n−4. As in the case of dual polar spaces, subspaces of such type are called parabolic [4]. The

3.2.
Apartments of half-spin Grassmannians and half-cube graphs. From this moment we will suppose that Π = (P, L) is a polar space of type D n and n ≥ 4.
Let B be a frame of Π. For each k ∈ {0, 1, . . . , n − 1} we denote by A k the associated apartment of G k (Π). The intersection is the apartment of the half-spin Grassmannian G δ (Π) associated with the frame B. It is the image of an isometric embedding of the n-dimensional half-cube graph So, the half-cube graph 1 2 H n can be identified with the restriction of Γ δ (Π) to the apartment A δ . This graph has precisely the following two types of maximal cliques (see, for example, [6,7]): • the star A δ ∩ [S δ , S ∈ A n−4 , which consists of 4 vertices; • the special subset A δ ∩ [U ] δ , U ∈ A −δ which consists of n vertices.
If n is even then the diameter of 1 2 H n is equal to n 2 and for every vertex there is unique opposite vertex. This collineation establishes a one-to-one correspondence between apartments. All apartments of [M δ are the images of isometric embeddings of 1 2 H m in Γ δ (Π). In the next section we will need the following lemmas. Proof. An easy verification shows that the intersection of all maximal cliques of  (1) The intersection of two distinct maximal cliques of 1 2 H n contains at most three vertices; this intersection contains precisely three vertices if and only if one of the cliques is a star and the other is a special subset.
(2) Every clique of 1 2 H n formed by three distinct vertices is contained in precisely one star and precisely one special subset.
(3) For any four distinct mutually adjacent vertices v, w, u, s of 1 2 H n there is a vertex adjacent with v, w, u and non-adjacent with s.

Proof. (1). Easy verification.
(2). It is not difficult to see that any 3-element clique is contained in a star and a special subset. The first statement guarantees that these star and special subset are unique.
(3). By (2), {v, w, u} is the intersection of two maximal cliques. One of these cliques does not contain s and it has a vertex non-adjacent with s.

Extensions of embeddings
Let f be an embedding of 1 2 H m , m ≥ 4 in Γ δ (Π), δ ∈ {+, −}. The embedding is not assumed to be isometric and its image will be denoted by I.  The image of every maximal clique of 1 2 H m is a clique in Γ δ (Π) and it is contained in a certain maximal clique of Γ δ (Π) (a star or a special subspace). The images of distinct maximal cliques of 1 2 H m are contained in distinct maximal cliques of Γ δ (Π) (otherwise, there exist adjacent elements of I whose pre-images are not adjacent). Proof. Suppose that the image of a certain maximal clique of 1 2 H m is contained in two distinct maximal cliques of Γ δ (Π). By Lemma 2.4, the intersection of these cliques is contained in a plane. Every maximal clique of 1 2 H m contains at least four vertices. Hence there are four distinct mutually adjacent vertices of 1 2 H m whose images belong to a plane, but this contradicts Lemma 4.2. Proof. Let f , as above, be an embedding of 1 2 H m in Γ δ (Π). Suppose that X 0 is a special subset whose image is contained in a special subspace. We take any star Y such that |X 0 ∩ Y| = 3 (see Lemma 3.3). By Lemma 4.1, the set f (X 0 ) ∩ f (Y) spans a plane. Since f (X 0 ) is contained in a special subspace, Lemma 2.4 implies that the maximal clique of Γ δ (Π) containing f (Y) is a star. The same arguments show that for any special subset X satisfying |X ∩ Y| = 3 the image f (X ) is contained in a special subspace. Now, let X be an arbitrary special subset. There exists a sequence X 0 , X 1 , . . . , X 2k = X such that |X i−1 ∩ X i | = 3 for all i ∈ {1, . . . , 2k} and X i is a special subset or a star if i is even or odd, respectively. By the arguments given above, f (X ) is contained in a special subspace. Similarly, we establish that the image of every star is a subset of a star. Using the same arguments, we show that the case (B) is realized if the image of a certain special subset is contained in a star.
An embedding of 1 2 H m in Γ δ (Π) will be called extendible if it can be extended to an embedding of H m in Γ n−1 (Π). Proof. We take any polar space of type D m . Let A be an apartment in the corresponding dual polar space. Denote by A + and A − the associated apartments in the half-spin Grassmannians. Recall that A is the disjoint sum of A + and A − . Let f : A + → G δ (Π) be an embedding of 1 2 H m in Γ δ (Π). Suppose that f is extendible. The associated extension also will be denoted by  Remark 5.1. Our proof of Theorem 5.1 is a modification of the proof of Theorem 3.1 given in [12]. We are not be able to show that for every extendible isometric embedding of 1 2 H m in Γ δ (Π) the associated embedding of H m in Γ n−1 (Π) is isometric. If this is possible then the required result follows from Theorem 3.1.

Proof of Theorem 5.1
Throughout the section we suppose that m is even and not less than 4. Proof. Let f be an isometric embedding of 1 2 H m in Γ δ (Π). We take any geodesic v 0 , v 1 , . . . , v m/2 in 1 2 H m . The distance between f (v 0 ) and f (v m/2 ) in Γ δ (Π) is equal to m 2 . Hence the distance between these vertices in Γ n−1 (Π) is m and can be extended to a geodesic of Γ n−1 (Π). Then the singular subspace This gives the claim, since every vertex of 1 2 H m is on a geodesic connecting v 0 with v m/2 (Lemma 3.1).
If M is an (n−m−1)-dimensional singular subspace of Π then [M n−m is a polar space of type D m and there is the natural collineation of [M δ to one of the halfspin Grassmann spaces of this polar space; this collineation establishes a one-to-one correspondence between apartments. Therefore, by Lemma 6.1, it is sufficient to prove Theorem 5.1 only in the case when m = n.
So, let m = n. Let also {p 1 , . . . , p 2n } be a frame of Π. Denote by A and A + , A − the associated apartments in the Grassmannians G n−1 (Π) and G + (Π), G − (Π), respectively. Let also f : A → G n−1 (Π) be an embedding of H n in Γ n−1 (Π) such that the restriction of f to A + is an isometric embedding of 1 2 H n in Γ δ (Π), δ ∈ {+, −}; in other words, f | A+ is an extendible isometric embedding of 1 2 H n in Γ δ (Π). For every i ∈ {1, . . . , 2n} we define This is an apartment in the parabolic subspace [p i n−1 and the restriction of Γ n−1 (Π) to A i is isomorphic to H n−1 .
Recall that the distance in the Grassmann graphs Γ n−1 (Π) and Γ δ (Π) is denoted by d and d δ , respectively.
Proof. Since the restriction of f to A + is an isometric embedding of 1 2 H n in Γ δ (Π), we have which implies (1). Lemma 6.3. The restriction of f to every A i is an isometric embedding of H n−1 in Γ n−1 (Π).
Proof. We need to show that (1) holds for all X, Y ∈ A i . It follows from Lemma 6.2 if X, Y both belong to A + .
Consider the case when X ∈ A + and Y ∈ A − . Let (2) X = X 0 , X 1 , . . . , X k = Y be a geodesic of Γ n−1 (Π) contained in A. Since n is even, d(X, Y ) = k < n and there exists X k+1 ∈ A + such that is a geodesic of Γ n−1 (Π). We have X 0 , X k+1 ∈ A + and, by Lemma 6.2, ) is a geodesic in Γ n−1 (Π). The latter guarantees that d(f (X 0 ), f (X k )) is equal to k and we get (1). Now, suppose that X, Y ∈ A − . As above, let (2) be a geodesic of Γ n−1 (Π) contained in A. Since X, Y ∈ A − both contain the point p i (they are elements of A i ), the intersection of X and Y contains a line and d(X, Y ) ≤ n − 2.
Remark 6.1. Lemma 6.3 shows that (1) holds in the case when d(X, Y ) < n. We can not prove (1) if X, Y ∈ A − and d(X, Y ) = n.
Theorem 3.1 and Lemma 6.3 imply the existence of points q 1 , . . . , q 2n such that each f (X i ) is an apartment in the parabolic subspace [q i n−1 . Note that (4) q i ⊥ q j if j = σ(i) (if j = σ(i) then there is X ∈ A containing p i , p j and f (X) contains q i , q j ).
It is clear that d(X, Y ) = 2. Since f is an embedding of H n in Γ n−1 (Π), Now, assume that q i ⊥ q σ(i) . Then (4) and the first part of Lemma 2.1 imply that q i ⊥ f (Y ). Since f (Y ) is a maximal singular subspace, we get q i ∈ f (Y ) (by the second part of Lemma 2.1). So, q i belongs to f (X) ∩ f (Y ) which is impossible (because q i , q i1 , . . . , q in−2 form an independent subset).
It follows from Lemma 6.5 and (4) that q 1 , . . . , q 2n form a frame. Then any subset of {q 1 , . . . , q 2n } is independent and, as in the proof of Lemma 6.4, we establish that every element of f (A) is spanned by a subset of this frame.