Embeddings of orthogonal Grassmannians

In my talk I will report on my recent work with Ilaria Cardinali [CP1], [CP2], devoted to grassmannians of orthogonal polar spaces of Dynkin type \( B_n \).

Let \( V := V(2n+1,{\mathbb{F}}) \) for a field \( {\mathbb{F}} \) and let \( q \) be a non singular quadratic form of \( V \) of Witt index \( n \). Let \( \Delta \) be the building of type \( B_n \) where the elements of type \( k = 1, 2,..., n \) are the \( k \)-dimensional subspaces of \( V \) totally singular for \( q \).

The \( k \)-grassmannian of \( \Delta \) is the point-line geometry where the points are the \( k \)-subspaces of \( V \) totally singular for \( q \). When \( k \le n \) the lines of \( \Delta_k \) are the sets \( \{Z \mid X \subset Z\subset Y , ~\operatorname{dim}(Z) = k\} \) for totally singular subspaces \( X\subset Y \) of \( V \) with \( \operatorname{dim}(X) = k-1 \) and \( \operatorname{dim}(Y) = k+1 \). When \( k = n \) the lines of \( \Delta_n \) are the sets \( l_{X}:=\{Z\mid X \subset Z\subset X^{\perp} , ~\operatorname{dim}(Z) = n, ~ Z ~\text{totally singular}\} \) where \( X \) is a totally singular \( (n-1) \)-subspace of \( V \) and \( X^\perp \) is the orthogonal of \( X \) with respect to \( q \). Note that the points of \( l_{X} \) form a non-singular conic in the plane \( \operatorname{PG}(X^\perp/X) \).

With \( k \) as above, let \( W_k :=\bigwedge^k V \) and let \( \varepsilon_k \) be the mapping from \( \Delta_k \) to \( \operatorname{PG}(W_k) \) sending every point \( X = \langle v_1,..., v_k\rangle \) of \( \Delta_k \) to the 1-subspace \( \langle v_1\wedge...\wedge v_k\rangle \) of \( W_k \). Clearly \( \varepsilon_k \) is injective. Wen \( k \le n \) it maps lines of \( \Delta_k \) onto lines of \( \operatorname{PG}(W_k) \). In this case \( \varepsilon_k \) is a (full) projective embedding of \( \Delta_k \) in the subspace of \( \operatorname{PG}(W_k) \) spanned by \( \varepsilon_k(\Delta_k) \). We call it the natural embedding of \( \Delta_k \), also the grassmann embedding of \( \Delta_k \).

If \( k=n \) then \( \varepsilon_n \) maps lines of \( \Delta_n \) onto conics of \( \operatorname{PG}(W_n) \). So, \( \varepsilon_n \) is not a projective embedding in the usual sense. We say that an injective mapping \( \varepsilon:{\cal S}\rightarrow{\cal P} \) from the point-set of a point-line geometry \( \cal S \) to the set of points of a projective space \( \cal P \) is a veronesean embedding of \( \cal S \) if it maps lines of \( \cal S \) onto non-singular conics of \( \cal P \) (compare [TVM]). Likewise for projective embeddings, we assume that \( \varepsilon({\cal S}) \) spans \( \cal P \) and we put \( \operatorname{dim}(\varepsilon) = \operatorname{dim}({\cal P}) + 1 \) (vector dimension of \( \varepsilon \)). Thus, \( \varepsilon_n \) is a veronesean embedding of \( \Delta_n \). We call it the grassmann veronesean embedding of \( \Delta_n \), also the grassmann embedding of \( \Delta_n \), for short.

We recall that \( \Delta_n \) admits a projective embedding which deserves to be called the natural embedding of \( \Delta_n \), namely the spin embedding, henceforth denoted by \( \operatorname{spin}_n \). It lives in the so-called spin module, namely the Weyl module \( V(\omega_n) \) (see below).

A veronesean embedding \( \sigma_n \) of \( \Delta_n \) can also be defined as follows. Let \( \eta_N \) be the usual veronesean embedding of \( \operatorname{PG}(N-1,{\mathbb{F}}) \) in \( \operatorname{PG}({{N+1}\choose 2}-1,{\mathbb{F}}) \), with \( N = 2^n \). Then \( \sigma_n := \eta_N\circ \operatorname{spin}_n \) is a veronesean embedding of \( \Delta_n \). One might wonder if \( \varepsilon_n \cong \sigma_n \). This is indeed one of the points to be discussed in my talk. However, before to come to problems and results we need to state a few more conventions.

Henceforth \( G := \operatorname{SO}(2n+1,{\mathbb{F}}) \) \( (= \operatorname{PSO}(2n+1,{\mathbb{F}}) \)) is the stabilizer of the form \( q \) in \( \operatorname{SL}(V) = \operatorname{SL}(2n+1,{\mathbb{F}}) \). The group \( G \) acts faithfully on \( W_k \) as follows: \( g\in G \) sends \( v_1\wedge...\wedge v_k \) to \( g(v_1)\wedge...\wedge g(v_k) \). Note that \( G \) is a Chevalley group of adjoint type. The universal Chevvaley group of type \( B_n \) is the spin group \( \widetilde{G} = \operatorname{Spin}(2n+1,{\mathbb{F}}) \). If \( \operatorname{char}({\mathbb{F}}) = 2 \) then \( \widetilde{G} = G \), otherwise \( \widetilde{G} = 2^\cdot G \) (two-fold non-split extension).

Let \( \omega_1, \omega_2,..., \omega_n \) be the fundamental dominant weights for the root system of type \( B_n \), numbered in the usual way (see the picture at the beginning of this abstract). For \( k \le n \) let \( V(\omega_k) \) be the Weyl module for \( G \) with \( \omega_k \) as the highest weight. We recall that \( \operatorname{dim}(V(\omega_k)) = {{2n+1}\choose k} \). When \( k = n \) we consider the Weyl module \( V(2\omega_n) \) for \( G \), with highest weight \( 2\omega_n \), and the spin module \( V(\omega_n) \) for \( \widetilde{G} \). Recall that \( \operatorname{dim}(V(2\omega_n)) = {{2n+1}\choose n} \), while \( \operatorname{dim}(V(\omega_n)) = 2^n \). For \( \lambda = \omega_k \) or \( \lambda = 2\omega_n \) we say that an embedding \( \varepsilon \) of \( \Delta_k \) lives in \( V(\lambda) \) (also that \( V(\lambda) \) hosts \( \varepsilon \)) if \( \operatorname{PG}(V(\lambda)) \) can be taken as the codomain of \( \varepsilon \) (whence \( \operatorname{dim}(\varepsilon) = \operatorname{dim}(V(\lambda)) \)).

I can now turn to results and conjectures.

Theorem 1    Let \( \operatorname{char}({\mathbb{F}}) \neq 2 \). Then:

1. \( \sigma_n \cong \varepsilon_n \).
2. \( V(2\omega_n) \) hosts \( \varepsilon_n \) and \( V(\omega_k) \) host \( \varepsilon_k \) for every \( k \le n \). Hence \( \operatorname{dim}(\varepsilon_k) = {{2n+1}\choose k} \), for every \( k = 1,...,n \).
3. For every \( k \le n \) the embedding \( \varepsilon_k \) is absolutely universal.

As for claim (3), recall that \( \Delta_k \) admits the absolutely universal projective embedding for any \( k = 1,..., n \) (Kasikova and Shult [KS]). Hence a projective embedding of \( \Delta_k \) is absolutely universal if it is its own hull (i.e., it is relatively universal). The case of \( k = n \) is not considered in (3) of Theorem 1. When \( k = n \) the projective embedding to consider is \( \operatorname{spin}_n \). Actually, if \( \operatorname{char}({\mathbb{F}}) \neq 2 \) then \( \operatorname{spin}_n \) is universal (Shult and Thas [ST]). On the other hand, as proved by Blok, Cardinali and De Bruyn [BCDB], if \( {\mathbb{F}} \) is a perfect field of characteristic 2 then \( \operatorname{spin}_n \) is a quotient of the symplectic grassmann embedding \( {\bar{\varepsilon}}_n \) (see below). Hence it is not universal.

Relative universality can be defined for veronesean embeddings just like for projective embeddings. Indeed, given a veronesean embedding \( \varepsilon \), we can define its linear hull \( \tilde{\varepsilon} \) as in [Pa]. We say that \( \varepsilon \) is relatively universal if \( \tilde{\varepsilon}\cong \varepsilon \). Thus, it makes sense to conjecture the following:

Conjecture 1    When \( \operatorname{char}({\mathbb{F}})\neq 2 \) the embedding \( \varepsilon_n \) is relatively universal.

However, now we are not allowed to jump from relative universality to absolute universality as we can do when dealing with projective embeddings of \( \Delta_k \). Indeed we do not know if \( \Delta_n \) admits an absolutely universal veronesean embedding.

We shall now turn to the case of \( \operatorname{char}({\mathbb{F}}) = 2 \), but firstly we recall a few facts on symplectic grassmannians and their natural embeddings. Put \( \overline{V} := V(2n,{\mathbb{F}}) \), let \( \overline{\Delta} \) be the building of type \( C_n \) associated to the symplectic group \( \operatorname{Sp}(2n,{\mathbb{F}}) \) and, for \( k = 1, 2,..., n \), let \( {\overline{\Delta}}_{k} \) be the \( k \)-grassmannian of \( \overline{\Delta} \). Put \( \overline{W}_{k}:=\bigwedge^k\overline{V} \) and let \( {\bar{\varepsilon}}_{k} \) be the \( k \)-grassmann embedding of \( {\overline{\Delta}}_{k} \), namely the projective embedding sending every totally isotropic \( k \)-subspace \( \langle v_1,..., v_k\rangle \) of \( \overline{V} \) to the point \( \langle v_1\wedge...\wedge v_k\rangle \) of \( \operatorname{PG}(\overline{W}_k) \). It is well known that \( \operatorname{dim}({\bar{\varepsilon}}_{k}) = {{2n}\choose k}-{{2n}\choose{k-2}} \).

Let \( \operatorname{char}({\mathbb{F}}) = 2 \) and let \( f_q \) be the bilinear form associated to \( q \). Then the radical of \( f_q \) is a non-singular 1-dimensional subspace \( N_0 \) of \( V \). We can recover \( \Delta \) in \( V/N_0 \) by replacing every element \( X \) of \( \Delta \) with \( (X+N_0)/N_0 \). In this way, when \( {\mathbb{F}} \) is perfect we obtain an isomorphism \( \Delta\cong \overline{\Delta} \).

Theorem 2    Let \( \operatorname{char}({\mathbb{F}}) = 2 \). Then \( W_k \) admits a \( G \)-invariant subspace \( {\cal N}_k \) contained in \( \langle \varepsilon_k(\Delta_k)\rangle \) (the latter being now regarded as a subspace of \( W_k \)) such that all the following hold:

1. \( \operatorname{dim}({\cal N}_k) \leq {{2n}\choose{k-1}}-{{2n}\choose{k-3}} \) and \( \operatorname{dim}(\langle\varepsilon_k(\Delta_k)\rangle/{\cal N}_k) \leq{{2n}\choose k}-{{2n}\choose{k-2}} \) (with the usual convention that \( {m\choose i} = 0 \) when \( i \le 0 \)). Consequently, \( \operatorname{dim}(\varepsilon_k) \leq {{2n+1}\choose k}-{{2n+1}\choose{k-2}} \).
   If \( {\mathbb{F}} \) is perfect then \( \operatorname{dim}({\cal N}_k) = {{2n}\choose{k-1}}-{{2n}\choose{k-3}} \), \( \operatorname{dim}(\langle\varepsilon_k(\Delta_k)\rangle/{\cal N}_k) = {{2n}\choose k}-{{2n}\choose{k-2}} \) and \( \operatorname{dim}(\varepsilon_k) = {{2n+1}\choose k}-{{2n+1}\choose{k-2}} \).
2. If \( k \ge 1 \) then \( {\cal N}_k \) hosts a projective embedding \( \iota_{k-1} \) of \( \Delta_{k-1} \). Moreover, \( \iota_{k-1}\cong{\bar{\varepsilon}}_{k-1} \) when \( {\mathbb{F}} \) is perfect.
3. \( {\cal N}_k \) defines a quotient \( \varepsilon_k/{\cal N}_k \) of \( \varepsilon_k \). If either \( k \le n \) or \( {\mathbb{F}} \) is perfect, then \( \varepsilon_k/{\cal N}_k \) is a full projective embedding of \( \Delta_k \). If \( {\mathbb{F}} \) is non-perfect then \( \varepsilon_n/{\cal N}_n \) is a lax embedding, namely the image by \( \varepsilon_n/{\cal N}_n \) of a line of \( \Delta_n \) is properly contained in a projective line of \( \langle\varepsilon_n(\Delta_n)\rangle/{\cal N}_n \).
   Let \( {\mathbb{F}} \) be perfect. Then \( \varepsilon_k/{\cal N}_k\cong {\bar{\varepsilon}}_{k} \), for every \( k = 1,..., n \).

Corollary 3    Let \( {\mathbb{F}} \) be a perfect field of characteristic 2 and \( k \le n \). Then \( {\bar{\varepsilon}}_{k} \) is not universal.

The restriction \( k \le n \) is essential here. Indeed the isomorphism \( {\bar{\varepsilon}}_{n} \cong \varepsilon_n/{\cal N}_n \) gives no information on the linear hull of \( {\bar{\varepsilon}}_{n} \), since \( \varepsilon_n \) is not a projective embedding. In fact, if \( |{\mathbb{F}}| = 2^r \) with \( r \ge 1 \) then \( {\bar{\varepsilon}}_{n} \) is universal (Cooperstein [Co1]). On the other hand, if \( {\mathbb{F}} = {\mathbb{F}}_2 \) then \( {\bar{\varepsilon}}_{n} \) is not universal (Li [L], Blokhuis and Brouwer [BB]). We do not know if \( {\bar{\varepsilon}}_{n} \) is universal when \( {\mathbb{F}} \) is an infinite field of characteristic 2.

The assumption that \( {\mathbb{F}} \) is perfect is likely to be superfluous in Corollary 3. It is certainly superfluous when \( k = 1 \) (see [DBP]). We warn that when \( {\mathbb{F}} \) is non-perfect the linear hull of \( {\bar{\varepsilon}}_{1} \) is defined in an infinite dimensional vector space, due to the fact that any perfect extension of \( {\mathbb{F}} \) has infinite degree over \( {\mathbb{F}} \).

To finish, we propose two more conjectures.

Conjecture 2    The inequalities of claim (1) of Theorem 2 are actually equalities even if \( \mathbb{F} \) is non-perfect.

Conjecture 3    Let \( \operatorname{char}({\mathbb{F}}) = 2 \). If \( k \le n \) (or \( k = n \)) then the hull of \( \varepsilon_k \) lives in \( V(\omega_k) \) (respectively \( V(2\omega_n \)). Consequently, if \( k \ge 1 \) then \( \varepsilon_k \) is not relatively universal. Moreover, the veronesean embedding \( \sigma_n \) is the hull of \( \varepsilon_n \).

In view of [Co2], the above conjecture holds true when \( k = 2 \) and \( {\mathbb{F}} = {\mathbb{F}}_2 \).

References

BCDB

R. Blok, I. Cardinali and B. De Bruyn. On the nucleus of the grassmann embedding of symplectic dual polar spaces \( DSp(2n,F), \operatorname{char}(F)=2 \). European J. Combin. 30 (2009), no. 2, 468–472.

BB

A. Blokhuis and A. E. Brouwer. The universal embedding dimension of the binary symplectic dual polar space. Discrete Math. 264 (2003), 3–11.

CP1

I. Cardinali and A. Pasini. Orthogonal grassmannians, preprint October 2011.

CP2

I. Cardinali and A. Pasini. Veronesean embeddings of dual polar spaces of type \( B_n \), in preparation.

Co1

B. N. Cooperstein. On the generation of dual polar spaces of symplectic type over finite fields. J. Combin. Theory Ser. A 83 (1998), 221–232.

Co2

B. N. Cooperstein. Generating long root subgroup geometries of classical groups over finite prime fields. Bull. Belg. Math. Soc. Simon Stevin 5 (1998), no. 4, 531–548.

DBP

B. De Bruyn and A. Pasini. On symplectic polar spaces over non-perfect fields of characteristic 2. Linear Multilinear Algebra 57 (2009), 567–575.

KS

A. Kasikova and E. E. Shult. Absolute embeddings of point-line geometries. J. Algebra 238 (2001), 265–291.

L

P. Li. On the universal embedding of the \( Sp_{2n}(2) \) dual polar space. J. Combin. Theory Ser. A 94 (2001), 100–117.

Pa

A. Pasini. Embeddings and expansions. Bull. Belg. Math. Soc. Simon Stevin 10 (2003), 585–626.

ST

E. E. Shult and J. A. Thas. Hyperplanes of dual polar spaces and the spin module. Arch. Math. 59 (1992), 610–623.

TVM

J. A. Thas and H. Van Maldeghem. Generalized Veronesean embeddings of projective spaces. Combinatorica (to appear).