The generating rank of a polar Grassmannian

Abstract

In this paper we compute the generating rank of k-polar Grassmannians defined over commutative division rings. Among the new results, we compute the generating rank of k-Grassmannians arising from Hermitian forms of Witt index n defined over vector spaces of dimension N > 2n. We also study generating sets for the 2-Grassmannians arising from quadratic forms of Witt index n defined over V(N, 𝔽 q ) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 and anisotropic defect (polar corank) d ≠ 2 they can be generated over the prime subfield, thus determining their generating rank.

Sets of generators and chains of subspaces

“…The main results of this subsection are a remake of Section 2.3 of Cardinali, Giuzzi and Pasini [3]. We shall state them in § §4.2.2 and 4.2.3, but before to come to them we need to recall a few basics and well known theorems on projective embeddings of polar spaces.…”

“…In fact, in general rk gen (Γ) < rk WO (Γ) (compare Example 1.7). However, as shown in [3], we can bypass this obstacle by considering only subspaces which contain a pair of mutually disjoint maximal singular subspaces.…”

“…As in [3], we call these subspaces nice subspaces. Let W nice (Γ) be the family of well ordered chains of nice subspaces and assume that Γ has finite polar rank n = prk(Γ).…”

“…Remark 9 Only the commutative case is considered in Lemma 2.3 of [3] but the arguments used to prove that Lemma also work when K is non-commutative provided that, for every nice subspace S of Γ, the polar space e(S) can be described by a reflexive sesquilinear form defined over (the underlying vector space of) [e(S)]. In view of Theorem 4.3, this is the case when char(K) = 2.…”

“…Remark 10 The polar corank crk(Γ) is called anisotropic defect in [3], in view of the fact that [e(M )∪e(M ′ )] ⊥ ∩e(P ) = ∅. However the word 'defect' is usually given a different meaning in the literature.…”