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, Fq) for q = 4, 8, 9 and 2n ≤ N ≤ 2n + 2. We prove that for N > 6 they can be generated over the prime subfield, thus determining their generating rank.

On the generation of some Lie-type geometries

“…As said in Section 1.1, the case of Gr 1,n (A n (K)) has been already considered in [2], but the proof we shall give in this paper is different and simpler than that of [2]. Theorem 1.1 also contains a proof of a conjecture presented in [6,Conjecture 5.11].…”

“…Probably, what makes these cases so difficult is the fact that the special case Gr 1,3 (A 3 (K)) ∼ = Gr +,− (D 3 (K)) of Gr 1,n (A n (K)) somehow enters the game in any attempt to compute the generating rank of Gr k (B n (K)) or Gr k (D n (K)) and, as we have seen above, as far as generation is concerned, Gr 1,n (A n (K)) can behave wildly. Nevertheless, in [6] we have shown that for K = F 4 , F 8 or F 9 the Grassmannians Gr 2 (B n (K)) (n ≥ 3) and Gr 2 (D n (K)) (n > 3) are generated over the corresponding prime subfields F 2 or F 3 . The generating ranks of Gr 2 (B n (K 0 )) and Gr 2 (D n (K 0 )), for K 0 a finite field of prime order, are known to be equal to 2n+1 2 and 2n 2 respectively (Cooperstein [7]).…”

“…When K is a field it is known as the long root geometry for SL(n + 1, K). In [2] it is proved that if n > 2 then Gr 1,n (A n (K)) is not K 0 -generated, for any proper sub-division ring K 0 of K (see also [6,Theorem 5.10] for an alternative proof in the special case where n = 3 and K is a field). However, when K is a field and is generated by K 0 ∪ {a 1 , ..., a t } for suitable elements a 1 , ..., a t ∈ K \ K 0 , then Gr 1,n (A n (K)) can be generated by adding at most t elements to Gr 1,n (A n (K 0 )) (Blok and Pasini [2]).…”

“…Assume firstly that n = 3. We have discussed this case in [6, Theorem 5.10] but we turn back to it here, using an argument different from that of [6].…”

Grassmann embeddings of polar Grassmannians

“…[4, Theorem 1.5] for 1 < k ≤ 3, k < n and [5, Theorem 5] for k = n = 2). In a recent paper [10], the authors have investigated the generating rank of polar Grassmannians; in particular, for H k (n, d 0 , 0; F) with d 0 ≥ 0 and k < n it is shown that the Grassmann embedding of H k is universal; see [10,Corollary 2]. For k = 2, and k = 3 < n for d 0 ≤ 1, the Grassmann embedding of Q k (n, d 0 , 0, F) is universal; see [4].…”