2013/03/22 by Ilaria Cardinali, Cardinali, Ilaria, Antonio Pasini +1
Computer Science · Mathematics · #20C33 #20G15 #51B25 #51E24 #Advanced Topics in Algebra #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT) #math.RT #msc:20C33 #msc:20G15 #msc:51B25 #msc:51E24
paper · pdf · doi:10.48550/arxiv.1303.5559
arxiv created 2013/03/22 · openalex publication_date 2013/03/22 · arxiv updated 2013/03/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a point-line geometry P and a pappian projective space S,a veronesean embedding of P in S is an injective map e from the point-set of P to the set of points of S mapping the lines of P onto non-singular conics of S and such that e(P) spans S. In this paper we study veronesean embeddings of the dual polar space Δn associated to a non-singular quadratic form q of Witt index n >= 2 in V = V(2n + 1; F). Three such embeddings are considered,namely the Grassmann embedding grn,the composition vsn of the spin (projective) embedding of Δn in PG(2n-1; F) with the quadric veronesean map of V(2n; F) and a third embedding wn defined algebraically in the Weyl module V (2λn),where λn is the fundamental dominant weight associated to the n-th simple root of the root system of type Bn. We shall prove that wn and vsn are isomorphic. If char(F) is different from 2 then V (2λn) is irreducible and wn is isomorphic to grn while if char(F) = 2 then grn is a proper quotient of wn. In this paper we shall study some of these submodules. Finally we turn to universality,focusing on the case of n = 2. We prove that if F is a finite field of odd order q > 3 then sv2 is relatively universal. On the contrary,if char(F) = 2 then vs2 is not universal. We also prove that if F is a perfect field of characteristic 2 then vsn is not universal,for any n>=2.