On certain forms and quadrics related to symplectic dual polar spaces in characteristic 2
Let V be a 2n-dimensional vector space over a field and xi a non-degenerate alternating form defined on V. Let Delta be the building of type C (n) formed by the totally xi-isotropic subspaces of V and, for 1 a parts per thousand currency sign k a parts per thousand currency sign n, let and Delta (k) be the k-grassmannians of PG(V) and Delta, embedded in and in a subspace respectively, where . This paper is a continuation of Cardinali and Pasini (Des. Codes. Cryptogr., to appear). In Cardinali and Pasini (to appear), focusing on the case of k = n, we considered two forms alpha and beta related to the notion of 'being at non maximal distance' in and Delta (n) and, under the hypothesis that , we studied the subspaces of W (n) where alpha and beta coincide or are opposite. In this paper we assume that . We determine which of the quadrics associated to alpha or beta are preserved by the group in its action on W (n) and we study the subspace of W (n) formed by vectors v such that alpha(v, x) = beta(v, x) for every . Finally, we show how properties of can be exploited to investigate the poset of G-invariant subspaces of V (k) for k = n - 2i and .
symplectic dual polar spaces grassmannians weyl modules