Two forms related to the symplectic dual polar space in odd characteristic
Let V be a 2n-dimensional vector space over a field F equipped with a nondegenerate alternating form xi. Let G(n) be the n-grassmannian of PG(V) and Delta(n) the dual of the polar space Delta associated to xi. Then G(n) and Delta(n) are naturally embedded in the vector space W-n = boolean AND V-n and V-n subset of W-n respectively, where dim(W-n) = ((2n)(n)) and dim(V-n) = ((2n)(n)) - ((2n)(n-2)). The spaces Wn and Vn can be regarded as modules for the symplectic group Sp(2n, F). If char(F)not equal 2, we will define two forms alpha and beta of W-n which coincide on V-n and we will investigate the relation between these two forms and the collineation of W-n naturally induced by xi. We will obtain a description of the module W-n in terms of the two subspaces of Wn where the linear functionals induced by alpha and beta are equal and respectively opposite.
symplectic dual polar spaces weyl modules