Let G be a symplectic group Sp(N) or an orthogonal group SO(N) and let P be the maximal parabolic subgroup of G associated to a simple root \(\alpha_ d\). Let \(\lambda_ d\) be the fundamental weight associated to \(\alpha_ d\) and let L be the associated ample line bundle on G/P. By realizing G as the fixed point set of a certain involution on the special linear group, the author obtains an identification of the corresponding Weyl groups as subgroups of the symmetric group. From this identification a canonical set \(W^ p\) of representatives for \(W/W_ P\) may be described as d-tuples of integers. This description is then applied to so-called ’admissible pairs’ of representatives in \(W^ P\) which can be used to describe a basis for the fundamental representation \(H^ 0(G/P,L)\). The description of the bases in terms of these d-tuples is carried out in a sequel to this article (cf. the following review).