A 14-dimensional Module for the Symplectic Group: Orbits on Vectors

Abstract

Let 𝔽 be a field, V a 6-dimensional 𝔽-vector space and f a nondegenerate alternating bilinear form on V. We consider a 14-dimensional module for the symplectic group Sp(V, f) ≅ Sp(6, 𝔽) associated with (V, f), and classify the orbits on vectors. For characteristic distinct from 2, this module is irreducible and isomorphic to the Weyl module of Sp(V, f) for the fundamental weight λ3. If the characteristic is 2, then the module is reducible as it contains an 8-dimensional submodule isomorphic to the spin module of Sp(V, f).