Canonical representation of tangent vectors of Grassmannians

The structure of the tangent bundle of the real Grassmann manifold $G^+_{p,n}$ under the Plücker embedding (in the exterior algebra of the initial Euclidean space) is studied. Explicit expressions for the relation between decompositions of a tangent vector with respect to different bases of the tangent space are obtained, and a constructive method yielding the canonical (= simplest) decomposition is presented.