Ela on Relations of Invariants for Vector - Valued Forms

An algorithm is given for computing explicit formulas for the generators of relations among the invariant rational functions for vector-valued bilinear forms. These formulas have applications in the geometry of Riemannian submanifolds and in CR geometry. 1.1. Introduction. Vector-valued forms play a key role in the study of higher codimensional geometries. For example, they occur naturally in the study of Rieman-nian submanifolds (as the second fundamental form) and in CR geometry (as the Levi form). In each of these there are natural group actions acting on the vector-valued forms, taking care of different choices of local coordinates and the such. The algebraic invariants of these forms under these group actions provide invariants for the given geometries. In Riemannian geometry, for example, the scalar curvature can be expressed as an algebraic invariant of the second fundamental form. But before these invariants can be used, their algebraic structure must be known. In [5], an explicit list of the generators is given. In that paper, though, there is no hint as to the relations among these generators. In this paper, a method is given for producing a list of the generators for the relations of the invariants. In [5], the problem of finding the rational invariants of bilinear maps from a complex vector space V of dimension n to a complex vector space W of dimension k, on which the group GL(n, C) × GL(k, C) acts, is reduced to the problem of finding invariant one-dimensional subspaces of the vector spaces (V ⊗ V ⊗ W *) ⊗r , for each positive integer r. From this, it is shown that the invariants can be interpreted as being generated by (Invariants for GL(n, C) of V ⊗ V) ⊗ (Invariants for GL(k, C) of W *), each component of which had been computed classically. In this paper we extend this type of result, showing how to compute the relations of bilinear forms from knowledge of the relations for V ⊗ V under the action of GL(n, C) and the relations for W * under the action of GL(k, C). In particular, in a way that will be made more precise later, we show that the relations can be interpreted as being generated by: