Gradient property of standard representation for classical orthogonal groups

Abstract

We prove that an orthogonal representation in a certain class, which includes the standard (real) representation of any classical orthogonal groupSO N ,SU N ,O N ,U N , possesses gradient property, and, moreover, a particularly simple structure of scalars and vectors, independent of the group considered. We also show this result is true under weaker conditions if the representation acts in a odd-dimensional space, due to topological properties. The result is of particular interest in connection with bifurcation theory.