Jacobians and invariants
Let \(f\) be a homogeneous polynomial of degree \(d\) on \(\mathbb{C}^ n\), the Jacobian \(J(f)\) is the span of all the partial derivatives in space of polynomials of degree \(d-1\). Let \(\rho\) be a representation of a connected semi-simple group \(G\) on \(\mathbb{C}^ n\). Then \(f\) is invariant if \(\rho(g)^*f=f\) for all \(g\) in \(G\). Now by the chain rule, one has \(J(\rho(g)^*f)=\rho(g)^*J(f)\) for all \(g \in G\). Hence \(J(f)\) is invariant under the \(\rho(g)^*\) action of \(G\) if \(f\) is invariant. In this paper, some converse result is obtained, namely Theorem: Assume that either a) \(f=0\) is projectively smooth or b) \(\rho\) is irreducible. Then \(f\) is invariant if \(J(f)\) is invariant and \(d \geq 3\).
scientific article; zbMATH DE number 447105