Secant varieties and degrees of invariants

Abstract

The ring of invariant polynomials ${\mathbb C}[V]^G$ over a given finite dimensional representation space $V$ of a complex reductive group $G$ is known, by a famous theorem of Hilbert, to be finitely generated. The general proof being nonconstructive, the generators and their degrees have remained a subject of interest. In this article we determine certain divisors of the degrees of the generators. Also, for irreducible representations, we provide lower bounds for the degrees, determined by the geometric properties of the unique closed projective $G$-orbit $\mathbb X$, and more specifically its secant varieties. For a particular class of representations, where the secant varieties are especially well behaved, we exhibit an exact correspondence between the generating invariants and the secant varieties intersecting the semistable locus.