## Journal of Geometry and Symmetry in Physics

### Secant Varieties and Degrees of Invariants

Valdemar V. Tsanov

#### Abstract

The ring of invariant polynomials $\mathbb{C}[V]^G$ over a given finite dimensional representation space $V$ of a connected 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. \linebreak 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.

#### Article information

Source
J. Geom. Symmetry Phys., Volume 51 (2019), 73-85.

Dates
First available in Project Euclid: 26 April 2019

https://projecteuclid.org/euclid.jgsp/1556244029

Digital Object Identifier
doi:10.7546/jgsp-51-2019-73-85

Mathematical Reviews number (MathSciNet)
MR0

Zentralblatt MATH identifier
0

#### Citation

Tsanov, Valdemar V. Secant Varieties and Degrees of Invariants. J. Geom. Symmetry Phys. 51 (2019), 73--85. doi:10.7546/jgsp-51-2019-73-85. https://projecteuclid.org/euclid.jgsp/1556244029