In this talk I explain the techniques needed to deal with the orbit closures in representations with finitely many orbits. These representations were classified by Kac in 1980.
The goals are to decide the normality and Cohen-Macaulay property of the orbit closure, the Hilbert function of the coordinate ring, the defining equations and even in some cases the free resolution of the polynomial ring.
I will discuss the techniques and will give examples of orbit closures in the space of skew symmetric 3-tensors in 7 variables.
This is based on the joint work with Kraskiewicz (on the arxive, arXiv:1201.1102 and arXiv:1301.0720) and on the work of Federico Galetto (arXiv:1210.6410).