Dealing with Orbit Closures in Representations with Finitely Many Orbits
summary
Classifies irreducible representations whose action has finitely many orbits, identifies the relevant graded piece G1 (examples like ∧^3 C^7), and converts each orbit closure into a vector-bundle desingularization over a Grassmannian to study normality, Hilbert series, rational singularities and minimal free resolutions. Methods combine cohomology calculations, Euler characteristic bounds, and computer algebra (Macaulay2) plus Eisenbud’s exactness criterion to prove expected complexes in many E6/E7 cases; E8 remains too large for current computations.
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