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Fundamental invariants of orbit closures
Zitatschlüssel BI-Fundamental-Invariants-Of-Orbit-Closures
Autor Peter Bürgisser and Christian Ikenmeyer
Seiten 390–434
Jahr 2017
DOI 10.1016/j.jalgebra.2016.12.035
Journal Journal of Algebra
Jahrgang 477
Monat 05
Notiz Available online, to appear soon
Zusammenfassung For several objects of interest in geometric complexity theory, namely for the determinant, the permanent, the product of variables, the power sum, the unit tensor, and the matrix multiplication tensor, we introduce and study a fundamental SL-invariant function that relates the coordinate ring of the orbit with the coordinate ring of its closure. For the power sums we can write down this fundamental invariant explicitly in most cases. Our constructions generalize the two Aronhold invariants on ternary cubics. For the other objects we identify the invariant function conditional on intriguing combinatorial problems much like the well-known Alon-Tarsi conjecture on Latin squares. We provide computer calculations in small dimensions for these cases. As a main tool for our analysis, we determine the stabilizers, and we establish the polystability of all the mentioned forms and tensors (including the generic ones).
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