Note on the decomposition of states
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We derive a sharp decomposition formula for the state polytope of the Hilbert point and the Hilbert-Mumford index of reducible varieties by using the decomposition of characters and basic convex geometry. This proof captures the essence of the decomposition of the state polytopes in general, and considerably simplifies an earlier proof by the authors which uses a careful analysis of initial ideals of reducible varieties.
Abstract
We derive a sharp decomposition formula for the state polytope of the Hilbert point and the Hilbert-Mumford index of reducible varieties by using the decomposition of characters and basic convex geometry. This proof captures the essence of the decomposition of the state polytopes in general, and considerably simplifies an earlier proof by the authors which uses a careful analysis of initial ideals of reducible varieties.
참고문헌 (4)
- D. Bayer, "Standard bases and geometric invariant theory. I. Initial ideals and state polytopes", J. Symbolic Comput., 6(2-3), pp.209~217 (1988)[DOI]
- D. Hyeon, "A state polytope decomposition formula", Proc. Edinb. Math. Soc. (2), 59(3), pp.759~776 (2016)
- G. R. Kempf, "Instability in invariant theory", Ann. of Math., 108(2), pp.299~316 (1978)[DOI]
- D. Mumford, "Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34" (1965), Springer-Verlag