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Askold G. Khovanskii

  1. Moment polytopes, semigroup of representations and Kazarnovskii's theorem.

    Втр, 03/02/2010 - 16:01 — SomNambul
    Authors: Kiumars Kaveh, Askold G. Khovanskii
    Subjects: Representation Theory
    Abstract

    Two representations of a reductive group G are spectrally equivalent if the
    same irreducible representations appear in both of them. The semigroup of
    finite dimensional representations of G with tensor product and up to spectral
    equivalence is a rather complicated object. We show that the Grothendieck group
    of this semigroup is more tractable and give a description of it in terms of
    moment polytopes of representations. As a corollary, we give a proof of the
    Kazarnovskii theorem on the number of solutions in G of a system f_1(x) = ...

  2. Convex bodies associated to actions of reductive groups.

    Чт, 01/28/2010 - 16:01 — SomNambul
    Authors: Kiumars Kaveh, Askold G. Khovanskii
    Subjects: Algebraic Geometry
    Abstract

    We associate convex bodies to a wide class of graded G-algebras where G is a
    connected reductive group. These convex bodies give information about the
    Hilbert function as well as the multiplicities of irreducible representations
    appearing in the graded algebra. We extend the notion of Duistermaat-Heckman
    measure to graded G-algebras and prove a Fujita type approximation theorem as
    well as a Brunn-Minkowski inequality for this measure. This in particular
    applies to arbitrary G-line bundles giving an equivariant version of the theory
    of volumes of line bundles.

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