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Anton Khoroshkin

  1. Anick-type resolutions and consecutive pattern avoidance.

    Tue, 02/16/2010 - 16:01 — SomNambul
    Authors: Vladimir Dotsenko, Anton Khoroshkin
    Subjects: Combinatorics
    Abstract

    For permutations avoiding consecutive patterns from a given set, we present a
    combinatorial formula for the multiplicative inverse of the corresponding
    exponential generating function. The formula comes from homological algebra
    considerations in the same sense as the corresponding inversion formula for
    avoiding word patterns comes from the well known Anick's resolution.

  2. Free resolutions via Gr\"obner bases.

    Fri, 12/25/2009 - 16:01 — SomNambul
    Authors: Vladimir Dotsenko, Anton Khoroshkin
    Subjects: K-Theory and Homology
    Abstract

    In many different settings (associative algebras, commutative algebras,
    operads, dioperads), it is possible to develop the machinery of Gr\"obner
    bases; it allows to find a "monomial replacement" for every object in the
    corresponding category. The main goal of this article is to demonstrate how
    this machinery can be used for the purposes of homological algebra. More
    precisely, we define combinatorial resolutions in the monomial case and then
    show how they can be adjusted to be used in the general homogeneous case. We
    also discuss a way to make our monomial resolutions minimal.

  3. Gr\"obner bases for operads.

    Thu, 09/10/2009 - 10:12 — SomNambul
    Authors: Vladimir Dotsenko, Anton Khoroshkin
    Subjects: Quantum Algebra
    Abstract

    We define a new monoidal category on collections (shuffle composition).
    Monoids in this category (shuffle operads) turn out to bring a new insight in
    the theory of symmetric operads. For this category, we develop the machinery of
    Gr\"obner bases for operads, and present operadic versions of Bergman's Diamond
    Lemma and Buchberger's algorithm. This machinery can be applied to study
    symmetric operads. In particular, we obtain an effective algorithmic version of
    Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads.

  4. Gr\"obner bases for operads.

    Thu, 09/10/2009 - 10:12 — SomNambul
    Authors: Vladimir Dotsenko, Anton Khoroshkin
    Subjects: Quantum Algebra
    Abstract

    We define a new monoidal category on collections (shuffle composition).
    Monoids in this category (shuffle operads) turn out to bring a new insight in
    the theory of symmetric operads. For this category, we develop the machinery of
    Gr\"obner bases for operads, and present operadic versions of Bergman's Diamond
    Lemma and Buchberger's algorithm. This machinery can be applied to study
    symmetric operads. In particular, we obtain an effective algorithmic version of
    Hoffbeck's PBW criterion of Koszulness for (symmetric) quadratic operads.

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