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G. Dupont

  1. Friezes and a construction of the euclidean cluster variables.

    Втр, 03/02/2010 - 16:01 — SomNambul
    Authors: G. Dupont, I. Assem
    Subjects: Representation Theory
    Abstract

    Let $Q$ be an euclidean quiver. Using friezes in the sense of
    Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical)
    cluster character associated to any object in the cluster category of $Q$. In
    particular, this algorithm allows to compute all the cluster variables in the
    cluster algebra associated to $Q$. It also allows to compute the sum of the
    Euler characteristics of the quiver grassmannians of any module $M$ over the
    path algebra of $Q$.

  2. Generic cluster characters.

    Пт, 02/05/2010 - 16:01 — SomNambul
    Authors: G. Dupont
    Subjects: Representation Theory
    Abstract

    Let $\mathcal C$ be a Hom-finite triangulated 2-Calabi-Yau category with
    constructible cones and let $T$ be a cluster-tilting object in $\mathcal C$. We
    introduce a set $\mathcal G^T(\mathcal C)$ of generic values of the cluster
    character associated to $T$ parameterized by the Grothendieck group $K_0(\add
    T)$. We prove that the set $\mathcal G^T(\mathcal C)$ naturally contains the
    cluster monomials of the cluster algebra associated to the Gabriel quiver of
    the cluster-tilted algebra $\End_{\CC}(T)^{\op}$.

  3. Generalized Chebyshev Polynomials and Positivity for Regular Cluster Characters.

    Чт, 11/05/2009 - 16:01 — SomNambul
    Authors: G. Dupont
    Subjects: Representation Theory
    Abstract

    We introduce a positive character on the category of finite dimensional
    representations of the $\mathbb A$-double-infinite quiver. We prove several
    interactions between this character and generalized Chebyshev polynomials
    finding applications to acyclic cluster algebras.

    If $Q$ is any representation-infinite acyclic quiver, we prove that the
    positivity of the cluster character of an indecomposable regular $\kQ$-module
    can be deduced from the positivity of the cluster characters of its
    quasi-composition factors.

  4. Transverse Quiver Grassmannians and Bases in Affine Cluster Algebras.

    Пт, 10/30/2009 - 16:01 — SomNambul
    Authors: G. Dupont
    Subjects: Representation Theory
    Abstract

    Sherman-Zelevinsky and Cerulli constructed canonically positive bases in
    cluster algebras associated to affine quivers having at most three vertices.
    These constructions involve cluster monomials and Chebyshev polynomials of the
    first kind evaluated at a certain "imaginary" element in the cluster algebra.

  5. Cluster multiplication in regular components via generalized Chebyshev polynomials.

    Пт, 10/16/2009 - 04:10 — SomNambul
    Authors: G. Dupont
    Subjects: Representation Theory
    Abstract

    We introduce a multivariate generalization of normalized Chebyshev
    polynomials of the second kind. We prove that these polynomials arise in the
    context of cluster characters associated to Dynkin quivers of type $\mathbb A$
    and representation-infinite quivers. This allows to obtain a simple
    combinatorial description of cluster algebras of type $\mathbb A$. We also
    provide explicit multiplication formulas for cluster characters associated to
    regular modules over the path algebra of any representation-infinite quiver.

  6. Quantized Chebyshev polynomials and cluster characters with coefficients.

    Пт, 08/28/2009 - 23:56 — SomNambul
    Authors: G. Dupont
    Subjects: Representation Theory
    Abstract

    We introduce quantized Chebyshev polynomials as deformations of generalized
    Chebyshev polynomials previously introduced by the author in the context of
    acyclic coefficient-free cluster algebras. We prove that these quantized
    polynomials arise in cluster algebras with principal coefficients associated to
    acyclic quivers of infinite representation types and equioriented Dynkin
    quivers of type $\mathbb A$. We also study their interactions with bases and
    especially canonically positive bases in affine cluster algebras.

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