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Algèbres amassées associées aux variétés de Richardson ouvertes : un algorithme de calcul de graines initiales

Abstract : Cluster algebras are integral domains with a particular combinatorial structure. This structure consists in the data of a family of seeds linked together by an operation called mutation. Each seed consists in two parts : a cluster and a quiver. Richardson open varieties are some strata of the flag variety associated to a simple linear algebraic group of simply-laced type. These are the intersection of Schubert cells with respect to two opposite Borel subgroups. In [Lec16] a cluster subalgebra of maximal rank on the coordinate ring of an open Richardson variety has been constructed and this subalgebra is conjectured to be equal to the whole ring. The construction of this cluster algebra comes from a Frobenius category C v,w of modules over the preprojective algebra, defined as the intersection of two categories C w and C v already studied by Geiss, Leclerc, Schröer and Buan, Iyama, Reiten and Scott. The bond between cluster algebras and cluster structures is given by the cluster character defined in [GLS06]. In this thesis we build an algorithm which, given the parameters defining a Richardson open variety, compute an explicit maximal rigid module of the associated Frobenius category and its quiver. This algorithm has an initial seed for the cluster structure on C w defined by a representative w of an element w of the Weyl group as a starting datum. By a combinatorially defined sequence of mutation on this initial seed we obtain a maximal rigid module of C w which is, up to deletion of some direct summands is a maximal rigid module of C v,w . In addition, the subquiver of the mutated quiver is exactly the quiver of the endomorphism algebra of the C v,w -maximal rigid module, giving then the complete description of an initial seed for the cluster structure on C v,w .
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https://hal-normandie-univ.archives-ouvertes.fr/tel-03237853
Contributor : Etienne Ménard Connect in order to contact the contributor
Submitted on : Monday, July 12, 2021 - 10:28:24 AM
Last modification on : Tuesday, October 19, 2021 - 11:33:54 PM

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  • HAL Id : tel-03237853, version 1

Citation

Etienne Menard. Algèbres amassées associées aux variétés de Richardson ouvertes : un algorithme de calcul de graines initiales. Théorie des représentations [math.RT]. ComUE Normandie Université, 2021. Français. ⟨tel-03237853⟩

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