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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 thedata of a family of seeds linked together by an operation called mutation. Each seed consists in two parts : acluster and a quiver.Richardson open varieties are some strata of the flag variety associated to a simple linear algebraic groupof 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 beenconstructed and this subalgebra is conjectured to be equal to the whole ring. The construction of this clusteralgebra comes from a Frobenius category C v,w of modules over the preprojective algebra, defined as the intersectionof two categories C w and C v already studied by Geiss, Leclerc, Schröer and Buan, Iyama, Reiten and Scott. Thebond 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, computean explicit maximal rigid module of the associated Frobenius category and its quiver. This algorithm has aninitial seed for the cluster structure on C w defined by a representative w of an element w of the Weyl group as astarting datum. By a combinatorially defined sequence of mutation on this initial seed we obtain a maximal rigidmodule 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 rigidmodule, giving then the complete description of an initial seed for the cluster structure on C v,w .
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Submitted on : Friday, June 11, 2021 - 11:13:11 AM
Last modification on : Tuesday, October 19, 2021 - 11:33:54 PM
Long-term archiving on: : Sunday, September 12, 2021 - 6:01:33 PM


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Etienne Menard. Algèbres amassées associées aux variétés de Richardson ouvertes : un algorithme de calcul de graines initiales. Algèbre commutative [math.AC]. Normandie Université, 2021. Français. ⟨NNT : 2021NORMC211⟩. ⟨tel-03213985⟩



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