237 research outputs found

    A survey on maximal green sequences

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    Maximal green sequences appear in the study of Fomin-Zelevinsky's cluster algebras. They are useful for computing refined Donaldson-Thomas invariants, constructing twist automorphisms and proving the existence of theta bases and generic bases. We survey recent progress on their existence and properties and give a representation-theoretic proof of Greg Muller's theorem stating that full subquivers inherit maximal green sequences. In the appendix, Laurent Demonet describes maximal chains of torsion classes in terms of bricks generalizing a theorem by Igusa.Comment: 15 pages, submitted to the proceedings of the ICRA 18, Prague, comments welcome; v2: misquotation in section 6 corrected; v3: minor changes, final version; v4: reference to Jiarui Fei's work added, post-final version; v4: formulation of Remark 4.3 corrected; v5: misquotation of Hermes-Igusa's 2019 paper corrected; v5: reference to Kim-Yamazaki's paper adde

    Minimal Length Maximal Green Sequences and Triangulations of Polygons

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    We use combinatorics of quivers and the corresponding surfaces to study maximal green sequences of minimal length for quivers of type A\mathbb{A}. We prove that such sequences have length n+tn+t, where nn is the number of vertices and tt is the number of 3-cycles in the quiver. Moreover, we develop a procedure that yields these minimal length maximal green sequences.Comment: 22 pages, 1 figur
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