Reddening Sequences for Banff Quivers and the Class $\mathcal{P}$

Eric Bucher ; John Machacek

Symmetry, integrability and geometry: methods and applications, Tome 16 (2020)

Résumé

We show that a reddening sequence exists for any quiver which is Banff. Our proof is combinatorial and relies on the triangular extension construction for quivers. The other facts needed are that the existence of a reddening sequence is mutation invariant and passes to induced subquivers. Banff quivers define locally acyclic cluster algebras which are known to coincide with their upper cluster algebras. The existence of reddening sequences for these quivers is consistent with a conjectural relationship between the existence of a reddening sequence and a cluster algebra's equality with its upper cluster algebra. Our result completes a verification of the conjecture for Banff quivers. We also prove that a certain subclass of quivers within the class $\mathcal{P}$ define locally acyclic cluster algebras.

Détail

Keywords: cluster algebras, reddening sequences.
Mots-clés : quiver mutation

Bibliographie

[1] Alim M., Cecotti S., Córdova C., Espahbodi S., Rastogi A., Vafa C., “BPS quivers and spectra of complete $\mathcal{N}=2$ quantum field theories”, Comm. Math. Phys., 323 (2013), 1185–1227, arXiv: 1109.4941 | DOI | MR | Zbl

[2] Berenstein A., Fomin S., Zelevinsky A., “Cluster algebras. III Upper bounds and double Bruhat cells”, Duke Math. J., 126 (2005), 1–52, arXiv: math.RT/0305434 | DOI | MR | Zbl

[3] Brüstle T., Dupont G., Pérotin M., “On maximal green sequences”, Int. Math. Res. Not., 2014 (2014), 4547–4586, arXiv: 1205.2050 | DOI | MR | Zbl

[4] Bucher E., “Maximal green sequences for cluster algebras associated to orientable surfaces with empty boundary”, Arnold Math. J., 2 (2016), 487–510, arXiv: 1412.3713 | DOI | MR | Zbl

[5] Bucher E., Machacek J., Shapiro M., “Upper cluster algebras and choice of ground ring”, Sci. China Math., 62 (2019), 1257–1266, arXiv: 1802.04835 | DOI | MR | Zbl

[6] Bucher E., Mills M. R., “Maximal green sequences for cluster algebras associated with the $n$-torus with arbitrary punctures”, J. Algebraic Combin., 47 (2018), 345–356, arXiv: 1503.06207 | DOI | MR | Zbl

[7] Canakci I., Lee K., Schiffler R., “On cluster algebras from unpunctured surfaces with one marked point”, Proc. Amer. Math. Soc. Ser. B, 2 (2015), 35–49, arXiv: 1407.5060 | DOI | MR | Zbl

[8] Cao P., Li F., “Uniform column sign-coherence and the existence of maximal green sequences”, J. Algebraic Combin., 50 (2019), 403–417, arXiv: 1712.00973 | DOI | MR | Zbl

[9] Derksen H., Weyman J., Zelevinsky A., “Quivers with potentials and their representations II: applications to cluster algebras”, J. Amer. Math. Soc., 23 (2010), 749–790 | DOI | MR | Zbl

[10] Fomin S., Zelevinsky A., “Cluster algebras. I Foundations”, J. Amer. Math. Soc., 15 (2002), 497–529, arXiv: math.RT/0104151 | DOI | MR | Zbl

[11] Ford N., Serhiyenko K., “Green-to-red sequences for positroids”, J. Combin. Theory Ser. A, 159 (2018), 164–182, arXiv: 1610.01695 | DOI | MR | Zbl

[12] Garver A., Musiker G., “On maximal green sequences for type $\mathbb A$ quivers”, J. Algebraic Combin., 45 (2017), 553–599, arXiv: 1403.6149 | DOI | MR | Zbl

[13] Goncharov A., Shen L., “Donaldson–Thomas transformations of moduli spaces of G-local systems”, Adv. Math., 327 (2018), 225–348, arXiv: 1602.06479 | DOI | MR | Zbl

[14] Goodearl K. R., Yakimov M. T., Cluster algebra structures on Poisson nilpotent algebras, arXiv: 1801.01963 | MR

[15] Gross M., Hacking P., Keel S., Kontsevich M., “Canonical bases for cluster algebras”, J. Amer. Math. Soc., 31 (2018), 497–608, arXiv: 1411.1394 | DOI | MR | Zbl

[16] Keller B., “On cluster theory and quantum dilogarithm identities, in Representations of Algebras and Related Topics”, EMS Ser. Congr. Rep., 2011, 85–116, Eur. Math. Soc., Zürich, arXiv: 1102.4148 | DOI | MR

[17] Keller B., “Cluster algebras and derived categories”, Derived Categories in Algebraic Geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2012, 123–183, arXiv: 1202.4161 | DOI | MR | Zbl

[18] Kontsevich M., Soibelman Y., Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv: 0811.2435 | MR

[19] Ladkani S., On cluster algebras from once punctured closed surfaces, arXiv: 1310.4454

[20] Lam T., Speyer D. E., Cohomology of cluster varieties. I. Locally acyclic case, arXiv: 1604.06843

[21] Lawson J. W., Mills M. R., “Properties of minimal mutation-infinite quivers”, J. Combin. Theory Ser. A, 155 (2018), 122–156, arXiv: 1610.08333 | DOI | MR | Zbl

[22] Mills M. R., “Maximal green sequences for quivers of finite mutation type”, Adv. Math., 319 (2017), 182–210, arXiv: 1606.03799 | DOI | MR | Zbl

[23] Mills M. R., On the relationship between green-to-red sequences, local-acyclicity, and upper cluster algebra, arXiv: 1804.00479

[24] Muller G., “Locally acyclic cluster algebras”, Adv. Math., 233 (2013), 207–247, arXiv: 1111.4468 | DOI | MR | Zbl

[25] Muller G., “${\mathcal A}={\mathcal U}$ for locally acyclic cluster algebras”, SIGMA, 10 (2014), 094, 8 pp., arXiv: 1308.1141 | DOI | MR | Zbl

[26] Muller G., “The existence of a maximal green sequence is not invariant under quiver mutation”, Electron. J. Combin., 23 (2016), 2.47, 23 pp., arXiv: 1503.04675 | DOI | MR | Zbl

[27] Muller G., Speyer D. E., “Cluster algebras of Grassmannians are locally acyclic”, Proc. Amer. Math. Soc., 144 (2016), 3267–3281, arXiv: 1401.5137 | DOI | MR | Zbl

[28] Postnikov A., Total positivity, Grassmannians, and networks, arXiv: math.CO/0609764

[29] Seven A. I., “Maximal green sequences of skew-symmetrizable $3\times3$ matrices”, Linear Algebra Appl., 440 (2014), 125–130, arXiv: 1207.6265 | DOI | MR | Zbl