×

zbMATH — the first resource for mathematics

Cambrian frameworks for cluster algebras of affine type. (English) Zbl 1423.13131
Summary: We give a combinatorial model for the exchange graph and \( \mathbf {g}\)-vector fan associated to any acyclic exchange matrix \( B\) of affine type. More specifically, we construct a reflection framework for \( B\) in the sense of [N. Reading and D. E. Speyer, “Combinatorial frameworks for cluster algebras”] and establish good properties of this framework. The framework (and in particular the \( \mathbf {g}\)-vector fan) is constructed by combining a copy of the Cambrian fan for \( B\) with an antipodal copy of the Cambrian fan for \( -B\).

MSC:
13F60 Cluster algebras
20F55 Reflection and Coxeter groups (group-theoretic aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Armstrong, Drew, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc., 202, 949, x+159 pp., (2009) · Zbl 1191.05095
[2] Bourbaki, Nicolas, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), xii+300 pp., (2002), Springer-Verlag, Berlin · Zbl 0983.17001
[3] Brady, Thomas; Watt, Colum, A partial order on the orthogonal group, Comm. Algebra, 30, 8, 3749-3754, (2002) · Zbl 1018.20040
[4] Brady, Thomas; Watt, Colum, \(K(π,1)\)’s for Artin groups of finite type, Geom. Dedicata. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 94, 225-250, (2002) · Zbl 1053.20034
[5] Br\"ustle, Thomas; Dupont, Gr\'egoire; P\'erotin, Matthieu, On maximal green sequences, Int. Math. Res. Not. IMRN, 16, 4547-4586, (2014) · Zbl 1346.16009
[6] Carter, R. W., Conjugacy classes in the Weyl group, Compositio Math., 25, 1-59, (1972) · Zbl 0254.17005
[7] Demonet L. Demonet, Mutations of group species with potentials and their representations. Applications to cluster algebras, Preprint, 2010 (arXiv:1003.5078). · Zbl 1210.16017
[8] Deodhar, Vinay V., A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel), 53, 6, 543-546, (1989) · Zbl 0688.20028
[9] Dyer, Matthew, Reflection subgroups of Coxeter systems, J. Algebra, 135, 1, 57-73, (1990) · Zbl 0712.20026
[10] Dyer, M. J.; Lehrer, G. I., Reflection subgroups of finite and affine Weyl groups, Trans. Amer. Math. Soc., 363, 11, 5971-6005, (2011) · Zbl 1243.20051
[11] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. I. Foundations, J. Amer. Math. Soc., 15, 2, 497-529, (2002) · Zbl 1021.16017
[12] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras: notes for the CDM-03 conference. Current developments in mathematics, 2003, 1-34, (2003), Int. Press, Somerville, MA · Zbl 1119.05108
[13] Fomin, Sergey; Zelevinsky, Andrei, Cluster algebras. IV. Coefficients, Compos. Math., 143, 1, 112-164, (2007) · Zbl 1127.16023
[14] GHKK M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, Preprint, 2014. <span class=”texttt”>a</span>rXiv:1411.1394 · Zbl 1446.13015
[15] Humphreys, James E., Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, xii+204 pp., (1990), Cambridge University Press, Cambridge · Zbl 0725.20028
[16] Igusa, Kiyoshi; Orr, Kent; Todorov, Gordana; Weyman, Jerzy, Cluster complexes via semi-invariants, Compos. Math., 145, 4, 1001-1034, (2009) · Zbl 1180.14047
[17] IOTW2 K. Igusa, K. Orr, G. Todorov, and J. Weyman, Modulated semi-invariants, Preprint, 2015. <span class=”texttt”>a</span>rXiv:1507.03051
[18] Ingalls, Colin; Paquette, Charles; Thomas, Hugh, Semi-stable subcategories for Euclidean quivers, Proc. Lond. Math. Soc. (3), 110, 4, 805-840, (2015) · Zbl 1378.16026
[19] Ingalls, Colin; Thomas, Hugh, Noncrossing partitions and representations of quivers, Compos. Math., 145, 6, 1533-1562, (2009) · Zbl 1182.16012
[20] Kac, Victor G., Infinite-dimensional Lie algebras, xxii+400 pp., (1990), Cambridge University Press, Cambridge · Zbl 0716.17022
[21] Keller, Bernhard, On cluster theory and quantum dilogarithm identities. Representations of algebras and related topics, EMS Ser. Congr. Rep., 85-116, (2011), Eur. Math. Soc., Z\"urich · Zbl 1307.13028
[22] Macdonald, I. G., Affine root systems and Dedekind’s \(η\)-function, Invent. Math., 15, 91-143, (1972) · Zbl 0244.17005
[23] Muller, Greg, The existence of a maximal green sequence is not invariant under quiver mutation, Electron. J. Combin., 23, 2, Paper 2.47, 23 pp., (2016) · Zbl 1339.05163
[24] Reading, Nathan, Cambrian lattices, Adv. Math., 205, 2, 313-353, (2006) · Zbl 1106.20033
[25] Reading, Nathan, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc., 359, 12, 5931-5958, (2007) · Zbl 1189.05022
[26] Reading, Nathan, Sortable elements and Cambrian lattices, Algebra Universalis, 56, 3-4, 411-437, (2007) · Zbl 1184.20038
[27] Reading, Nathan, Universal geometric cluster algebras, Math. Z., 277, 1-2, 499-547, (2014) · Zbl 1328.13034
[28] Reading, Nathan; Speyer, David E., Cambrian fans, J. Eur. Math. Soc. (JEMS), 11, 2, 407-447, (2009) · Zbl 1213.20038
[29] Reading, Nathan; Speyer, David E., Sortable elements in infinite Coxeter groups, Trans. Amer. Math. Soc., 363, 2, 699-761, (2011) · Zbl 1231.20036
[30] Reading, Nathan; Speyer, David E., Sortable elements for quivers with cycles, Electron. J. Combin., 17, 1, Research Paper 90, 19 pp., (2010) · Zbl 1215.20039
[31] Reading, Nathan; Speyer, David E., Combinatorial frameworks for cluster algebras, Int. Math. Res. Not. IMRN, 1, 109-173, (2016) · Zbl 1330.05167
[32] Reading, Nathan; Speyer, David E., A Cambrian framework for the oriented cycle, Electron. J. Combin., 22, 4, Paper 4.46, 21 pp., (2015) · Zbl 1329.05166
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.