Cambrian frameworks for cluster algebras of affine type

Reading, Nathan; Speyer, David

doi:10.1090/tran/7193

Cambrian frameworks for cluster algebras of affine type
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by Nathan Reading and David E. Speyer PDF

Trans. Amer. Math. Soc. 370 (2018), 1429-1468

Abstract:

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 $-B$.

References

Drew Armstrong, Generalized noncrossing partitions and combinatorics of Coxeter groups, Mem. Amer. Math. Soc. 202 (2009), no. 949, x+159. MR 2561274, DOI 10.1090/S0065-9266-09-00565-1
Nicolas Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR 1890629, DOI 10.1007/978-3-540-89394-3
Thomas Brady and Colum Watt, A partial order on the orthogonal group, Comm. Algebra 30 (2002), no. 8, 3749–3754. MR 1922309, DOI 10.1081/AGB-120005817
Thomas Brady and Colum Watt, $K(\pi ,1)$’s for Artin groups of finite type, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 225–250. MR 1950880, DOI 10.1023/A:1020902610809
Thomas Brüstle, Grégoire Dupont, and Matthieu Pérotin, On maximal green sequences, Int. Math. Res. Not. IMRN 16 (2014), 4547–4586. MR 3250044, DOI 10.1093/imrn/rnt075
R. W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1–59. MR 318337
L. Demonet, Mutations of group species with potentials and their representations. Applications to cluster algebras, Preprint, 2010 (arXiv:1003.5078).
Vinay V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel) 53 (1989), no. 6, 543–546. MR 1023969, DOI 10.1007/BF01199813
Matthew Dyer, Reflection subgroups of Coxeter systems, J. Algebra 135 (1990), no. 1, 57–73. MR 1076077, DOI 10.1016/0021-8693(90)90149-I
M. J. Dyer and G. I. Lehrer, Reflection subgroups of finite and affine Weyl groups, Trans. Amer. Math. Soc. 363 (2011), no. 11, 5971–6005. MR 2817417, DOI 10.1090/S0002-9947-2011-05298-0
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. MR 1887642, DOI 10.1090/S0894-0347-01-00385-X
Sergey Fomin and Andrei Zelevinsky, Cluster algebras: notes for the CDM-03 conference, Current developments in mathematics, 2003, Int. Press, Somerville, MA, 2003, pp. 1–34. MR 2132323
Sergey Fomin and Andrei Zelevinsky, Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), no. 1, 112–164. MR 2295199, DOI 10.1112/S0010437X06002521
M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, Preprint, 2014. arXiv:1411.1394
James E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, vol. 29, Cambridge University Press, Cambridge, 1990. MR 1066460, DOI 10.1017/CBO9780511623646
Kiyoshi Igusa, Kent Orr, Gordana Todorov, and Jerzy Weyman, Cluster complexes via semi-invariants, Compos. Math. 145 (2009), no. 4, 1001–1034. MR 2521252, DOI 10.1112/S0010437X09004151
K. Igusa, K. Orr, G. Todorov, and J. Weyman, Modulated semi-invariants, Preprint, 2015. arXiv:1507.03051
Colin Ingalls, Charles Paquette, and Hugh Thomas, Semi-stable subcategories for Euclidean quivers, Proc. Lond. Math. Soc. (3) 110 (2015), no. 4, 805–840. MR 3335288, DOI 10.1112/plms/pdv002
Colin Ingalls and Hugh Thomas, Noncrossing partitions and representations of quivers, Compos. Math. 145 (2009), no. 6, 1533–1562. MR 2575093, DOI 10.1112/S0010437X09004023
Victor G. Kac, Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. MR 1104219, DOI 10.1017/CBO9780511626234
Bernhard Keller, On cluster theory and quantum dilogarithm identities, Representations of algebras and related topics, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2011, pp. 85–116. MR 2931896, DOI 10.4171/101-1/3
I. G. Macdonald, Affine root systems and Dedekind’s $\eta$-function, Invent. Math. 15 (1972), 91–143. MR 357528, DOI 10.1007/BF01418931
Greg Muller, The existence of a maximal green sequence is not invariant under quiver mutation, Electron. J. Combin. 23 (2016), no. 2, Paper 2.47, 23. MR 3512669
Nathan Reading, Cambrian lattices, Adv. Math. 205 (2006), no. 2, 313–353. MR 2258260, DOI 10.1016/j.aim.2005.07.010
Nathan Reading, Clusters, Coxeter-sortable elements and noncrossing partitions, Trans. Amer. Math. Soc. 359 (2007), no. 12, 5931–5958. MR 2336311, DOI 10.1090/S0002-9947-07-04319-X
Nathan Reading, Sortable elements and Cambrian lattices, Algebra Universalis 56 (2007), no. 3-4, 411–437. MR 2318219, DOI 10.1007/s00012-007-2009-1
Nathan Reading, Universal geometric cluster algebras, Math. Z. 277 (2014), no. 1-2, 499–547. MR 3205782, DOI 10.1007/s00209-013-1264-4
Nathan Reading and David E. Speyer, Cambrian fans, J. Eur. Math. Soc. (JEMS) 11 (2009), no. 2, 407–447. MR 2486939, DOI 10.4171/JEMS/155
Nathan Reading and David E. Speyer, Sortable elements in infinite Coxeter groups, Trans. Amer. Math. Soc. 363 (2011), no. 2, 699–761. MR 2728584, DOI 10.1090/S0002-9947-2010-05050-0
Nathan Reading and David E. Speyer, Sortable elements for quivers with cycles, Electron. J. Combin. 17 (2010), no. 1, Research Paper 90, 19. MR 2661393
Nathan Reading and David E. Speyer, Combinatorial frameworks for cluster algebras, Int. Math. Res. Not. IMRN 1 (2016), 109–173. MR 3514060, DOI 10.1093/imrn/rnv101
Nathan Reading and David E. Speyer, A Cambrian framework for the oriented cycle, Electron. J. Combin. 22 (2015), no. 4, Paper 4.46, 21. MR 3441666