Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 067, 9 pages      arXiv:1604.06286

Exchange Relations for Finite Type Cluster Algebras with Acyclic Initial Seed and Principal Coefficients

Salvatore Stella a and Pavel Tumarkin b
a) IN$d$AM - Marie Curie Actions fellow, Università ''La Sapienza'', Roma, Italy
b) Department of Mathematical Sciences, Durham University, UK

Received April 22, 2016, in final form July 06, 2016; Published online July 09, 2016

We give an explicit description of all the exchange relations in any finite type cluster algebra with acyclic initial seed and principal coefficients.

Key words: cluster algebras; exchange relations.

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  1. Caldero P., Keller B., From triangulated categories to cluster algebras, Invent. Math. 172 (2008), 169-211, math.RT/0506018.
  2. Chapoton F., Fomin S., Zelevinsky A., Polytopal realizations of generalized associahedra, Canad. Math. Bull. 45 (2002), 537-566, math.CO/0202004.
  3. Felikson A., Shapiro M., Tumarkin P., Cluster algebras and triangulated orbifolds, Adv. Math. 231 (2012), 2953-3002, arXiv:1111.3449.
  4. Felikson A., Tumarkin P., Bases for cluster algebras from orbifolds, arXiv:1511.08023.
  5. Fomin S., Shapiro M., Thurston D., Cluster algebras and triangulated surfaces. I. Cluster complexes, Acta Math. 201 (2008), 83-146, math.RA/0608367.
  6. Fomin S., Thurston D., Cluster algebras and triangulated surfaces. II. Lambda lengths, arXiv:1210.5569.
  7. Fomin S., Zelevinsky A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), 497-529, math.RT/0104151.
  8. Fomin S., Zelevinsky A., Cluster algebras. II. Finite type classification, Invent. Math. 154 (2003), 63-121, math.RA/0208229.
  9. Fomin S., Zelevinsky A., Cluster algebras. IV. Coefficients, Compos. Math. 143 (2007), 112-164, math.RA/0602259.
  10. Musiker G., Williams L., Matrix formulae and skein relations for cluster algebras from surfaces, Int. Math. Res. Not. 2013 (2013), 2891-2944, arXiv:1108.3382.
  11. Nakanishi T., Stella S., Diagrammatic description of $c$-vectors and $d$-vectors of cluster algebras of finite type, Electron. J. Combin. 21 (2014), Paper 1.3, 107, arXiv:1210.6299.
  12. Nakanishi T., Zelevinsky A., On tropical dualities in cluster algebras, in Algebraic Groups and Quantum Groups, Contemp. Math., Vol. 565, Amer. Math. Soc., Providence, RI, 2012, 217-226, arXiv:1101.3736.
  13. Reading N., Universal geometric cluster algebras from surfaces, Trans. Amer. Math. Soc. 366 (2014), 6647-6685, arXiv:1209.4095.
  14. Schiffler R., A geometric model for cluster categories of type $D_n$, J. Algebraic Combin. 27 (2008), 1-21, math.RT/0608264.
  15. Stella S., Polyhedral models for generalized associahedra via Coxeter elements, J. Algebraic Combin. 38 (2013), 121-158, arXiv:1111.1657.
  16. Yang S.-W., Zelevinsky A., Cluster algebras of finite type via Coxeter elements and principal minors, Transform. Groups 13 (2008), 855-895, arXiv:0804.3303.

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