Special folding of quivers and cluster algebras


  • Dani Kaufman




We give a precise definition of folded quivers and folded cluster algebras. We define a special folding of a quiver as one which cannot be associated with a skew-symmetrizable exchange matrix. We give many examples of including some with finite mutation structure that do not have analogues in the unfolded cases. We relate these examples to the finite mutation type quivers $X_6$ and $X_7$. We also construct a folded cluster algebra associated to punctured surfaces which allow for self-folded triangles. We give a simple construction of a folded cluster algebra for which the cluster complex is a generalized permutohedron.


Barot, M., and Geiss, C., Tubular cluster algebras I: categorification, Math. Z. 271 (2012), no. 3–4, 1091-1115. https://doi.org/10.1007/s00209-011-0905-8

Felikson, A., Shapiro, M., and Tumarkin, P., Cluster algebras and triangulated orbifolds, Adv. Math. 231 (2012), no. 5, 2953–3002. https://doi.org/10.1016/j.aim.2012.07.032

Felikson, A., Shapiro, M., and Tumarkin, P., Cluster algebras of finite mutation type via unfoldings, Int. Math. Res. Not. IMRN 2012, no. 8, 1768–1804. https://doi.org/10.1093/imrn/rnr072

Felikson, A., Shapiro, M., and Tumarkin, P., Skew-symmetric cluster algebras of finite mutation type, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1135–1180. https://doi.org/10.4171/JEMS/329

Fock, V., and Goncharov, A., Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. No. 103 (2006), 1–211. https://doi.org/10.1007/s10240-006-0039-4

Fomin, S., and Reading, N., Root systems and generalized associahedra, Geometric combinatorics, 63–131, IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007. https://doi.org/10.1090/pcms/013/03

Fomin, S., Shapiro, M., and Thurston, D., Cluster algebras and triangulated surfaces. I: Cluster complexes, Acta Math. 201 (2008), no. 1, 83–146. https://doi.org/10.1007/s11511-008-0030-7

Fomin, S., and Thurston, D., Cluster algebras and triangulated surfaces. II: Lambda lengths., Mem. Amer. Math. Soc. 255 (2018), no. 1223, v+97 pp. https://doi.org/10.1090/memo/1223

Fomin, S., and Zelevinsky, A., Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529. https://doi.org/10.1090/S0894-0347-01-00385-X

Fraser, C., Cyclic symmetry loci in Grasssmannians, arXiv:2010.05972. https://doi.org/10.48550/arXiv.2010.05972

Kaufman, D., Special folding of quivers and cluster algebras, arXiv:2304.07510 https://doi.org/10.48550/arXiv.2304.07510

Maxwell, G., The normal subgroups of finite and affine Coxeter groups, Proc. Lond. Math. Soc. (3) 76 (1998), no. 2, 359–382. https://doi.org/10.1112/S0024611598000112

Scott, J. S., Grassmannians and cluster algebras, Proc. Lond. Math. Soc. (3) 92 (2006), no. 2, 345–380. https://doi.org/10.1112/S0024611505015571



How to Cite

Kaufman, D. (2024). Special folding of quivers and cluster algebras. MATHEMATICA SCANDINAVICA, 130(2). https://doi.org/10.7146/math.scand.a-143446