Skip to main content

Relation Between f-Vectors and d-Vectors in Cluster Algebras of Finite Type or Rank 2


We study f-vectors, which are the maximal degree vectors of F-polynomials in cluster algebra theory. For a cluster algebra of finite type, we find that positive f-vectors correspond with d-vectors, which are exponent vectors of denominators of cluster variables. Furthermore, using this correspondence and properties of d-vectors, we prove that cluster variables in a cluster are uniquely determined by their f-vectors when the cluster algebra is of finite type or rank 2.

This is a preview of subscription content, access via your institution.

Fig. 1


  1. 1.

    Buan, A. B., Iyama, O., Reiten, I., Scott, J., Cluster structures for 2-Calabi-Yau categories and unipotent groups, Compos. Math., 145, (2009), no. 4, 1035–1079,

  2. 2.

    Buan, A. B., Marsh, R., Reineke, M., Reiten, I., Todorov, G., Tilting theory and cluster combinatorics, Adv. Math. 204 (2006), 572–618

  3. 3.

    Cao, P., Li, F., The enough \(g\)-pairs property and denominator vectors of cluster algebras, Trans. Amer. Math. Soc. 377 (2020), 1547–1572

  4. 4.

    Ceballos, C., Pilaud, V., Denominator vectors and compatibility degrees in cluster algebras of finite type, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1421–1439

  5. 5.

    Çanakçı, Í., Schiffler, R., Snake graphs and continued fractions, 2017. preprint, arXiv:1711.02461

  6. 6.

    Fock, V. V., Goncharov, A. B., Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Ecole Normale. Sup. 42 (2009), no. 6, 865–930

  7. 7.

    Fu, C., Geng, S., On indecomposable \(\tau \)-rigid modules over cluster-tilted algebras of tame type, 2017. preprint, arXiv:1705.10939

  8. 8.

    Fujiwara, S., Gyoda, Y., Duality between final-seed and initial-seed mutations in cluster algebras, SIGMA, 15 (2019), 24 pages

  9. 9.

    Fu, C., Geng, S., Liu, P., Cluster algebras arising from cluster tubes I: integer vectors, 2018. preprint, arXiv:1801.00709

  10. 10.

    Fu, C., Keller, B., On cluster algebras with coefficients and 2-Calabi-Yau categories, Trans. Amer. Math. Soc. 362 (2010), no. 2, 859–895

  11. 11.

    Fomin, S., Shapiro, M., Thurston, D., Cluster algebras and triangulated surfaces. part I: Cluster complexes, Acta Math. 201 (2008), 83–146

  12. 12.

    Fomin, S., Zelevinsky, A., Cluster Algebra I: Foundations, J. Amer. Math. Soc. 15 (2002), 497–529

  13. 13.

    Fomin, S., Zelevinsky, A., Cluster algebras II: Finite type classification, Invent. Math. 154 (2003), 63–121

  14. 14.

    Fomin, S., Zelevinsky, A., Cluster Algebra IV: Coefficients, Comp. Math. 143 (2007), 112–164

  15. 15.

    Geng, S., Peng, L., The dimension vectors of indecomposable modules of cluster-tilted algebras and the Fomin-Zelevinsky denominators conjecture, Acta. Math. 28 (2012), no. 3, 581–586

  16. 16.

    Gyoda, Y., Yurikusa, T., \(F\)-matrices of cluster algebras from triangulated surfaces, Ann. Comb. 24 (2020), no. 4, 649–695

  17. 17.

    Inoue, R., Iyama, O., Kuniba, A., Nakanishi, T., Suzuki, J., Periodicities of T-systems and Y-systems, Nagoya Math. J. 197 (2010), 59–174

  18. 18.

    Inoue, R., Iyama, O., Keller, B., Kuniba, A., Nakanishi, T., Suzuki, J., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type \(B_r\), Publ. RIMS. 49 (2013), 1–42

  19. 19.

    Inoue, R., Iyama, O., Keller, B., Kuniba, A., Nakanishi, T., Suzuki, J., Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Type \(C_r,F_4,G_2\), Publ. RIMS. 49 (2013), 43–85

  20. 20.

    Lee, K., Li, L., Zelevinsky, A., Greedy elements in rank 2 cluster algebras, Selecta Mathematica. 20 (2014), no. 1, 57–82

  21. 21.

    Lee, K., Schiffler, R., A combinatorial formula for rank \(2\) cluster variables, J. Algebraic Combin. 37 (2013), no. 1, 67–85

  22. 22.

    Nakanishi, T., Dilogarithm identities for conformal field theories and cluster algebras: simply laced case, Nagoya Math. J. 202 (2011), 23–43

  23. 23.

    Nakanishi, T., Stella, S., Diagrammatic description of \(c\)-vectors and \(d\)-vectors of cluster algebras of finite type, Electron. J. Comb. 21 (2014), no. 2, 107 pages

  24. 24.

    Ringel, C. M., Cluster-concealed algebras, Adv. Math. 226 (2011), no. 2, 1513–1537

  25. 25.

    Rabideau, M., Schiffler, R., Continued fractions and orderings on the Markov numbers, 2018. preprint, arXiv:1801.07155

  26. 26.

    Reading, N., Stella, S., Initial-seed recursions and dualities for d-vectors, Pacific J. Math. 293 (2018), 179–206

  27. 27.

    Sherman, P., Zelevinsky, A., Positivity and canonical bases in rank 2 cluster algebras of finite and affine types, Moscow Math. J. 4 (2004), no. 4, 947–974

Download references


The author would like to express his gratitude to Bernhard Keller for insightful comments about Theorem 1.8. The author appreciates important remarks about Conjecture 1.10 by Changjian Fu. Toshiya Yurikusa gives helpful advice about Theorem 1.11. The author received generous support from Tomoki Nakanishi. The author also thanks Haruhisa Enomoto, Yoshiki Aibara, and Naohiro Tsuzu. This work was supported by JSPS KAKENHI Grant number JP20J12675.

Author information



Corresponding author

Correspondence to Yasuaki Gyoda.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by Alexander Yong

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Gyoda, Y. Relation Between f-Vectors and d-Vectors in Cluster Algebras of Finite Type or Rank 2. Ann. Comb. 25, 573–594 (2021).

Download citation

Mathematics Subject Classification

  • 13F60


  • Cluster algebra
  • Mutation
  • F-vector
  • D-vector