Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated 22 min ago
Peer-reviewedCC BY 4.0Forum of Mathematics, Sigma

Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel

  • Peigen Cao1
  • Antoine de Saint Germain2
  • Jiang-Hua Lu2
  1. 1University of Science and Technology of China
  2. 2The University of Hong Kong
Open at sourceDOICite

Abstract

We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.

Related

  1. Discriminants of Hermitian forms, maximal degenerations and Kontsevich's tropical approach to the Hodge conjecture
    Patrick Brosnan
    arXivPreprintCC BY 4.0Full textAlgebra & number theoryarXiv
  2. A Tropical Viewpoint on the Nagata--Biran--Szemberg Conjecture
    Juliette Bruce
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Finite Cotangent Power Sums Weighted by Additive Characters With Application to Mean Values of Additive-and-Multiplicatively Twisted L-Functions
    Abdelmejid Bayad, Salah Eddine Rihane
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Halving the original Kalton--Roberts upper bound for nearly additive set functions
    Boon Suan Ho, Tomasz Kania
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. The Fibonacci numbers are not an additive uniqueness set for multiplicative functions
    Poo-Sung Park
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeXRIS
BibTeX
@article{cao2026tropical,
  title = {{Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel}},
  author = {Peigen Cao and Antoine de Saint Germain and Jiang-Hua Lu},
  journal = {Forum of Mathematics, Sigma},
  year = {2026},
  month = oct,
  doi = {10.1017/fms.2026.10291},
  url = {https://www.cambridge.org/core/product/identifier/S2050509426102916/type/journal_article},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Tropical friezes and cluster-additive functions via Fock-Goncharov duality and a conjecture of Ringel
AU  - Peigen Cao
AU  - Antoine de Saint Germain
AU  - Jiang-Hua Lu
PY  - 2026
DA  - 2026/10/20
JO  - Forum of Mathematics, Sigma
DO  - 10.1017/fms.2026.10291
UR  - https://www.cambridge.org/core/product/identifier/S2050509426102916/type/journal_article
AB  - We study tropical friezes and cluster-additive functions associated to symmetrizable generalized Cartan matrices in the framework of Fock-Goncharov duality in cluster algebras. In particular, we generalize and prove a conjecture of C. M. Ringel on cluster-additive functions associated to arbitrary Cartan matrices of finite type. For a Fock-Goncharov dual pair of positive spaces of finite type, we use tropical friezes and cluster-additive functions to explicitly express the Fock-Goncharov pairing between their tropical points and the bijections between global monomials on one and tropical points of the other.
SN  - 2050-5094
ER  -