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Log-concavity and unimodality of cluster monomials in fan coordinates of type An

  • Zhichao Chen
  • Zhe Sun
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Abstract

We prove that every cluster monomial of type An is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. In even rank, we reduce log-concavity in fan coordinates to log-concavity along integer lines. In odd rank, we use convolution to handle the coefficient sums arising from the fan substitution. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type An.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026logconcavity,
  title = {{Log-concavity and unimodality of cluster monomials in fan coordinates of type $A_n$}},
  author = {Zhichao Chen and Zhe Sun},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09312},
  eprint = {2610.09312},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09312},
}
RIS
TY  - UNPB
TI  - Log-concavity and unimodality of cluster monomials in fan coordinates of type A_n
AU  - Zhichao Chen
AU  - Zhe Sun
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09312
UR  - https://arxiv.org/abs/2610.09312
AB  - We prove that every cluster monomial of type A_n is log-concave and unimodal with respect to its Laurent expansion in a fan initial cluster. We derive a coefficient formula for laminar interval products arising from compatible diagonals in the polygon model. In even rank, we reduce log-concavity in fan coordinates to log-concavity along integer lines. In odd rank, we use convolution to handle the coefficient sums arising from the fan substitution. This gives partial affirmative answers to the log-concavity conjecture of Chen-Huang-Sun and the unimodality conjecture of Chen. Our result also complements Shen's study on the supports of products of canonical basis elements for cluster X-varieties of type A_n.
ER  -