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PreprintCC BY 4.0arXiv

q-Zeta positivity for root and Ferrers-cell posets

  • Qihang Wang
  • Weiye Li
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Abstract

We prove that Chapoton's q-Zeta numerator has nonnegative coefficients for the positive-root poset of every finite reduced irreducible crystallographic root system, graded by root height minus one and normalized by Chapoton's prescribed denominator. The same holds for every interval of the root poset with zero adjoined. The key observation is that adjoining a minimum changes the numerator only by the substitution t↦ qt. The numerator thereby becomes a nonnegative combination of flag h-numbers of a lower-semimodular lattice, and a classical labelling of the dual lattice gives a positive descent expansion. To make the coefficients explicit, we construct signed edge-lexicographic (EL) labellings of Ferrers-cell posets with nonincreasing boundaries. They give ballot-word formulas in types A, B, and C, which prove an observation of Chapoton in type B, and a coloured path formula in type D. For trapezoidal boundaries, one-boundary major-index enumerators evaluate the path sums in Gaussian-binomial form.

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Cite

BibTeXRIS
BibTeX
@misc{wang2026qzeta,
  title = {{$q$-Zeta positivity for root and Ferrers-cell posets}},
  author = {Qihang Wang and Weiye Li},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.24541},
  eprint = {2609.24541},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24541},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - q-Zeta positivity for root and Ferrers-cell posets
AU  - Qihang Wang
AU  - Weiye Li
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.24541
UR  - https://arxiv.org/abs/2609.24541
AB  - We prove that Chapoton's q-Zeta numerator has nonnegative coefficients for the positive-root poset of every finite reduced irreducible crystallographic root system, graded by root height minus one and normalized by Chapoton's prescribed denominator. The same holds for every interval of the root poset with zero adjoined. The key observation is that adjoining a minimum changes the numerator only by the substitution t↦ qt. The numerator thereby becomes a nonnegative combination of flag h-numbers of a lower-semimodular lattice, and a classical labelling of the dual lattice gives a positive descent expansion. To make the coefficients explicit, we construct signed edge-lexicographic (EL) labellings of Ferrers-cell posets with nonincreasing boundaries. They give ballot-word formulas in types A, B, and C, which prove an observation of Chapoton in type B, and a coloured path formula in type D. For trapezoidal boundaries, one-boundary major-index enumerators evaluate the path sums in Gaussian-binomial form.
ER  -