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Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids

  • Matthew H. Y. Xie
  • Zuo-Ru Zhang
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Abstract

We prove that the Kazhdan--Lusztig polynomial of a loopless sparse paving matroid is unimodal. When the rank is at least five and the corank is positive, we show that the coefficient sequence is strictly unimodal except for three explicitly determined parameter triples, each giving two equal maximal coefficients. We also determine the modes and classify the degenerate loopless sparse paving matroids of positive rank and corank.

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Cite

BibTeXRIS
BibTeX
@misc{xie2026unimodality,
  title = {{Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids}},
  author = {Matthew H. Y. Xie and Zuo-Ru Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.40084},
  eprint = {2609.40084},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.40084},
}
RIS
TY  - UNPB
TI  - Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids
AU  - Matthew H. Y. Xie
AU  - Zuo-Ru Zhang
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2609.40084
UR  - https://arxiv.org/abs/2609.40084
AB  - We prove that the Kazhdan--Lusztig polynomial of a loopless sparse paving matroid is unimodal. When the rank is at least five and the corank is positive, we show that the coefficient sequence is strictly unimodal except for three explicitly determined parameter triples, each giving two equal maximal coefficients. We also determine the modes and classify the degenerate loopless sparse paving matroids of positive rank and corank.
ER  -