PreprintarXiv licencearXiv
Prescribed Turán sign patterns for inverse Kazhdan-Lusztig polynomials of matroids
Abstract
Gao and Xie conjectured that the coefficients of the ordinary, unnormalized inverse Kazhdan-Lusztig polynomial of every matroid are log-concave with no internal zeros. We give counterexamples and prove locality and composition formulas for deletions of projective charts. For every finite field and every nonempty finite set S contained in {2, 3, ...}, these formulas yield simple 3-connected matroids whose negative internal Turán determinants occur exactly at S. Their coefficients are positive and strictly decreasing. The polynomial degree can be prescribed above max S, and all selected Turán ratios can tend to zero simultaneously. The universal first-index inequality remains open.
Related
- Real-rootedness of Kazhdan--Lusztig polynomials of sparse paving matroids
- Unimodality of Kazhdan--Lusztig polynomials of sparse paving matroids
- Relative Rouquier linearity and inverse hybrid Kazhdan-Lusztig positivity
- Kazhdan--Lusztig Polynomials for permutations with Pid,w(1)=3 and Irreducible Singular Locus
- Real-rootedness of the Z-polynomials of sparse paving matroids and strict interlacing for uniform matroids
BibTeX
@misc{badalov2026prescribed,
title = {{Prescribed Turán sign patterns for inverse Kazhdan-Lusztig polynomials of matroids}},
author = {Elias Badalov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13417},
eprint = {2609.13417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13417},
}
RIS
TY - UNPB
TI - Prescribed Turán sign patterns for inverse Kazhdan-Lusztig polynomials of matroids
AU - Elias Badalov
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13417
UR - https://arxiv.org/abs/2609.13417
AB - Gao and Xie conjectured that the coefficients of the ordinary, unnormalized inverse Kazhdan-Lusztig polynomial of every matroid are log-concave with no internal zeros. We give counterexamples and prove locality and composition formulas for deletions of projective charts. For every finite field and every nonempty finite set S contained in {2, 3, ...}, these formulas yield simple 3-connected matroids whose negative internal Turán determinants occur exactly at S. Their coefficients are positive and strictly decreasing. The polynomial degree can be prescribed above max S, and all selected Turán ratios can tend to zero simultaneously. The universal first-index inequality remains open.
ER -