Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids
Matthew H. Y. Xie, Philip B. Zhang
TL;DR
The paper proves that for paving matroids, the Hadamard transform $\mathcal{B}(Q_M(t))$, obtained by multiplying the inverse Kazhdan-Lusztig polynomial $Q_M(t)$ by $ (1+t)^n $ in a coefficient-wise fashion, has only real roots. This real-rootedness is established via an interlacing framework built from multiplier sequences, Wronskian signs, and the Hermite-Kakeya-Obreschkoff theorem, allowing a decomposition into interlacing components derived from uniform matroids. Once $\mathcal{B}(Q_M(t))$ is real-rooted, Newton’s inequalities yield that the coefficients of $Q_M(t)$ are log-concave and have no internal zeros for paving matroids, with higher-order Turán inequalities also holding. The work extends the evidence for Gao and Xie’s conjecture on log-concavity and highlights a robust method based on real-rooted polynomials to study matroid invariants. The conclusions suggest broader conjectures about the real-rootedness of $\mathcal{B}(Q_M(t))$ in general matroids and the interplay between the degrees and positivity of Kazhdan-Lusztig polynomials.
Abstract
Gao and Xie (2021) conjectured that the inverse Kazhdan-Lusztig polynomial of any matroid is log-concave. Although the inverse Kazhdan-Lusztig polynomial may not always have only real roots, we conjecture that the Hadamard product of an inverse Kazhdan-Lusztig polynomial of degree $n$ and $(1+t)^n$ has only real roots. Using interlacing polynomials and multiplier sequences, we confirm this conjecture for paving matroids. This result allows us to confirm the log-concavity conjecture for these matroids by applying Newton's inequalities.