Log-concavity of inverse Kazhdan-Lusztig polynomials of paving matroids

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.