Log‐concavity of inverse Kazhdan–Lusztig polynomials of paving matroids

Gao and Xie conjectured that the inverse Kazhdan–Lusztig polynomial of any matroid is log‐concave. Although these polynomials are not necessarily real‐rooted, we conjecture that the Hadamard product of an inverse Kazhdan–Lusztig polynomial of degree with is real‐rooted. Using the theory of interlacing polynomials and multiplier sequences, we confirm this conjecture for paving matroids. As a consequence, the log‐concavity of inverse Kazhdan–Lusztig polynomials for paving matroids follows from Newton's inequalities.