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arXiv · 2510.01086

Inverse Kazhdan-Lusztig polynomials of matroids under deletion

Tom Braden·Luis Ferroni·Jacob P. Matherne·Nutan Nepal
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Abstract

We provide a deletion formula for the inverse Kazhdan--Lusztig polynomial and the inverse $Z$-polynomial of a matroid. Our formulas provide analogues to the deletion formulas of Braden--Vysogorets for Kazhdan--Lusztig and $Z$-polynomials. We discuss several consequences, which include closed formulas and recursions for these invariants on uniform matroids, projective geometries, glued cycles, and arbitrary matroids of corank $2$. As a relevant application of our deletion formula, we show the existence of a matroid of rank $19$ which disproves a conjecture of Xie and Zhang concerning a real-rootedness property of inverse Kazhdan--Lusztig polynomials.

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Tom Braden, Luis Ferroni, Jacob P. Matherne, Nutan Nepal. 2025-10-01. Inverse Kazhdan-Lusztig polynomials of matroids under deletion. https://arxiv.org/abs/2510.01086

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