Matroid relaxations and Kazhdan–Lusztig non-degeneracy
Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 745-769.

In this paper we study the interplay between the operation of circuit-hyperplane relaxation and the Kazhdan–Lusztig theory of matroids. We obtain a family of polynomials, not depending on the matroids but only on their ranks, that relate the Kazhdan–Lusztig, the inverse Kazhdan–Lusztig and the Z-polynomial of each matroid with those of its relaxations. As an application of our main theorem, we prove that all matroids having a free basis are non-degenerate. Additionally, we obtain bounds and explicit formulas for all the coefficients of the Kazhdan–Lusztig, inverse Kazhdan–Lusztig and Z-polynomial of all sparse paving matroids.

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Accepted:
Published online:
DOI: 10.5802/alco.244
Classification: 05B35,  05E10,  52B40,  11B83
Keywords: Kazhdan–Lusztig polynomials of matroids, Circuit-hyperplane relaxations, Geometric lattices, Real-rooted polynomials
Ferroni, Luis 1; Vecchi, Lorenzo 2

1 KTH Royal Institute of Technology Department of Mathematics Stockholm Sweden
2 Università di Bologna Dipartimento di Matematica Bologna Italy
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ferroni, Luis; Vecchi, Lorenzo. Matroid relaxations and Kazhdan–Lusztig non-degeneracy. Algebraic Combinatorics, Volume 5 (2022) no. 4, pp. 745-769. doi : 10.5802/alco.244. https://alco.centre-mersenne.org/articles/10.5802/alco.244/

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