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Matroid relaxations and Kazhdan–Lusztig non-degeneracy
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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.
Title: Matroid relaxations and Kazhdan–Lusztig non-degeneracy
Description:
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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