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Proudfoot, Xu, and Young introduced the Z-polynomial for any matroid and conjectured the polynomial has only real roots. Recently, Ferroni, Nasr, and Vecchi introduced the γ-polynomial of a matroid. In this paper, we prove that both the Z-polynomials and γ-polynomials of uniform matroids are ultra log-concave, which due to Newton's inequality partially supports the real-rootedness conjecture. We also give an alternative formula for the γ-polynomials of uniform matroids. As an application, we use this formula to provide a new proof of the γ-positivity of sparse paving matroids. [ABSTRACT FROM AUTHOR]

Elias, et al. (2016) conjectured that the Kazhdan-Lusztig polynomial of any matroid is log-concave. Inspired by a computer proof of Moll's log-concavity conjecture given by Kauers and Paule, the authors use a computer algebra system to prove the conjecture for arbitrary uniform matroids. [ABSTRACT FROM AUTHOR]

Motivated by the concepts of the inverse Kazhdan--Lusztig polynomial and the equivariant Kazhdan--Lusztig polynomial, Proudfoot defined the equivariant inverse Kazhdan--Lusztig polynomial for a matroid. In this paper, we show that the equivariant inverse Kazhdan--Lusztig polynomial of a matroid is very useful for determining its equivariant Kazhdan--Lusztig polynomials, and we determine the equivariant inverse Kazhdan--Lusztig polynomials for Boolean matroids and uniform matroids. As an application, we give a new proof of Gedeon, Proudfoot, and Young's formula for the equivariant Kazhdan--Lusztig polynomials of uniform matroids. Inspired by Lee, Nasr, and Radcliffe's combinatorial interpretation for the ordinary Kazhdan--Lusztig polynomials of uniform matroids, we further present a new formula for the corresponding equivariant Kazhdan--Lusztig polynomials. [ABSTRACT FROM AUTHOR]

The equivariant Kazhdan-Lusztig polynomial of a matroid was introduced by Gedeon, Proudfoot, and Young. Gedeon conjectured an explicit formula for the equivariant Kazhdan-Lusztig polynomials of thagomizer matroids with an action of symmetric groups. In this paper, we discover a new formula for these polynomials which is related to the equivariant Kazhdan- Lusztig polynomials of uniform matroids. Based on our new formula, we confirm Gedeon's conjecture by the Pieri rule. [ABSTRACT FROM AUTHOR]