Papers › The Kazhdan-Lusztig polynomials of uniform matroids
The Kazhdan-Lusztig polynomials of uniform matroids
Alice L. L. Gao, Linyuan Lu, Matthew H. Y. Xie, Arthur L. B. Yang, Philip B. Zhang
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The Kazhdan-Lusztig polynomial of a matroid was introduced by Elias, Proudfoot, and Wakefield [{\it Adv. Math. 2016}]. Let U_(m,d) denote the uniform matroid of rank d on a set of m+d elements. Gedeon, Proudfoot, and Young [{\it J. Combin. Theory Ser. A, 2017}] pointed out that they can derive an explicit formula of the Kazhdan-Lusztig polynomials of U_(m,d) using equivariant Kazhdan-Lusztig polynomials. In this paper we give two alternative explicit formulas, which allow us to prove the real-rootedness of the Kazhdan-Lusztig polynomials of U_(m,d) for 2≤m≤15 and all d's. The case m=1 was previously proved by Gedeon, Proudfoot, and Young [{\it S\'{e}m. Lothar. Combin. 2017}]. We further determine the Z-polynomials of all U_(m,d)'s and prove the real-rootedness of the Z-polynomials of U_(m,d) for 2≤m≤15 and all d's. Our formula also enables us to give an alternative proof of Gedeon, Proudfoot, and Young's formula for the Kazhdan-Lusztig polynomials of U_(m,d)'s without using the equivariant Kazhdan-Lusztig polynomials.
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