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The Kazhdan-Lusztig polynomials of uniform matroids
Advances in Applied Mathematics ( IF 1.0 ) Pub Date : 2021-01-01 , DOI: 10.1016/j.aam.2020.102117
Alice L.L. Gao , Linyuan Lu , Matthew H.Y. Xie , Arthur L.B. Yang , Philip B. Zhang

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 $2\leq m\leq 15$ 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 $2\leq m\leq 15$ 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.

中文翻译:

均匀拟阵的 Kazhdan-Lusztig 多项式

拟阵的 Kazhdan-Lusztig 多项式由 Elias、Proudfoot 和 Wakefield [{\it Adv. 数学。2016}]。让 $U_{m,d}$ 表示在一组 $m+d$ 元素上秩为 $d$ 的统一拟阵。Gedeon、Proudfoot 和 Young [{\it J. Combin. 理论系列 A, 2017}] 指出,他们可以使用等变 Kazhdan-Lusztig 多项式推导出 $U_{m,d}$ 的 Kazhdan-Lusztig 多项式的显式公式。在本文中,我们给出了两个可供选择的显式公式,它们使我们能够证明 $U_{m,d}$ 的 Kazhdan-Lusztig 多项式对于 $2\leq m\leq 15$ 和所有 $d$ 的实根性. $m=1$ 的情况先前已由 Gedeon、Proudfoot 和 Young [{\it S\'{e}m. 洛萨。结合。2017}]。我们进一步确定所有 $U_{m,d}$ 的 $Z$-多项式并证明 $U_{m 的 $Z$-多项式的实根性,d}$ 为 $2\leq m\leq 15$ 和所有 $d$。我们的公式还使我们能够在不使用等变 Kazhdan-Lusztig 多项式的情况下,为 $U_{m,d}$ 的 Kazhdan-Lusztig 多项式提供 Gedeon、Proudfoot 和 Young 公式的替代证明。
更新日期:2021-01-01
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