报告题目：Induced log-concavity of equivariant matroid invariants
报告人：解红叶博士，天津理工大学
报告时间：2024年4月12日（周五）14:30-16:30
报告地点：腾讯会议ID：426 508 071
报告摘要：Inspired by the notion of equivariant log-concavity, we introduce the concept of induced log-concavity for a sequence of representations of a finite group. For an equivariant matroid equipped with a symmetric group action or a finite general linear group action, we transform the problem of proving the induced log-concavity of matroid invariants to that of proving the Schur positivity of symmetric functions. We prove the induced log-concavity of the equivariant Kazhdan-Lusztig polynomials of $q$-niform matroids equipped with the action of a finite general linear group, as well as that of the equivariant Kazhdan-Lusztig polynomials of uniform matroids equipped with the action of a symmetric group. As a consequence of the former, we obtain the log-concavity of Kazhdan-Lusztig polynomials of $q$-niform matroids, thus providing further positive evidence for Elias, Proudfoot and Wakefield's log-concavity conjecture on the matroid Kazhdan-Lusztig polynomials. From the latter we obtain the log-concavity of Kazhdan-Lusztig polynomials of uniform matroids, which was recently proved by Xie and Zhang by using a computer algebra approach. We also establish the induced log-concavity of the equivariant characteristic polynomials and the equivariant inverse Kazhdan-Lusztig polynomials for $q$-niform matroids and uniform matroids. This is a joint work with Alice L.L. Gao, Ethan Y.H. Li, Arthur L.B. Yang, and Zhong-Xue Zhang.
报告人简介：解红叶，男，天津理工大学讲师，2018年博士毕业于南开大学组合数学中心，研究方向是对称函数和单峰型问题。近几年主要研究拟阵Kazhdan-Lusztig多项式的计算和单峰型性质。在Journal of Combinatorial Theory, Series B、Journal of Combinatorial Theory, Series A、Advances in Applied Mathematics等期刊上发表学术论文若干。主持国家自然科学基金青年科学基金项目和面上项目各一项。