中科院数学与系统科学研究院
数学研究所
学术报告
表示论研讨班
报告人：桂弢 博士（北京国际数学中心）
题 目：Top-heaviness of lower Bruhat intervals
时 间：2023.11.09（星期四）14:30-15:30
地 点：数学院南楼N803
摘 要：Bjorner and Ekedahl (Ann. of math., 2008) proved that lower Bruhat intervals of crystallographic Coxeter groups are top-heavy using Hodge theory. Using Soergel bimodules and their Hodge theory established by Elias and Williamson (Ann. of math., 2014), one can prove the same thing for general Coxeter groups. I will explain the proof and discuss the relation between top-heaviness and Kazhdan—Lusztig polynomials. I will also discuss the relation to Huh—Wang’s work (Acta math., 2017).
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题 目：Asymptotic log-concavity of dominant lower Bruhat intervals
时 间：2023.11.09（星期四）15:45-16:45
地 点：数学院南楼N803
摘 要：The unimodality of lower Bruhat intervals for the “upper half” remains as an open problem. For affine Weyl group W with corresponding finite Weyl group W_f, we prove that lower W_f-parabolic Bruhat intervals correspondinging to dominant coroot lattice elements are asymptotically log-concave. Our approach, via the Brunn—Minkowski inequality, builds a bridge between the discrete nature of Betti numbers of parabolic affine Schubert varieties and the continuous nature of the geometry of convex polytopes. Joint with Gaston Burrull and Hongsheng Hu.