Abstract

Björner and Ekedahl [Ann. of Math. (2), 170(2): 799-817, 2009] pioneered the study of length-enumerating sequences associated with parabolic lower Bruhat intervals in crystallographic Coxeter groups. In this talk, we study the asymptotic behavior of these sequences in affine Weyl groups. We prove that the length-enumerating sequences associated with the dominant intervals corresponding to a dominant coroot lattice element are ``asymptotically'' log-concave. More precisely, we prove that a certain sequence of discrete measures naturally constructed from the length-enumerating sequences converges weakly to a continuous measure constructed from a certain polytope. Moreover, a certain sequence of step functions naturally constructed from the length-enumerating sequences uniformly converges to the density function of that continuous measure, which implies the weak convergence and that the sequences of numbers of elements in each layer of the dilated dominant interval converges to a sequence of volumes of hyperplane sections of the polytope. By the Brunn--Minkovski inequality, the density function is log-concave. Our approach relies on the ``dominant lattice formula'', which yields a new bridge between the discrete nature of Betti numbers of parabolic affine Schubert varieties and the continuous nature of the geometry of convex polytopes. Our technique can be seen as a refinement in our context of the classical Ehrhart's theory relating the volume of a polytope and the number of lattice points the polytope contains, by replacing the volume by volumes of transversal sections and the number the total lattice points by the number of lattice points of a given length. Joint with Gaston Burrull and Hongsheng Hu.

Speaker Intro

I got my Ph. D. in 2023 from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences. Currently I am a postdoc of the Beijing International Center for Mathematical Research, Peking University. My research interests are Lie theory, geometric/combinatorial representation theory, and combinatorial Hodge theory. And I have broad interests in topological, geometric, and combinatorial problems related to representation theory.