Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via Brunn--Minkowski Inequality

2023/11/29 by Burrull, Gaston, Gui, Tao, Hu, Hongsheng
#05E14 #05E16 (Primary) #51F15 (Secondary) #52A27 #52A38 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2311.17980

Abstract

Björner and Ekedahl [Ann. of Math. (2), 170.2(2009), pp. 799--817] pioneered the study of length-counting sequences associated with parabolic lower Bruhat intervals in crystallographic Coxeter groups. In this paper, we study the asymptotic behavior of these sequences in affine Weyl groups. Let W be an affine Weyl group with corresponding Weyl group W_f and ^fW be the set of minimal representatives for the right cosets W_f \backslash W. Let t_λ be the translation by a dominant coroot lattice element λ and ^fb_i^t_λ be the number of elements of length i below t_λ in the Bruhat order on ^fW. We show that the sequence (^fb_i^t_λ)_i is ''asymptotically log-concave'' in the following sense: The sequence of discrete measures (\mathfrakm_k)_k constructed from the k-fold dilated sequence (^fb_i^t_kλ)_i, as k tends to infinity, converges weakly to a continuous measure obtained from a polytope P^λ. Moreover, the sequence of step functions (S_k)_k of (^fb_i^t_kλ)_i converges uniformly to the density function of this continuous measure. By Brunn--Minkowski inequality, this density is log-concave.

Related

• On the shape of Bruhat intervals
• On the size of Bruhat intervals
• A universal bound in the dimensional Brunn-Minkowski inequality for log-concave measures
• Increasing Subsequences and Kronecker Coefficients
• Log-concavity in one-dimensional Coulomb gases and related ensembles