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

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 ${}^{f}W$ be the set of minimal representatives for the right cosets $W_f\mathrm{\backslash }W$. Let $t_{\lambda }$ be the translation by a dominant coroot lattice element $\lambda$ and ${}^f{b}_i^{t_{\lambda }}$ be the number of elements of length $i$ below $t_{\lambda }$ in the Bruhat order on ${}^{f}W$. We show that the sequence $({}^f{b}_i^{t_{\lambda }}{)}_i$ is ‘‘asymptotically log-concave’’ in the following sense: The sequence of discrete measures $({\mathfrak{m}}_k{)}_k$ constructed from the $k$-fold dilated sequence $({}^f{b}_i^{t_{k\lambda }}{)}_i$, as $k$ tends to infinity, converges weakly to a continuous measure obtained from a polytope $P^{\lambda }$. Moreover, the sequence of step functions $(S_k{)}_k$ of $({}^f{b}_i^{t_{k\lambda }}{)}_i$ converges uniformly to the density function of this continuous measure. By Brunn–Minkowski inequality, this density is log-concave.