Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via Brunn--Minkowski Inequality
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2023
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| _version_ | 1866908475898986496 |
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| author | Burrull, Gaston Gui, Tao Hu, Hongsheng |
| author_facet | Burrull, Gaston Gui, Tao Hu, Hongsheng |
| contents | 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 $W_f \backslash W$. Let $t_λ$ be the translation by a dominant coroot lattice element $λ$ and ${}^{f}{b}_i^{t_λ}$ be the number of elements of length $i$ below $t_λ$ in the Bruhat order on ${}^{f}{W}$. We show that the sequence $({}^{f}{b}_i^{t_λ})_i$ is ''asymptotically log-concave'' in the following sense: The sequence of discrete measures $(\mathfrak{m}_k)_k$ constructed from the $k$-fold dilated sequence $({}^{f}{b}_i^{t_{kλ}})_i$, 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 $({}^{f}{b}_i^{t_{kλ}})_i$ converges uniformly to the density function of this continuous measure. By Brunn--Minkowski inequality, this density is log-concave. |
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arxiv_https___arxiv_org_abs_2311_17980 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via Brunn--Minkowski Inequality Burrull, Gaston Gui, Tao Hu, Hongsheng Combinatorics Representation Theory 05E16 (Primary), 05E14, 52A27, 52A38, 51F15 (Secondary) 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 $W_f \backslash W$. Let $t_λ$ be the translation by a dominant coroot lattice element $λ$ and ${}^{f}{b}_i^{t_λ}$ be the number of elements of length $i$ below $t_λ$ in the Bruhat order on ${}^{f}{W}$. We show that the sequence $({}^{f}{b}_i^{t_λ})_i$ is ''asymptotically log-concave'' in the following sense: The sequence of discrete measures $(\mathfrak{m}_k)_k$ constructed from the $k$-fold dilated sequence $({}^{f}{b}_i^{t_{kλ}})_i$, 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 $({}^{f}{b}_i^{t_{kλ}})_i$ converges uniformly to the density function of this continuous measure. By Brunn--Minkowski inequality, this density is log-concave. |
| title | Asymptotic Log-concavity of Dominant Lower Bruhat Intervals via Brunn--Minkowski Inequality |
| topic | Combinatorics Representation Theory 05E16 (Primary), 05E14, 52A27, 52A38, 51F15 (Secondary) |
| url | https://arxiv.org/abs/2311.17980 |