Séminaire Lotharingien de Combinatoire, 91B.94 (2024), 12 pp.
Gaston Burrull, Tao Gui and Hongsheng Hu
Asymptotic Log-Concavity of Dominant Lower Bruhat Intervals via Brunn-Minkowski Inequality
Abstract.
Björner and Ekedahl [Ann. of Math. (2), 170.2 (2009), 799-817]
pioneered the study of length-counting sequences associated with
parabolic lower Bruhat intervals in crystallographic Coxeter
groups. In this extended abstract, we study the asymptotic behavior of
these sequences in affine Weyl groups.
Let W be an affine Weyl group with corresponding Weyl
group Wf,
and fW be the set of minimal representatives for the
right cosets Wf &backslash; W.
Let tλ be the translation
by a dominant coroot lattice element λ and
fbitλ
be the number of elements of
length i below tλ in the Bruhat order on
fW.
We show that the sequence
{fbitλ}i
is "asymptotically log-concave" in the following sense:
the ``shape'' of the k-fold dilated sequence
{fbitkλ}i
as k tends to infinity,
converges to a continuous function obtained from a certain polytope
Pλ;
by the Brunn-Minkowski inequality, this function is log-concave.
Received: November 15, 2023.
Accepted: February 15, 2024.
Final version: April 1, 2024.
The following versions are available: