Soergel Bimodules and the Shape of Bruhat Intervals

Given an element w of a Coxeter group, let ai(w) be the number of elements less than w in Bruhat order. A theorem of Björner and Ekedahl states that if W is crystallographic, then ai(w) ≤ aj(w) for all 0 ≤ i < j ≤ `(w) − i. Their proof uses the hard Lefschetz property in intersection cohomology. In this note we extend Björner and Ekedahl’s theorem to all Coxeter groups using the hard Lefschetz theorem for Soergel bimodules recently proved by Elias and Williamson. As we explain, the parabolic case remains open.