The Electronic Journal of Combinatorics

v17i1n27 — Comment by the author, May 23, 2010.

In September 2009 Reiner, Woo and Yong discussed a correspondence between Fulton's essential set of a given permutation and minimal (in Bruhat order) elements of the set of permutations which are not strictly below it. It is interesting that every element in the latter set is necessarily bigrassmannian. See:
V. Reiner, A. Woo and A. Yong, Presenting the cohomology of a Schubert variety, Sections 1, 2, 4, arXiv:math.CO/08092981v2, 24pp., 2009, to appear in Trans. Amer. Math. Soc.