Sách Bijection between bigrassmannian permutations maximal below a permutation and its essential set

Abstract
Bigrassmannian permutations are known as permutations which have precisely
one left descent and one right descent. They play an important role in the study
of Bruhat order. Fulton introduced the essential set of a permutation and studied
its combinatorics. As a consequence of his work, it turns out that the essential set
of bigrassmannian permutations consists of precisely one element. In this article,
we generalize this observation for essential sets of arbitrary permutations. Our
main theorem says that there exists a bijection between bigrassmanian permutations
maximal below a permutation and its essential set. For the proof, we make use
of two equivalent characterizations of bigrassmannian permutations by Lascoux-
Sch¨utzenberger and Reading.