Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài:Bijection between bigrassmannian permutations maximal below a permutation and its essential set. | Bijection between bigrassmannian permutations maximal below a permutation and its essential set Masato Kobayashi Department of Mathematics the University of Tennessee Knoxville TN 37996 USA kobayashi@ Submitted Aug 23 2009 Accepted May 10 2010 Published May 20 2010 Mathematics Subject Classification 20F55 20B30 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-Schutzenberger and Reading. 1 Introduction Bigrassmannian elements play an important role in study of the Bruhat order on Coxeter groups. They are known as elements which have precisely one left descent and one right descent. In particular in the symmetric group type A bigrassmannian permutations have two other equivalent characterizations one as join-irreducible permutations and one as monotone triangles with some minimal condition we will see detail of these in Fact . Here let us recall the definitions of join and join-irreducibility from poset theory. Definition . Let P c be a finite poset and Q c P. Then consider the set x G P x y for all y G Q . If this set has a unique minimal element we call it the join of Q denoted by V Q. Define the meet of Q A Q order dually. P is said to be a lattice if V Q and A Q exist for all Q. We say that x G P is join-irreducible if whenever x V Q then x G Q. THE ELECTRONIC JOURNAL OF .