Richard Stanley
A Survey of the Bruhat Order of the Symmetric Group
The Bruhat order is a certain partial ordering of the elements of the symmetric group Sn. It was shown by Lascoux and Schützenberger that the number of elements of its MacNeille completion is the number of n × n alternating sign matrices. We will discuss this result and some further properties of Bruhat order, including: (1) its connection with the Bruhat decomposition of the complete flag variety, (2) topological properties based on lexicographic shellability, and (3) a weighted enumeration of the maximal chains of Bruhat order based on the theory of Schubert polynomials. No prior knowledge of Bruhat order, MacNeille completion, flag varieties, lexicographic shellability, or Schubert polynomials will be assumed.