Enumeration of chains in the Bruhat order of the symmetric groupRichard P. StanleyMIT
April 28,

ABSTRACT


The Bruhat order is a partial ordering of the symmetric group S_{n}, first defined by Ehresmann, that arises naturally in algebraic geometry and representation theory. (It can be extended to any Coxeter group, but we only consider S_{n}.) A result implicit in the work of Chevalley and stated explicitly by Stembridge gives a formula for the number of maximal chains in S_{n} (under the Bruhat order), where the maximal chains are counted with a certain weight. The approach of Chevalley connects this counting with the degree of the complete flag variety F_{n} of type A_{n1}. We consider the more general question of counting weighted saturated chains between any two elements u<v in S_{n}. There are close connectionw with the theory of Schubert polynomials and the degrees of Schubert varieties in F_{n}. This lecture is based on joint work with Alexander Postnikov. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

