Enumeration of chains in the Bruhat order of the symmetric group

Richard P. Stanley


April 28,
refreshments at 3:45pm


The Bruhat order is a partial ordering of the symmetric group Sn, 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 Sn.) A result implicit in the work of Chevalley and stated explicitly by Stembridge gives a formula for the number of maximal chains in Sn (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 Fn of type An-1. We consider the more general question of counting weighted saturated chains between any two elements u<v in Sn. There are close connectionw with the theory of Schubert polynomials and the degrees of Schubert varieties in Fn. This lecture is based on joint work with Alexander Postnikov.

Speaker's Contact Info: rstan(at-sign)math.mitedu

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Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)math.mit.edu

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