Speaker:  Louis Billera
Title:   The complete cd-index of a Bruhat interval
Time: 3:00 PM, Monday, April 14, 2008   Note special time.
Place:  Malott 205

Abstract:  In recent work with Francesco Brenti (Quasisymmetric functions and Kazhdan-Lusztig polynomials, arXiv:0710.3965), we have defined, for a Bruhat interval in an arbitrary Coxeter group, a nonhomogeneous extension of the usual cd-index, called the complete cd-index.  This noncommutative, nonhomogeneous polynomial has many interesting algebraic and combinatorial properties.  For example, it specializes to the R-tilde-polynomial of the interval, yet contains enough information about the interval to give a simple representation of its Kazhdan-Lusztig polynomial.

While the definition of the complete cd-index was given in terms of a certain quasisymmetric function associated to a Bruhat interval, it can be given more directly in terms of paths in the corresponding Bruhat graph of the interval.  Here we have to consider descents in terms of an arbitrary reflection ordering on the group.  We review this definition, describe some of the properties of the complete cd-index and discuss some open problems.


April 10, 2008