Flip Posets of Bruhat Intervals

Abstract

In this paper we introduce a way of partitioning the paths of shortest lengths in the Bruhat graph B(u,v) of a Bruhat interval [u,v] into rank posets Pi in a way that each Pi has a unique maximal chain that is rising under a reflection order. In the case where each Pi has rank three, the construction yields a combinatorial description of some terms of the complete cd -index as a sum of ordinary cd -indices of Eulerian posets obtained from each of the Pi .