Shortest Path Poset Of Bruhat Intervals And The Complete Cd-Index

Abstract

Let (W, S ) be a Coxeter system, [u, v ] be a Bruhat interval and B (u, v ) be its corresponding Bruhat graph. The combinatorial and topological structures of the longest u-v paths of B (u, v ) have been studied extensively and is well-known. Nevertheless, not much is known of the remaining paths. Here we define the shortest path poset of [u, v ], denoted by SP (u, v ), which arises from the shortest u-v paths of B (u, v ). If W is finite, then SP (e, w0 ) is the union of Boolean posets, where w0 is the longest-length word of W . Furthermore, if SP (u, v ) has a unique rising chain under a reflection order, then SP (u, v ) is EL-shellable. The complete cd-index of a Bruhat interval is a non-homogeneous polynomial that encodes the descent-set distribution, under a reflection order, of paths of B (u, v ). The highest-degree terms of the complete cd-index correspond to the cd-index of [u, v ] (as an Eulerian poset). We study properties of the complete cd-index and compute it for some intervals utilizing an extension of the CL¨ labeling of Bjorner and Wachs that can be defined for dihedral intervals (which we characterize by their complete cd-index) and intervals in a universal Coxeter system. We also describe the lowest-degree terms of the complete cd-index for some intervals.