Saúl Blanco Rodríguez
First PositionVisiting assistant professor of mathematics at DePaul University
DissertationShortest Path Poset of Bruhat Intervals and the Complete<strong>cd</strong>-Index
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.