# Saúl Blanco Rodríguez

### First Position

Visiting assistant professor of mathematics at DePaul University### Dissertation

*Shortest Path Poset of Bruhat Intervals and the Complete<strong>cd</strong>-Index*

### Advisor

### Research Area

### 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*, *w*_{0}) is the union of Boolean posets, where *w*_{0} 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.