Shortest path poset of Bruhat intervals

We define the shortest path poset SP(u,v) of a Bruhat interval u,v], by considering the shortest u–v paths in the Bruhat graph of a Coxeter group W, where u,v∈W. We consider the case of SP(u,v) having a unique rising chain under a reflection order and show that in this case SP(u,v) is a Gorenstein^? poset. This allows us to derive the nonnegativity of certain coefficients of the complete cd-index. We furthermore show that the shortest path poset of an irreducible, finite Coxeter group exhibits a symmetric chain decomposition.