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.