Subword complexes and edge subdivisions

For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair $(\mathbf Q,\pi)$, where $\mathbf Q$ is a word in the alphabet of simple reflections and $\pi$ is a group element. We discuss the transformations of such a complex that are induced by braid moves of the word $\mathbf Q$. We show that under certain conditions, such a transformation is a composition of edge subdivisions and inverse edge subdivisions. In this case, we describe how the $H$- and $\gamma$-polynomials change under the transformation. This case includes all braid moves for groups with simply laced Coxeter diagrams.