Subword Complexes and Nil-Hecke Moves

For a finite Coxeter group $W$, a subword complex is a simplicial complex associated with a pair $(\mathbf{Q}, \rho),$ where $\mathbf{Q}$ is a word in the alphabet of simple reflections, $\rho$ is a group element. We describe the transformations of such a complex induced by nil-moves and inverse operations on $\mathbf{Q}$ in the nil-Hecke monoid corresponding to $W$. If the complex is polytopal, we also describe such transformations for the dual polytope. For $W$ simply-laced, these descriptions and results of [5] provide an algorithm for the construction of the subword complex corresponding to $(\mathbf{Q}, \rho)$ from the one corresponding to $(\delta(\mathbf{Q}), \rho),$ for any sequence of elementary moves reducing the word $\mathbf{Q}$ to its Demazure product $\delta(\mathbf{Q})$. The former complex is spherical or empty if and only if the latter one is empty.
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