Subword Complexes and Nil-Hecke Moves

M. A. Gorsky

Modelirovanie i analiz informacionnyh sistem, Tome 20 (2013) no. 6, pp. 121-128

Résumé

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. The article is published in the author's wording.

Détail

Keywords: subword complexes, Coxeter groups, nil-Hecke monoids.