Rationally smooth Schubert varieties, inversion hyperplane arrangements, and Peterson translation
نویسنده
چکیده
We show that an element w of a finite Weyl group W is rationally smooth if and only if the hyperplane arrangement I(w) associated to the inversion set of w is inductively free, and the product (d1 +1) · · · (dl +1) of the coexponents d1, . . . , dl is equal to the size of the Bruhat interval [e, w]. We also use Peterson translation of coconvex sets to give a Shapiro-Steinberg-Kostant rule for the exponents of w. Résumé. Nous montrons qu’un élément w d’un groupe de Weyl fini est rationnellement lisse si et seulement si l’arrangement des hyperplans associé à l’ensemble d’inversion de w est libre, et le produit (d1 + 1) · · · (dl + 1) des coexposants d1, . . . , dl est égal à la cardinalité de l’intervalle [e, w] pour l’ordre de Bruhat. Nous donnons une règle de Shapiro-Steinberg-Kostant pour calculer les exposants de w en utilisant traduction de Peterson sur des sousensembles coconvexes.
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