Geodesic
The Steinberg torus of a Weyl group as a module over the Coxeter complex
Journal of Algebraic Combinatorics, Tome 42 (2015) no. 4, pp. 1135-1175.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Associated with each irreducible crystallographic root system $\varPhi$, there is a certain cell complex structure on the torus obtained as the quotient of the ambient space by the coroot lattice of $\varPhi$. This is the Steinberg torus. A main goal of this paper is to exhibit a module structure on (the set of faces of) this complex over the (set of faces of the) Coxeter complex of $\varPhi$. The latter is a monoid under the Tits product of faces. The module structure is obtained from geometric considerations involving affine hyperplane arrangements. As a consequence, a module structure is obtained on the space spanned by affine descent classes of a Weyl group, over the space spanned by ordinary descent classes. The latter constitute a subalgebra of the group algebra, the classical descent algebra of Solomon. We provide combinatorial models for the module of faces when $\varPhi$ is of type $A$ or $C$.
Classification : 05E15, 17B22, 20F55, 52C35
Keywords: hyperplane arrangement, Coxeter complex, Steinberg torus, crystallographic root system, Weyl group, Tits product, Solomon's descent algebra, affine descent 
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     author = {Aguiar, Marcelo and Petersen, T. Kyle},
     title = {The {Steinberg} torus of a {Weyl} group as a module over the {Coxeter} complex},
     journal = {Journal of Algebraic Combinatorics},
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Aguiar, Marcelo; Petersen, T. Kyle. The Steinberg torus of a Weyl group as a module over the Coxeter complex. Journal of Algebraic Combinatorics, Tome 42 (2015) no. 4, pp. 1135-1175. http://geodesic.mathdoc.fr/item/JAC_2015__42_4_a0/