Geometric combinatorial algebras: cyclohedron and simplex

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto-Reutenauer algebra of permutations and the Loday-Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra (one-sided) with basis all the faces of the simplices.Comment: 23 figures, new expanded section about Hopf algebra of simplices, with journal correction

The S_{n+1} Action on Spherical Models and Supermaximal models of Type A_{n−1}

In this paper we recall the construction of the De Concini–Procesi wonderful models of the braid arrangement: these models, in the case of the braid arrangement of type A_{n-1}, are equipped with a natural S_n action, but only the minimal model admits an ‘hidden’ symmetry, i.e. an action of S_{n+1} that comes from its moduli space interpretation. We show that this hidden action can be lifted to the face poset of the corresponding minimal real spherical model and we compute the number of its orbits. Then we provide a spherical version of the construction of the supermaximal model (see Callegaro, Gaiffi, On models of the braid arrangement and their hidden symmetries. Int. Math. Res. Not. (published online 2015). doi: 10.1093/imrn/rnv009), i.e. the smallest model that can be projected onto the maximal model and again admits the extended S_{n+1} action