On the Hadamard product of Hopf monoids

Combinatorial structures which compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free. The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco which applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species which is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the dimension sequence of a connected Hopf monoid satisfies the following condition: except for the first, all coefficients of the reciprocal of its generating function are nonpositive

Coxeter groups and Hopf algebras I

Key words and phrases. Coxeter group; descent; global descent; hyperplane arrangement; left regular band (LRB); projection map; projection poset; lunes; shuffles; bilinear forms; descent algebra; semisimple; Hopf algebra; (quasi) symmetric functions; (co)algebra axioms; (nested) set partitions; (nested) set compositions; (co)free (co)algebra. Foreword In the study of a mathematical system, algebraic structures allow for the discovery of more information. This is the motor behind the success of many areas of mathematics such as algebraic geometry, algebraic combinatorics, algebraic topology and others. This was certainly the motivation behind the observation of G.-C. Rota stating that various combinatorial objects possess natural product and coproduct structures. These structures give rise to a graded Hopf algebra, which is usually referred to as a combinatorial Hopf algebra. Typically, it is a graded vector space where the homogeneous components are spanned by finite sets of combinatorial objects of a given type and the algebraic structures are given by some constructions on those objects. Recent foundational work has constructed many interesting combinatorial Hopf algebra

Bimonoids for hyperplane arrangements

Develops a new theory, parallel to the classical theory of connected Hopf algebras, including a real hyperplane arrangement