Species and hyperplane arrangements

Contents
List of Figures
Introduction
Preliminaries
Polyhedral geometry
Polytopes
(Unbounded) Polyhedra
Hyperplane arrangements
Faces, flats and characteristic polynomial
The Tits semigroup
The braid arrangement
The type B Coxeter arrangement
Hopf monoids in the category of Species
Hopf monoids in a nutshell
Series and characters
From Hopf monoids to modules
Hopf monoids on linearly ordered sets
Characteristic elements for real hyperplane arrangements
The Tits algebra
Characteristic elements
Definition and basic properties
Relation to the characteristic polynomial
Functoriality
First applications
The fundamental recursion for the characteristic polynomial
An identity of Crapo
The characteristic polynomial on a product. An identity of Kung
Characteristic elements of parameters 1
The unit element
The Takeuchi element
Application: Zaslavsky's formulas
The Adams elements
Braid arrangement
Type B Coxeter arrangement
Graphic arrangement
Coordinate arrangement
Characteristic elements and valuations
Main construction: Intrinsic elements
Generalization to Cone angles
Proof of t:unit-affine
Deformations of a linear arrangement and exponential sequences
The action of a central arrangement on its deformations
The map f0
The map i
Characteristic and intrinsic elements
Exponential sequences of arrangements
Hopf monoid of flats and faces
Antipode
Characteristic elements and power series
Characters and polynomial invariants
The module of generalized zonotopes modulo McMullen relations
The polytope algebra
Definition and structure theorem
Subalgebra relative to a fixed polytope
The polytope algebra as a module
The module structure
Eulerian idempotents and diagonalization
Simultaneous diagonalization
First example: the cube and the coordinate arrangement
The zonotope module of a product of arrangements
The module of generalized permutahedra
The symmetric group and the Eulerian polynomial
The module Generalized permutahedra
Simultaneous-eigenbasis for the Adams element
The module of type B generalized permutahedra
The hyperoctahedral group and the Type B Eulerian polynomial
The module of type B Generalized permutahedra
Two bases for type B generalized permutahedra
Hopf monoid structure
The McMullen (co)ideal
Higher monoidal structures
Type B Hopf monoids
Type B species
Type B generating functions
Type B bimonoids
Type A species with an involution
The action of Sp on SpB
Type B objects, monoids, and comonoids
Type B bimonoids
Convolution modules and the antipode
Type A with an involution
Type B
Type B Hopf monoids and the antipode
Characters and polynomial invariants
Type A with an involution
Type B
Series
Substitution product of type B objects
The type B bimonoids of set compositions and set partitions
The free (commutative) monoid
Type B Boolean functions
Type B Submodular functions
Type B generalized permutahedra
Isomorphism with (bi)submodular functions
Symplectic matroids
Bibliography