Species and hyperplane arrangements

Description

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