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The commutative Hopf monoid of set compositions is a fundamental Hopf monoidinternal to vector species, having undecorated bosonic Fock space thecombinatorial Hopf algebra of quasisymmetric functions. We construct ageometric realization of this Hopf monoid over the adjoint of the(essentialized) braid hyperplane arrangement, which identifies the monomialbasis with signed characteristic functions of the interiors of permutohedraltangent cones. We show that the indecomposable quotient Lie coalgebra isobtained by restricting functions to chambers of the adjoint arrangement, i.e.by quotienting out the higher codimensions. The resulting functions arecharacterized by the Steinmann relations of axiomatic quantum field theory,demonstrating an equivalence between the Steinmann relations, tangent cones to(generalized) permutohedra, and having algebraic structure internal to species.Our results give a new interpretation of a construction appearing in themathematically rigorous formulation of renormalization by Epstein-Glaser,called causal perturbation theory. In particular, we show that operatorproducts of time-ordered products correspond to the H-basis of thecocommutative Hopf monoid of set compositions, and generalized retardedproducts correspond to a spanning set of its primitive part Lie algebra.

Hopf monoids, permutohedral cones, and generalized retarded functions