Valuations and the Hopf Monoid of Generalized Permutahedra

Abstract The goal of this paper is to show that valuation theory and Hopf are compatible on the class generalized permutahedra. We prove structure $\textbf {GP}^+$ these polyhedra descends, modulo inclusion-exclusion relations, an indicator monoid $\mathbb {I}(\textbf {GP}^+)$ permutahedra isomorphic weighted ordered set partitions. This quotient cofree. It terminal object in category monoids with polynomial characters; partially explains ubiquity monoids. theoretic framework offers a simple, unified explanation for many new old valuations their subfamilies. Examples include, matroids: Chern–Schwartz–MacPherson cycles, Eur’s volume polynomial, Kazhdan–Lusztig motivic zeta function, Derksen–Fink invariant; posets: order Poincaré poset Tutte polynomial; permutahedra: universal character corresponding Chow ring permutahedral variety. obtain several algebraic combinatorial corollaries; example, existence valuative group indecomposability nestohedron into smaller nestohedra.