Valuations and the Hopf Monoid of Generalized Permutahedra. (17th January 2022)

• Record Type: Journal Article
• Authors: Ardila, Federico; Sanchez, Mario
• Is Part Of: International mathematics research notices. Volume 2023:Number 5(2023)
• Page Start: 4149
• Page End: 4224
• Publication Date: 2022-01-17
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnab355
• Languages: English
• ISSNs: 1073-7928

Abstract

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