Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2212.14000

Braid arrangement bimonoids and the toric variety of the permutohedron

William Norledge
View original source ↗Read source document ↗Export metadata ↓

Abstract

We show that the toric variety of the permutohedron (=permutohedral space) has the structure of a cocommutative bimonoid in species, with multiplication/comultiplication given by embedding/projecting-onto boundary divisors. In terms of Losev-Manin's description of permutohedral space as a moduli space, multiplication is concatenation of strings of Riemann spheres and comultiplication is forgetting marked points. In this way, the bimonoid structure is an analog of the cyclic operad structure on the moduli space of genus zero marked curves. Covariant/contravariant data on permutohedral space is endowed with the structure of cocommutative/commutative bimonoids by pushing-forward/pulling-back data along the (co)multiplication. Many well-known combinatorial objects index data on permutohedral space. Moreover, combinatorial objects often have the structure of bimonoids, with multiplication/comultiplication given by merging/restricting objects in some way. We prove that the bimonoid structure enjoyed by these indexing combinatorial objects coincides with that induced by the bimonoid structure of permutohedral space. Thus, permutohedral space may be viewed as a fundamental underlying object which geometrically interprets many combinatorial Hopf algebras. Aguiar-Mahajan have shown that classical combinatorial Hopf theory is based on the braid hyperplane arrangement in a crucial way. This paper aims to similarly establish permutohedral space as a central object, providing an even more unified perspective. The main motivation for this work concerns Feynman amplitudes in the Schwinger parametrization, which become integrals over permutohedral space if one blows-up everything in the resolution of singularities. Then the Hopf algebra structure of Feynman graphs, first appearing in the work of Connes-Kreimer, coincides with that induced by the bimonoid structure of permutohedral space.

Explore related subjects

math.COmath-phmath.CTmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

William Norledge. 2022-12-28. Braid arrangement bimonoids and the toric variety of the permutohedron. https://arxiv.org/abs/2212.14000

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

On Perfect Divisibility of Bull-Free Graphs Without Long Paths

A graph $G$ is {\em perfectly divisible} if, for every induced subgraph $H$ of $G$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $ω(H[B])<ω(H)$. Chudnovsky and Sivaraman [J. Graph Theory \textbf{90} (2019) 54-60] proved that every ($P_5$, bull)-free graph is perfectly divisible, while Chen and Xu [Discrete Appl. Math. \textbf{372} (2025) 298-307] proved the same for ($P_7,C_5$, bull)-free graphs. We extend these results by proving that every ($P_8,C_5$, bull)-free graph is perfectly divisible and that, letting $F$ denote the Grötzsch graph, a ($P_6$, bull)-free graph is perfectly divisible if and only if it is $F$-free.

math.CO↗
arXiv2026

Covering graphs by isometric trees

A connected subgraph of a graph is isometric if it preserves distances. Recently, graphs admitting a vertex or edge covering by a small number of isometric paths have been studied. In this paper, we consider the analogous problem for isometric trees, focusing on the treewidth of graphs admitting a vertex or edge covering by a small number of such trees. Baste, De Meyer, Giocanti, Objois, and Picavet showed that for coverings by two isometric trees, the treewidth is bounded. We show that already for three isometric trees, the treewidth can be linear in the number of vertices. On the positive side, we show that for graphs of bounded degree coverable by a small number of isometric trees, the treewidth is sublinear in the number of vertices.

math.CO↗
arXiv2026

Tree-independence number of $P_5$-free graphs with no large bicliques

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties; however, the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, ${\{P_t,K_{\ell,\ell}\}}$-free graphs have bounded tree-independence number. We prove this conjecture for ${t=5}$ by showing that every ${\{P_5,K_{\ell,\ell}\}}$-free graph has tree-independence number at most ${4\ell-4}$. We also obtain related bounds for the weaker parameter of $α$-degeneracy and answer a question of Hilaire, Milanič, and Vasić whether tree-independence number of ${\{P_5,K_{\ell,\ell}\}}$-free graphs exceeds $\ell$ by at most an additive constant.

math.CO↗