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Algebra and Combinatorics of the Brion map on Generalized Permutahedra

Alvaro Cornejo, Mariel Supina

TL;DR

The paper develops and applies the Hopf monoid framework to generalized permutahedra, focusing on the Brion map that links polytopes to vertex-cone posets. It computes and analyzes Brion maps for three key families—permutahedra, associahedra, and orbit polytopes—and studies their dual Brion maps via primitive Lie monoids. A central result is that the primitives of the dual associahedra form the positive part of the Witt algebra, with explicit Lie brackets provided for the other dual submonoids. By leveraging the Cartier–Milnor–Moore theorem and the Fock functor, the work connects combinatorial geometry with Lie theory, offering structural insights and potential directions for further algebraic properties (e.g., Noetherian aspects) of these universal enveloping constructions.

Abstract

The Brion morphism maps a generalized permutahedron to a collection of posets associated to its vertices. We compute this map explicitly for the Hopf monoids of permutahedra, associahedra, and orbit polytopes, and we explore the dual Brion map of the primitive Lie monoids associated to these three Hopf monoids. We describe the Lie monoid structure of the primitives in this dual setting and in particular we show that the Lie monoid of primitives of associahedra is isomorphic to the positive part of the Witt Lie algebra

Algebra and Combinatorics of the Brion map on Generalized Permutahedra

TL;DR

The paper develops and applies the Hopf monoid framework to generalized permutahedra, focusing on the Brion map that links polytopes to vertex-cone posets. It computes and analyzes Brion maps for three key families—permutahedra, associahedra, and orbit polytopes—and studies their dual Brion maps via primitive Lie monoids. A central result is that the primitives of the dual associahedra form the positive part of the Witt algebra, with explicit Lie brackets provided for the other dual submonoids. By leveraging the Cartier–Milnor–Moore theorem and the Fock functor, the work connects combinatorial geometry with Lie theory, offering structural insights and potential directions for further algebraic properties (e.g., Noetherian aspects) of these universal enveloping constructions.

Abstract

The Brion morphism maps a generalized permutahedron to a collection of posets associated to its vertices. We compute this map explicitly for the Hopf monoids of permutahedra, associahedra, and orbit polytopes, and we explore the dual Brion map of the primitive Lie monoids associated to these three Hopf monoids. We describe the Lie monoid structure of the primitives in this dual setting and in particular we show that the Lie monoid of primitives of associahedra is isomorphic to the positive part of the Witt Lie algebra
Paper Structure (24 sections, 29 theorems, 51 equations, 18 figures)

This paper contains 24 sections, 29 theorems, 51 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Background on Hopf monoids
  3. Hopf Monoids
  4. Duality of Hopf Monoids
  5. The Cartier--Milnor--Moore Theorem
  6. Our examples of Hopf monoids
  7. The Hopf Monoid of Posets
  8. The Hopf monoid of Generalized Permutahedra
  9. Geometric background
  10. The Product and Coproduct of Generalized Permutahedra
  11. Submonoids of $\overline{\mathbf{GP}}$
  12. The Hopf Monoid of Permutahedra
  13. The Hopf Monoid of Associahedra
  14. The Hopf Monoid of Orbit Polytopes
  15. Lie monoids of Primitives
  16. ...and 9 more sections

Key Result

Proposition 2.1

Let $\mathbf{H}$ be a Hopf monoid which is the linearization of the Hopf monoid in set species $\mathrm{H}$ and finite dimensional. Suppose $I=S \sqcup T$ is some decomposition and that $x \in \mathbf{H}[S]$, $y \in \mathbf{H}[T]$, and $z \in \mathbf{H}[I]$ are basis elements. The product $\mu_{S,T} Consequently, the product and coproduct of $\mathbf{H}^*$ for arbitrary elements is also determined

Figures (18)

  • Figure 3.1: A pair of normally equivalent polytopes and their normal fan.
  • Figure 3.2: Examples of generalized permutahedra in $\mathbb{R}^{[3]}$
  • Figure 3.3: An example of the associahedron $a_{[3]}$
  • Figure 3.4: The orbit polytope $\mathcal{O}(x)$ for $x=(0,1,1) \in \mathbb{R}^{[n]}$
  • Figure 3.5: The ray generators of $\mathrm{cone}_{(2,1,3)}(\pi_{[3]})$, and the associated poset
  • ...and 13 more figures

Theorems & Definitions (51)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4: Proposition 1.26, AguiarMahajan
  • Theorem 2.5: Section 11.9.2, AguiarMahajan
  • Theorem 2.6: Cartier--Milnor--Moore Cartier2007MilnorMoore, Hopf monoid version AguiarMahajan2
  • Example 3.1: Coproduct in $\mathbf{P}$
  • Definition 3.2
  • Definition 3.3
  • ...and 41 more