The polytope algebra of generalized permutahedra

TL;DR

The paper develops a unified framework linking McMullen’s polytope algebra with the Tits algebra of hyperplane arrangements, by treating the polytope algebra of deformations of a zonotope as a module over the Tits algebra. It specializes to type A and type B generalized permutahedra, revealing that graded components decompose into simple modules indexed by flats and that simple invariants $\,\eta_X(M)$ encode permutation statistics such as excedances. In type A, the module $ abla(\pi_d)$ yields an exact correspondence: $\eta_X(Ξ_r(π_d))$ counts permutations with prescribed flats and excedances, and in type B a parallel theory ties $\eta_X(Ξ_r(π_d^B))$ to signed permutation statistics, including the type B Eulerian polynomials. The results provide a lower bound $2^{d-1}$ on the number of full-dimensional generators for type B generalized permutahedra and identify invariant generating families, alongside explicit bases and a Hopf-monoid compatibility that places generalized permutahedra within a rich algebraic-combinatorial framework. Overall, the work illuminates deep connections between polytope algebras, hyperplane-arrangement algebras, and combinatorial representation theory, with concrete consequences for the structure of generalized permutahedra in both types A and B.

Abstract

The polytope subalgebra of deformations of a zonotope can be endowed with the structure of a module over the Tits algebra of the corresponding hyperplane arrangement. We explore this construction and find relations between statistics on (signed) permutations and the module structure in the case of (type B) generalized permutahedra. In type B, the module structure surprisingly reveals that any family of generators (via signed Minkowski sums) for generalized permutahedra of type B will contain at least $2^{d-1}$ full-dimensional polytopes. We find a generating family of simplices attaining this minimum. Finally, we prove that the relations defining the polytope algebra are compatible with the Hopf monoid structure of generalized permutahedra.

Figures (9)

• Figure 1.1: Different expressions for the class of the trapezoid above in the polytope algebra $\Pi(\mathbb{R}^2)$.
• Figure 1.2: A 2-dimensional polytope ${\mathfrak{p}}$ and two of its faces ${\mathfrak{p}}_v,{\mathfrak{p}}_w$ maximized in directions $v,w$, respectively. On the right, the normal fan $\Sigma_{\mathfrak{p}}$ and the normal cones corresponding to the faces ${\mathfrak{p}}_v,{\mathfrak{p}}_w$ of ${\mathfrak{p}}$.
• Figure 2.1: Vectors $v_1,v_2,v_3$ lie in the same plane. The vectors $v_1,v_2$ are linearly independent and $\log[{\mathfrak{l}}_1]\log[{\mathfrak{l}}_2]$ represents the class of a half-open parallelogram. In contrast, the product $\log[{\mathfrak{l}}_1]\log[{\mathfrak{l}}_2]\log[{\mathfrak{l}}_3]$ is zero.
• Figure 3.1: A 2-dimensional arrangement $\mathcal{A}$ together with its poset of faces (left) and lattice of flats (right). The Möbius function of the lattice of flats is shown in blue. The characteristic polynomial of $\mathcal{A}$ is $\chi(\mathcal{A},t) = t^2 - 3t + 2$.
• Figure 3.2: Product of faces in arrangements in $\mathbb{R}^2$ (left) and $\mathbb{R}^3$ (right). The second arrangement has been intersected with a sphere around the origin, its central face is not visible.

Theorems & Definitions (54)

• Theorem 2.1: mcmullen89
• Example 2.2
• Lemma 2.3: mcmullen89
• Theorem 2.5: mcmullen89
• Theorem 2.6: mcmullen89
• Remark 2.7
• Theorem 2.8: mcmullen93simple
• Theorem 2.9: mcmullen93simple
• Example 3.1
• Proposition 4.1