# ALGEBRAIC COMBINATORICS

The polytope algebra of generalized permutahedra
Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 909-946.

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.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.185
Classification: 05Exx,  52C35,  52Bxx,  18M80
Keywords: Polytope algebra, zonotopes, generalized permutahedra, Eulerian polynomials.
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Bastidas, Jose. The polytope algebra of generalized permutahedra. Algebraic Combinatorics, Volume 4 (2021) no. 5, pp. 909-946. doi : 10.5802/alco.185. https://alco.centre-mersenne.org/articles/10.5802/alco.185/

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