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 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.
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Keywords: Polytope algebra, zonotopes, generalized permutahedra, Eulerian polynomials.
Bastidas, Jose 1
CC-BY 4.0
@article{ALCO_2021__4_5_909_0,
author = {Bastidas, Jose},
title = {The polytope algebra of generalized permutahedra},
journal = {Algebraic Combinatorics},
pages = {909--946},
publisher = {MathOA foundation},
volume = {4},
number = {5},
year = {2021},
doi = {10.5802/alco.185},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.185/}
}
TY - JOUR AU - Bastidas, Jose TI - The polytope algebra of generalized permutahedra JO - Algebraic Combinatorics PY - 2021 SP - 909 EP - 946 VL - 4 IS - 5 PB - MathOA foundation UR - https://alco.centre-mersenne.org/articles/10.5802/alco.185/ DO - 10.5802/alco.185 LA - en ID - ALCO_2021__4_5_909_0 ER -
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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