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.
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/
[1] Hopf monoids and generalized permutahedra (2017) (https://arxiv.org/abs/1709.07504)
[2] Characteristic elements for real hyperplane arrangements, Sém. Lothar. Combin., Volume 82B (2020), Paper no. R60, 12 pages | MR | Zbl
[3] Monoidal functors, species and Hopf algebras. With forewords by Kenneth Brown and Stephen Chase and André Joyal, CRM Monograph Series, 29, Amer. Math. Soc., Providence, RI, 2010, lii+784 pages | DOI | MR
[4] Hopf monoids in the category of species, Hopf algebras and tensor categories (Contemp. Math.), Volume 585, Amer. Math. Soc., Providence, RI, 2013, pp. 17-124 | DOI | MR
[5] Topics in hyperplane arrangements, Mathematical Surveys and Monographs, 226, Amer. Math. Soc., Providence, RI, 2017, xxiv+611 pages | DOI | MR | Zbl
[6] Matroid polytopes and their volumes, Discrete Comput. Geom., Volume 43 (2010) no. 4, pp. 841-854 | DOI | MR | Zbl
[7] Coxeter submodular functions and deformations of Coxeter permutahedra, Adv. Math., Volume 365 (2020), Paper no. 107039, 36 pages | DOI | MR | Zbl
[8] Valuations and the Hopf Monoid of Generalized Permutahedra (2020) (https://arxiv.org/abs/2010.11178)
[9] Combinatorial species and tree-like structures. With a foreword by Gian-Carlo Rota, Encyclopedia of Mathematics and its Applications, 67, Cambridge University Press, Cambridge, 1998, xx+457 pages (Translated from the 1994 French original by Margaret Readdy) | MR | Zbl
[10] Some combinatorial and algebraic properties of Coxeter complexes and Tits buildings, Adv. in Math., Volume 52 (1984) no. 3, pp. 173-212 | DOI | MR | Zbl
[11] Oriented matroids, Encyclopedia of Mathematics and its Applications, 46, Cambridge University Press, Cambridge, 1993, xii+516 pages | MR | Zbl
[12] -Eulerian polynomials arising from Coxeter groups, European J. Combin., Volume 15 (1994) no. 5, pp. 417-441 | DOI | MR | Zbl
[13] A class of -symmetric functions arising from plethysm. In memory of Gian-Carlo Rota, J. Combin. Theory Ser. A, Volume 91 (2000) no. 1-2, pp. 137-170 | DOI | MR | Zbl
[14] A pithy look at the polytope algebra, Algebraic and geometric combinatorics on lattice polytopes, World Sci. Publ., Hackensack, NJ, 2019, pp. 117-131 | MR | Zbl
[15] Submodular functions, matroids, and certain polyhedra, Combinatorial Structures and their Applications (Proc. Calgary Internat. Conf., Calgary, Alta., 1969) (1970), pp. 69-87 | MR | Zbl
[16] Eulerian polynomials: from Euler’s time to the present, The legacy of Alladi Ramakrishnan in the mathematical sciences, Springer, New York, 2010, pp. 253-273 | DOI | MR | Zbl
[17] Signed words and permutations. V. A sextuple distribution, Ramanujan J., Volume 19 (2009) no. 1, pp. 29-52 | DOI | MR | Zbl
[18] Convex polytopes. Prepared and with a preface by Volker Kaibel, Victor Klee and Günter M. Ziegler, Graduate Texts in Mathematics, 221, Springer-Verlag, New York, 2003, xvi+468 pages | DOI | MR | Zbl
[19] Enumeration of points, lines, planes, etc, Acta Math., Volume 218 (2017) no. 2, pp. 297-317 | DOI | MR | Zbl
[20] Une théorie combinatoire des séries formelles, Adv. in Math., Volume 42 (1981) no. 1, pp. 1-82 | DOI | MR | Zbl
[21] The polytope algebra, Adv. Math., Volume 78 (1989) no. 1, pp. 76-130 | DOI | MR | Zbl
[22] On simple polytopes, Invent. Math., Volume 113 (1993) no. 2, pp. 419-444 | DOI | MR | Zbl
[23] Separation in the polytope algebra, Beiträge Algebra Geom., Volume 34 (1993) no. 1, pp. 15-30 | MR | Zbl
[24] Shard polytopes (2020) (https://arxiv.org/abs/2007.01008)
[25] Faces of generalized permutohedra, Doc. Math., Volume 13 (2008), pp. 207-273 | MR | Zbl
[26] Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN (2009) no. 6, pp. 1026-1106 | DOI | MR | Zbl
[27] Non-crossing partitions for classical reflection groups, Discrete Math., Volume 177 (1997) no. 1-3, pp. 195-222 | DOI | MR | Zbl
[28] The Burnside algebra of a finite group, J. Combinatorial Theory, Volume 2 (1967), pp. 603-615 | MR | Zbl
[29] Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages | DOI | MR
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