# ALGEBRAIC COMBINATORICS

A property of the Birkhoff polytope
Algebraic Combinatorics, Volume 1 (2018) no. 2, p. 275-281
The Birkhoff polytope ${B}_{n}$ is the convex hull of all $n×n$ permutation matrices in ${ℝ}^{n×n}$. We compute the combinatorial symmetry group of the Birkhoff polytope.A representation polytope is the convex hull of some finite matrix group $G\le GL\left(d,ℝ\right)$. We show that the group of permutation matrices is essentially the only finite matrix group which yields a representation polytope with the same face lattice as the Birkhoff polytope.
Revised : 2017-10-12
Accepted : 2017-10-12
Published online : 2018-03-02
DOI : https://doi.org/10.5802/alco.6
Classification:  52B15,  05E18,  20B25,  20C15,  52B05,  52B12
Keywords: Birkhoff polytope, representation polytope, permutation polytope, combinatorial symmetry
@article{ALCO_2018__1_2_275_0,
author = {Baumeister, Barbara and Ladisch, Frieder},
title = {A property of the Birkhoff polytope},
journal = {Algebraic Combinatorics},
publisher = {MathOA foundation},
volume = {1},
number = {2},
year = {2018},
pages = {275-281},
doi = {10.5802/alco.6},
zbl = {06882342},
mrnumber = {3856525},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2018__1_2_275_0}
}

A property of the Birkhoff polytope. Algebraic Combinatorics, Volume 1 (2018) no. 2, pp. 275-281. doi : 10.5802/alco.6. https://alco.centre-mersenne.org/item/ALCO_2018__1_2_275_0/

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