Geodesic
On permutation polytopes: notions of equivalence
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 4, pp. 1103-1114.

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By assigning a permutation polytope to a group, we produce new interesting polytopes. For the effective use of this construction method, it is desirable to understand which groups are leading to affine equivalent polytopes. Therefore, the notion of effective equivalence has been introduced (Baumeister et al. in Adv Math 222(2):431-452, 2009). In this note, we clarify the notion of effective equivalence and characterize it geometrically. Moreover, we present examples showing that the effective equivalent permutation groups do not correspond to affinely equivalent polytopes. We also apply the characterization to the examples. In our approach we provide a framework for the use of representation theoretic methods in the study of permutation polytopes.
Classification : 52B05, 52B20
Keywords: affine equivalence of polytopes, Birkhoff polytopes, permutation polytopes, effective equivalence of groups, group representation 
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Baumeister, Barbara; Grüninger, Matthias. On permutation polytopes: notions of equivalence. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 4, pp. 1103-1114. http://geodesic.mathdoc.fr/item/JAC_2015__41_4_a2/