On permutation polytopes: notions of equivalence

Baumeister, Barbara; Grüninger, Matthias

doi:10.1007/s10801-014-0568-8

Cited by 3 publications

(3 citation statements)

References 8 publications

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“…In the preceding proof, we reduced to the case that the combinatorial isomorphism sends P (g) to D(g) (for any g ∈ S n ). If we could show that then P (g) → D(g) can be extended to an affine isomorphism, Lemma 4.1 would follow from a characterization of effective equivalence by Baumeister and Grüninger [1,Corollary 4.5]. But we do not know how to do this, or whether this is even true more generally (for combinatorial isomorphisms of this form between representation polytopes of arbitrary groups).…”

Section: H a R Ac T E R I Z At I O N O F T H E B I R K H O F F P O Ly...mentioning

confidence: 99%

A property of the Birkhoff polytope

Baumeister¹,

Ladisch²

2018

Algebraic Combinatorics

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The Birkhoff polytope Bn 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 GL(d, R). 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.

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Section: H a R Ac T E R I Z At I O N O F T H E B I R K H O F F P O Ly...mentioning

confidence: 99%

A property of the Birkhoff polytope

Baumeister¹,

Ladisch²

2018

Algebraic Combinatorics

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“…If two ideal characters of G are in the same orbit under Aut(G), then the corresponding orbit polytopes of G are affinely equivalent. (If the ideal characters belong to different orbits, then it can still happen that the corresponding orbit polytopes are affinely equivalent [3], but at least for the elementary abelian groups of orders 4, 8 and 16, this is not the case.) Thus the number of orbit polytopes up to affine equivalence is at most the number of Aut(G)-orbits on the set of ideal characters.…”

Section: S O M E O R B I T P O Ly T O P E S O F E L E M E N Ta Ry a B...mentioning

confidence: 99%

“…Clearly, the affine symmetry group of an orbit polytope P (G, v) always contains the symmetries induced by G. Depending on the group and on the point v, there may be v tsv tv t 2 sv t 2 v t 3…”

Section: Introductionmentioning

confidence: 99%

Affine Symmetries of Orbit Polytopes

Friese,

Ladisch

2014

Preprint

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An orbit polytope is the convex hull of an orbit under a finite group G GL(d, R). We develop a general theory of possible affine symmetry groups of orbit polytopes. For every group, we define an open and dense set of generic points such that the orbit polytopes of generic points have conjugated affine symmetry groups. We prove that the symmetry group of a generic orbit polytope is again G if G is itself the affine symmetry group of some orbit polytope, or if G is absolutely irreducible. On the other hand, we describe some general cases where the affine symmetry group grows.We apply our theory to representation polytopes (the convex hull of a finite matrix group) and show that their affine symmetries can be computed effectively from a certain character. We use this to construct counterexamples to a conjecture of Baumeister et al. on permutation polytopes [Advances in Math. 222 (2009), 431-452, Conjecture 5.4].

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Affine symmetries of orbit polytopes

Friese

Ladisch

2016

Advances in Mathematics

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On permutation polytopes: notions of equivalence

Cited by 3 publications

References 8 publications

A property of the Birkhoff polytope

A property of the Birkhoff polytope

Affine Symmetries of Orbit Polytopes

Affine symmetries of orbit polytopes

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