Affine symmetries of orbit polytopes

Friese, Erik; Ladisch, Frieder

doi:10.1016/j.aim.2015.10.021

Cited by 15 publications

(47 citation statements)

References 22 publications

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“…We call Sym(G, V ) the generic symmetry group of the kG-module V . Theorem A is a straightforward generalization of an earlier result [8] from the case k = R to arbitrary fields.…”

Section: Introductionmentioning

confidence: 60%

“…Let G = GL(Gv), where Gv spans V . Then we can view V as a k G-module, and we can speak of generic points for G. In our previous paper, we showed that when k = R and w is generic for G, then GL( Gw) = G [8,Corollary 5.4]. In particular, we can not get an infinitely increasing chain of generic symmetry groups.…”

Section: And We Must Havementioning

confidence: 99%

“…In the following, we characterize Sym(G, χ) and χ for ideal characters χ. The first statement of Proposition 5.4 is essentially contained in our earlier paper [8,Theorem 8.5], but we give a different proof here.…”

Section: Propositionmentioning

confidence: 99%

“…To obtain these results, we develop a general theory of linear symmetry groups of orbits, which is a natural continuation of our previous paper [8]. The methods are more algebraic than geometric, and in particular, we first work over an arbitrary field k. Thus let G be a finite group acting linearly on a vector space V over some field k. (Later, we will specialize to k = R, the field of real numbers.)…”

Section: Introductionmentioning

confidence: 99%

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Classification of affine symmetry groups of orbit polytopes

Friese

Ladisch

2017

J Algebr Comb

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A b s t r ac t . Let G be a finite group acting linearly on a vector space V . We consider the linear symmetry groups GL(Gv) of orbits Gv ⊆ V , where the linear symmetry group GL(S) of a subset S ⊆ V is defined as the set of all linear maps of the linear span of S which permute S. We assume that V is the linear span of at least one orbit Gv. We define a set of generic points in V , which is Zariskiopen in V , and show that the groups GL(Gv) for v generic are all isomorphic, and isomorphic to a subgroup of every symmetry group GL(Gw) such that V is the linear span of Gw. If the underlying characteristic is zero, "isomorphic" can be replaced by "conjugate in GL(V )". Moreover, in the characteristic zero case, we show how the character of G on V determines this generic symmetry group. We apply our theory to classify all affine symmetry groups of vertex-transitive polytopes, thereby answering a question of Babai (1977).2010 Mathematics Subject Classification. Primary 52B15, Secondary 05E15, 20B25, 20C15.

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“…We call Sym(G, V ) the generic symmetry group of the kG-module V . Theorem A is a straightforward generalization of an earlier result [8] from the case k = R to arbitrary fields.…”

Section: Introductionmentioning

confidence: 60%

Section: And We Must Havementioning

confidence: 99%

Section: Propositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of affine symmetry groups of orbit polytopes

Friese

Ladisch

2017

J Algebr Comb

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“…If D is an orthogonal representation, then the permutation g → g −1 is realized by transposing matrices, sending D(g) to D(g) t = D(g −1 ). The general case (which we will not need) can be reduced to the orthogonal case [5,Prop. 6.4].…”

Section: T H E C O M B I N At O R I a L S Y M M E T Ry G Ro U P O F Tmentioning

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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