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...all functions f such that∑ i f(i, j) = ∑ k f(j, k) for all j). We thank S. Fiorini for bringing this fact to our attention. Second, these spaces arise in the context of the analysis of rank data, see =-=[5]-=-. There the R( n 2) is considered as the space of pairs and V1 is the space of means. They correspond to the Babington Smith Model resp. the Bradley/TerryMallows model. In Section 7.4.3 in [5], there ...

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... if pi(i) > pi(j) 0 if pi(i) < pi(j) The n-th Linear Ordering Polytope Pn is defined as the convex hull of the n! vectors ppi := (kij(pi))1≤i<j≤n ∈ R (n 2 ). In this we follow the definition given in =-=[4]-=-. The linear ordering polytope is an important and well-studied object in combinatorial optimization, see for example Chapter 6 of [6] and the references therein. For its study we will also consider t...

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... unique. We also note that the existence of the Z2 × Sn-action on Pn−1 is known from [2], where this action is constructed from the Z2 × Sn−1-action and a certain class of automorphisms borrowed from =-=[1]-=-. The projection Pn → Pn−1 maps n vertices to one. In [2, Lemma 2] the trivial and 3-cycle facets are shown to be the only facets having the maximal number of n!/2 vertices. Thus the preimages of thes...

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...pi))1≤i<j≤n ∈ R (n 2 ). In this we follow the definition given in [4]. The linear ordering polytope is an important and well-studied object in combinatorial optimization, see for example Chapter 6 of =-=[6]-=- and the references therein. For its study we will also consider the n-th permutahedron; this is the polytope with n! vertices (pi(i))1≤i≤n ∈ R n for pi ∈ Sn. Both polytopes carry a natural Sn-action ...

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... the decomposition arises from representation theoretic considerations in [7]. In that paper it is also shown to be unique. We also note that the existence of the Z2 × Sn-action on Pn−1 is known from =-=[2]-=-, where this action is constructed from the Z2 × Sn−1-action and a certain class of automorphisms borrowed from [1]. The projection Pn → Pn−1 maps n vertices to one. In [2, Lemma 2] the trivial and 3-...

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... consider the other known families of facets (see for example [6]) from this point of view. The construction of the linear ordering polytope can be generalized to arbitrary finite Coxeter groups. See =-=[3]-=- for Type B and [7, Section 3.5] for the general case. There is still a decomposition into two invariant subspaces as in Theorem 1. The space V1 generalizes as expected: The action of the group is the...

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...[5], there are also a basis of V1 and an generating set of V2 given. The later contains the basis we give in Section 3. Third, the decomposition arises from representation theoretic considerations in =-=[7]-=-. In that paper it is also shown to be unique. We also note that the existence of the Z2 × Sn-action on Pn−1 is known from [2], where this action is constructed from the Z2 × Sn−1-action and a certain...