Large convex sets in oriented matroids

Büchi, J; Fenton, William E.

doi:10.1016/0095-8956(88)90074-3

Cited by 9 publications

(8 citation statements)

References 7 publications

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“…There is a classification of quasiparabolic subsets (and more generally, of elements of B(Φ)) in terms of elements of B(Φ + ) and additional combinatorial data, which won't be discussed here. In [7], analogues of quasiparabolic sets in (possibly infinite) oriented matroids are called large convex sets.…”

Section: 5mentioning

confidence: 99%

“…It would be interesting to know if Conjecture 2.5(c), for example, also has an analogue in that generality. There are (at least) two natural closure operators which one could use; the natural oriented matroid closure operator d (see [7] or [2, §6]), and an analogue of 2-closure constructed from d in a similar way as the 2-closure on root systems is defined in terms of their geometric d-closure. In view of the results of this paper in the case of finite root systems, it would be particularly interesting to see how the 2-closure behaves in simplicial geometries (it certainly does not have good properties for (possibly non-simplicial) oriented geometries in general).…”

Section: 7mentioning

confidence: 99%

“…One natural candidate closure operator we wish to informally describe requires notions of infinite oriented matroids, one definition of which can be found in [7]. We consider only a natural subclass, which we shall henceforth just call oriented geometries, corresponding to oriented geometries in the case of finite oriented matroids (see also [3,Exercise 3.13]).…”

Section: 7mentioning

confidence: 99%

“…Call such a structure an oriented geometry root system of W ; any standard root system gives rise to one in this more general sense. Each oriented geometry root system gives (by definition in [7]) a closure operator on Ψ; these include analogues of the d-closures as previously considered, but one might expect there could be many more (not for finite Coxeter groups, but at least for some infinite non-affine Coxeter ones). The closure operator on Ψ of interest (as a possibly natural one for use in the conjectures) is the one which has as closed sets the intersections of closed sets for these oriented geometry root system closure operators.…”

Section: 7mentioning

confidence: 99%

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On the Weak Order of Coxeter Groups

Dyer¹

2019

Can. J. Math.-J. Can. Math.

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This paper provides some evidence for conjectural relations between extensions of (right) weak order on Coxeter groups, closure operators on root systems, and Bruhat order. The conjecture focused upon here refines an earlier question as to whether the set of initial sections of reflection orders, ordered by inclusion, forms a complete lattice. Meet and join in weak order are described in terms of a suitable closure operator. Galois connections are defined from the power set of W to itself, under which maximal subgroups of certain groupoids correspond to certain complete meet subsemilattices of weak order. An analogue of weak order for standard parabolic subsets of any rank of the root system is defined, reducing to the usual weak order in rank zero, and having some analogous properties in rank one (and conjecturally in general). t t t t t t t t t t t J J J J J J J J J J J

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Section: 5mentioning

confidence: 99%

Section: 7mentioning

confidence: 99%

Section: 7mentioning

confidence: 99%

Section: 7mentioning

confidence: 99%

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On the Weak Order of Coxeter Groups

Dyer¹

2019

Can. J. Math.-J. Can. Math.

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“…Such root systems play crucial roles in understanding various mathematical structures, particularly those arising from Lie theory. They naturally determine oriented matroids in the sense of [6]. One may ask if different (possibly non-reduced) root system realizations yield non-isomorphic reduced oriented matroid structures when transferred to the abstract root system T ×{±1} with natural W -action, where T is the set of reflections in W (see [1,3]).…”

Section: Introductionmentioning

confidence: 99%

Oriented matroid structures from realized root systems

Dyer

Wang

2020

J Algebr Comb

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This paper investigates the question of uniqueness of the reduced oriented matroid structure arising from root systems of a Coxeter system in real vector spaces. We settle the question for finite Coxeter systems, irreducible affine Weyl groups and all rank three Coxeter systems. In these cases, the oriented matroid structure is unique unless W is of type A n , n ≥ 3, in which case there are three possibilities.

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On the relation between the subadditivity cone and the quantum entropy cone

He,

Hubeny,

Rota

2023

J. High Energ. Phys.

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Given a multipartite quantum system, what are the possible ways to impose mutual independence among some subsystems, and the presence of correlations among others, such that there exists a quantum state which satisfies these demands? This question and the related notion of a pattern of marginal independence (PMI) were introduced in [1], and then argued in [2] to be central in the derivation of the holographic entropy cone. Here we continue the general information theoretic analysis of the PMIs allowed by strong subadditivity (SSA) initiated in [1]. We show how the computation of these PMIs simplifies when SSA is replaced by a weaker constraint, dubbed Klein’s condition (KC), which follows from the necessary condition for the saturation of subadditivity (SA). Formulating KC in the language of partially ordered sets, we show that the set of PMIs compatible with KC forms a lattice, and we investigate several of its structural properties. One of our main results is the identification of a specific lower dimensional face of the SA cone that contains on its boundary all the extreme rays (beyond Bell pairs) that can possibly be realized by quantum states. We verify that for four or more parties, KC is strictly weaker than SSA, but nonetheless the PMIs compatible with SSA can easily be derived from the KC-compatible ones. For the special case of 1-dimensional PMIs, we conjecture that KC and SSA are in fact equivalent. To make the presentation self-contained, we review the key ingredients from lattice theory as needed.

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Large convex sets in oriented matroids

Cited by 9 publications

References 7 publications

On the Weak Order of Coxeter Groups

On the Weak Order of Coxeter Groups

Oriented matroid structures from realized root systems

On the relation between the subadditivity cone and the quantum entropy cone

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