Oriented Matroids -- Combinatorial Structures Underlying Loop Quantum Gravity
aa r X i v : . [ g r- q c ] N ov Oriented Matroids –Combinatorial Structures UnderlyingLoop Quantum Gravity
Johannes Brunnemann ∗ , David Rideout † Department of Mathematics, University of Paderborn, 33098 Paderborn, Germany Perimeter Institute for Theoretical Physics, N2L 2Y5, Waterloo, ON, CanadaNovember 6, 2018
Abstract
We analyze combinatorial structures which play a central role in determining spectral properties of thevolume operator [1] in loop quantum gravity (LQG). These structures encode geometrical information ofthe embedding of arbitrary valence vertices of a graph in 3-dimensional Riemannian space, and can berepresented by sign strings containing relative orientations of embedded edges. We demonstrate that thesesignature factors are a special representation of the general mathematical concept of an oriented matroid[2, 3]. Moreover, we show that oriented matroids can also be used to describe the topology (connectedness)of directed graphs. Hence the mathematical methods developed for oriented matroids can be applied tothe difficult combinatorics of embedded graphs underlying the construction of LQG. As a first applicationwe revisit the analysis of [4, 5], and find that enumeration of all possible sign configurations used thereis equivalent to enumerating all realizable oriented matroids of rank 3 [2, 3], and thus can be greatlysimplified. We find that for 7-valent vertices having no coplanar triples of edge tangents, the smallest non-zero eigenvalue of the volume spectrum does not grow as one increases the maximum spin j max at the vertex,for any orientation of the edge tangents. This indicates that, in contrast to the area operator, consideringlarge j max does not necessarily imply large volume eigenvalues. In addition we give an outlook to possiblestarting points for rewriting the combinatorics of LQG in terms of oriented matroids. Contents ∗ [email protected] † [email protected] Oriented Matroids in Loop Quantum Gravity 24
Any approach to quantum gravity is expected to exhibit a discrete nature of spacetime at distances close to thePlanck scale ( ∼ − m ). Taking this perspective, our traditional picture of continuous spacetime geometrycan only arise as a kind of (semi-)classical limit from a more fundamental combinatorial theory, linking matterand gravity. Loop quantum gravity (LQG) is supposed to give a canonical quantization of gravity, based onthe initial value formulation of general relativity (GR), in a manner which is independent of any backgroundgeometry. LQG provides a mathematically rigorous way to non-perturbatively quantize geometry itself bypromoting classical geometric quantities such as length, area or volume into operators in quantum theory.Remarkably these operators indeed turn out to have discrete spectra . During the last 20 years LQG hasalready produced results which are regarded as physically relevant: the absence of a cosmological Big-Bangsingularity [10], as well as the reproduction of the entropy-area law for black holes from first principles [11],just to mention a couple. Moreover for the first time there exists a proposal for implementing the Hamiltonconstraint, which encodes the dynamics, as an operator in quantum theory [12]. Even better, if geometry iskept dynamical, matter can be quantized without the occurrence of anomalies or UV-divergences, which plaguequantum field theories constructed on a fixed background [13].Nevertheless there are important questions regarding the contruction of the physical sector of LQG which remainunanswered so far. One obstacle is the construction of the theory in terms of a projective limit over a posetof arbitrarily complicated directed graphs embedded into three dimensional Riemannian space, the Cauchysurfaces which arise in the initial value formulation of GR. Without reference to a background geometry, thecharacterization of such embeddings by coordinates, edge lengths or angles is meaningless. Rather one isreferred to diffeomorphism invariant characterizations such as closed circuits in graphs or the local intersectionbehavior of edges at graph vertices. While the mathematical construction of the projective limit over the graphposet is well understood, the complicated combinatorics of generic elements of the poset still lack an effectivecharacterization method which makes a study of the full theory viable.In practical computations, the combinatorial difficulty is often circumvented by imposing simplifications to LQG.One possible simplification is based on the assumption that the underlying classical theory is not full GR, but asymmetry reduced, so-called cosmological model. This is equivalent to partially fixing a background geometry.The resulting model is quantized using techniques similar to LQG using a restricted set of graphs, adapted tothe symmetry reduction. In the resulting loop quantum cosmology (LQC) (see [14, 15] and references therein)it is then possible to address questions such as the Big-Bang singularity, which is far beyond the possibilitiesone has in full LQG at present. Another ansatz (called Algebraic Quantum Gravity, AQG) is to keep GR, thatis full background independence, but to modify the quantization procedure by restricting only to certain types Thus far this is a kinematical statement. See [6, 7] for a discussion on how this property can be transferred to the physicalsector of LQG. Additionally in [8, 9] a setup is given where these properties can be regarded as physical. In a symmetry reduced setup. Also called cycles in graph theory.
2f graphs and embeddings in the poset [16]. This makes it possible to analyze the classical limit of the resultingmodel [17] as well as to construct a physical sector of LQG in a reduced phase space quantization [9].Given the promising results of the outlined simplified models, it is highly desireable to get a better understandingof the physical implications of the underlying assumptions, as simplifications may modify dynamical propertiesof the analyzed model. For example, in the context of LQC it is found that the operator corresponding to theclassical inverse scale factor of the universe is bounded in LQC [14], but unbounded in full LQG [18]. Moreoverthe Hilbert space of LQC cannot be continuously embedded into the Hilbert space obtained in LQG [19]. InAQG, on the other hand, one special graph and its embedding is fixed, in order to compute the volume spectrum.It has been suggested that only this particular embedding gives a consistent semiclassical limit of the model[20]. The mathematical tools presented here may allow us to consider this question for arbitrary embeddedvertices.In full LQG, it turns out that the global circuit structure of generic graphs and their local embedding propertiesare relevant for the construction of the Hamilton constraint operator of [12], which encodes the dynamics of thetheory. This arises because a crucial ingredient of the Hamilton constraint operator consists in the quantumanalogue to the classical volume integral of a region in three dimensional Riemannian space, called the volumeoperator. In [4, 5] the spectral properties of this operator due to [1] were analyzed in full LQG, and it wasfound that they are highly sensitive to the local graph embedding. In particular the presence of a finite smallestnon-zero eigenvalue is characterized by the chosen embedding.In order to bring the results of the outlined simplified approaches in a closer context to the full theory, it isimportant to continue work on full LQG, and to develop techniques to better understand its combinatorics.This paper is intended as a first step toward developing such a method. It is based on the observation thatlocal embedded graph geometry and global graph topology have a common home in the mathematical field of(oriented) matroid theory [3, 2], which provides a precise combinatorial abstraction of simplices and directedgraphs.Oriented matroids have already been used in the physics literature. One example is quantum information theory[21]. Another is a considerable amount of work done by the author of [22]–[23] and collaborators, to investigateconnections between matroid theory and supergravity [22, 24], Chern-Simons theory [25], string / M- theory[26, 27] and 2D-gravity [28]. In the context of supergravity, possible implications on the Ashtekar formalismin higher dimesions have been outlined [23]. Their approach is mainly based on the observation that certainantisymmetry properties of tensors and forms in the according contexts can be captured by oriented matroids.We suggest a different approach here. We are going to treat the combinatorics inherent in the construction ofLQG by means of oriented matroids. As we will demonstrate, the occurring combinatorial structures match ina very natural way. This opens up a new mathematical arena to study LQG, and to overcome certain technicaland conceptual obstacles which, at present, make concrete analytical and numerical investigations of the fulltheory very difficult. In particular it becomes feasible to develop a systematic and unified treatment of global(connectedness) and local (embedding) properties of the graph poset underlying LQG.The plan of this paper is as follows. In the next section we will describe the occurrence of geometrical andtopological properties of graphs in the present formulation of LQG in more detail and characterize their com-binatorics. We then introduce the concept of a matroid and an oriented matroid and show how the latter canbe used in order to encode the information on global and local properties of directed graphs. Furthermorewe will comment on the issue of realizability of oriented matroids, which is directly related to computing thenumber of possible diffeomorphic local graph embeddings, as well the number of certain graph topologies. Wewill finally show how this can be applied to the results of [4, 5], which will be revisited and simplified. Ourfindings will be discussed in section 6. There we will also comment on the notion of a semiclassical limit ofthe volume operator in light of the revisited numerical computation. In the summary and outlook section weoutline upcoming work, and steps necessary to fully establish the oriented matroid formalism within LQG, aswell as its potential impact on further studies in full LQG.
In this section we will only outline the current construction of LQG as necessary for our work. For a detailedintroduction we refer to [29, 30].LQG is based on the initial value formulation of Einstein’s equations for GR, which is possible for any globally3yperbolic spacetime. According to a theorem of Geroch [31], the spacetime is then homeomorphic to theCartesian product of the real numbers R (‘time axis’) times an orientable spacelike three dimensional Cauchysurface Σ. An according 3 + 1 decomposition procedure introduced by Arnowitt, Deser and Misner (ADM) [33]then rewrites GR in terms of canonically conjugated variables. These are components of the induced spatialmetric tensor on Σ and its first order (“time-”) derivatives in the R -direction given by components of extrinsiccurvature. In this setup the dynamics of Einstein’s equations is encoded completely in constraints (three spatialdiffeomorphism and a scalar constraint) which restrict admissible initial data on a chosen Cauchy surface Σ,and assure background independence . The resulting theory can then be treated using Dirac’s approach toconstrained Hamiltonian systems [34], which proposes to first construct the canonical theory and second toimplement the constraints formulated in terms of the canonical variables on phase space. This results in thewell known Dirac algebra of Poisson brackets among the constraints.Inspired by the work of Sen, Ashtekar realized [35] that the canonical ADM-variables of GR can be extended interms of Lie-algebra valued densitized triads E (local orthonormal frames) and according connections A on aprincipal fibre bundle with basis Σ and compact structure group G . In practice G is taken to be SU (2), howeverthe construction described here works to a certain extent for general compact Lie groups G . As different triadscan encode identical information on the spatial metric, another constraint, called the Gauss contraint, is addedto the Dirac algebra.This opens a way to formulate the theory in terms of holonomies and fluxes, very similar to lattice gauge theory.Consider a one dimensional embedded directed path p ⊂ Σ. By the holonomy map h , a connection A ∈ A can be understood as a mapping of p to the gauge group G , A : p h → A ( p ) ∈ G , where A denotes the space ofsmooth connections . In order to make this construction more explicit one often writes h p ( A ) instead of A ( p ).The set P of all paths carries a groupoid structure with respect to composition . As can be shown, the set A is contained in the set A = Hom ( P , G ) of all homomorphisms (no continuity assumption) of the path groupoid P to the gauge group G . The set A is consequently called the space of generalized connections. A is equippedwith a topology as follows. One considers finite collections E ( γ ) := { e , . . . , e N } of non self-intersecting paths e k , called edges , which mutually intersect at most at their beginning b ( e k ) and end (final) points f ( e k ), calledvertices V ( γ ) := { b ( e k ) , f ( e k ) } e k ∈ E ( γ ) . Such a collection is called a graph γ , the set of all finite semianalyticgraphs is denoted by Γ. Now one considers the subgroupoid l ( γ ) ⊂ P generated by all elements contained in E ( γ ) together with their inverses and finite composition . The characteristic function of E ( γ ) is also calledthe support supp( l ( γ )).Certainly the label set L = { l ( γ ) } γ ∈ Γ carries a partial order, that is l ( γ ) ≺ l ′ ( γ ′ ) if l ( γ ) is a subgroupoid of l ′ ( γ ′ ). By construction X l ( γ ) := Hom ( l ( γ ) , G ), is understood as the set of all homomorphisms from E ( γ ) to G N . That is, one copy of G is associated to each edge via the holonomy map.Using the partial order of the set L one then constructs the projective limit X among the X l ( γ ) . It can be shownthat there is bijection between A and X . Therefore A can be equipped with a topology inherited by X andturns out to be compact Hausdorff by the compactness of G and Tychonoff’s theorem. This makes it possibleto construct a Hilbert space H = L ( A , dµ ) as an inductive limit of Hilbert spaces H l ( γ ) = L ( X l ( γ ) , dµ γ ) = L ( G N ( l ( γ )) , dµ H N )In each H l ( γ ) one considers functions f γ : G N ( l ( γ )) → C which are called cylindrical, because their support consistsof those N copies of G which are labelled by supp( l ( γ )). These functions are square integrable continuous withrespect to the Haar measure dµ H on each copy of G . An orthonormal basis on H is given by so called spinnetwork functions T γ~j ~m~n , labelled by a graph γ and irreducible representations ~j := ( j , . . . , j N ) of G (“spins”)plus a pair of matrix indices ~m := ( m , . . . , m N ), ~n := ( n , . . . , n N ) one for each edge contained in E ( γ ). Thisis derived from the Peter&Weyl theorem and rests on the compactness of G . Holonomies act as multiplicationoperators, fluxes as derivations on H . In fact it is even diffeomorphic to R × Σ, as proved in [32]. The physical content of the theory must not depend on the choice of Cauchy surfaces. A path is an equivalence class of semianalytic curves c with respect to re-parametrization and finite semianalytic retracings.A curve is a map c : R ⊃ [0 , ∋ t → c ( t ) ⊂ Σ. Here c (0) =: b ( c ) is called the beginning point, c (1) =: f ( c ) the end point of c .Semianalyticity is needed to ensure that two paths can only have a finite number of intersections. In the context of LQG a holonomy is understood as a parallel transport of the connection A along p , where p is not necessarilya closed loop. Two paths can only be composed if they have a common point. Modulo finite retracings. Strictly speaking an edge corresponds to a path which contains a representative curve which has noself intersections. It follows that l ( γ ) is in fact an equivalence class of graphs: l ( γ ) = l ( γ ′ ) if their edge / ground-sets E ( γ ′ ) and E ( γ ) differ onlyby a re-labelling of elements and/or reorientation of edge directions plus finite compositions. One only considers graphs over thefull n -element ground set, called maximal representatives. µ depends only on the topology of Σ via supp( l ( γ )). It does notrefer to a choice of coordinates on Σ. In this sense µ is defined background independently. In particular it canbe shown that finite (semi-) analytic diffeomorphisms on Σ can be unitarily implemented on H . Even more, µ is uniquely determined if unitarity of diffeomorphisms is imposed [36, 37].So far all these constructions are on the kinematical level. By the quantum analogue to Dirac’s procedure the classical constraint functions have to be implemented as constraint operators on H . Their common kernelgives the physical Hilbert space H phys . Indeed it is straightforward to obtain the set of solutions to the Gaussconstraint H Gauss as a subset of H .However, one cannot in general expect H phys ⊆ H , rather elements of H phys are likely to be distributionscontained in the algebraic dual D ∗ of H .This is what happens when one constructs the space H diff of solutions to the spatial diffeomorphism constraint.Nevertheless it is possible to rigorously construct H diff via a rigging map construction [38]. Elements of H diff are then labelled by equivalence classes of supp( l ( γ )) with respect to finite semianalytic diffeomorphisms. Asdescribed in the introduction, each such equivalence class carries global (topological) information about theconnectedness of the representatives γ , that is their circuit structure, as well as local (geometric) informationabout the intersection behavior of their edges at the vertices of γ .Despite the successful solution of Gauss and diffeomorphism constraints, the scalar or Hamilton constraint,encoding the dynamics, has not been solved so far. However there is a proposal to implement it in quantumtheory on H [12] as well as a composite operator called the master constraint on H diff [39]. Moreover in thecontext of [39, 40] a rigorous program for constructing an inner product on H phys was proposed for the firsttime. Completing this step is crucial in order to construct the physical sector of LQG and to make physicalpredictions.A major difficulty in the treatment of the Hamilton- respectively master constraint operator in the full theorycomes from the fact that both operators are graph changing, due to the presence of holonomies in the classicalconstraint expression which are promoted to multiplication operators as mentioned above . Illustrativelyspeaking, constructing H phys is equivalent to finding eigenstates to graph changing operators at a projectivelimit.Nevertheless operators only containing fluxes, such as the volume operator, can in principle be discussed at thelevel of H , just by evaluating their action on a function f γ cylindrical over a graph γ . If cylindrical consis-tency can be shown, then the according operator is automatically defined on all of H .In the next section we will introduce the concept of matroids and oriented matroids and show how the connect-edness of graphs as well as their local geometric embedding properties can be described within this framework. In this section we will give a brief introduction to the subject of matroids and oriented matroids. The presen-tation is mostly based on [41] and [3]. Partly we have also used [42, 43]. We have tried to stay close to thenotation used there. Many definitions and proofs have been taken directly from these books. At some places wegive page numbers. However we would like to emphasize that, in order to keep this introduction short, we havechanged the manner of presentation of the subject. In the literature one starts from different axiom systems,e.g. for chirotopes, (oriented) matroids in terms of (signed) circuits, (signed) bases, etc. and proves that theseare equivalent. We take this equivalence for granted and use these objects equivalently in our presentation. Wehave extended the explanation where we felt it would be for the benefit of the reader not familiar with (oriented)matroids.In general matroids [41] and oriented matroids [3] can be thought of as a combinatorial abstraction of lineardependencies in a real vector space. In order to make the presentation more accessible to the reader, we will Also referred to as refined algebraic quantization. Consider for example eigen“functions” of the position operator in quantum mechanics, given by Dirac’s δ -distribution. When a cylindrical function f γ is multiplied by a holonomy h e supported on an edge e not contained in E ( γ ) then the supportof h e · f γ is given by l ( γ ∪ { e } ). That is, one has to show that the action of this operator on a cylindrical function f e γ ∈ H l ( e γ ) is equal to its action on acylindrical function f γ ∈ H l ( γ ) with γ ⊂ e γ , if H l ( e γ ) is restricted to its subspace H l ( γ ) supported on γ . E be a given finite set of n elements. We willalso write | E | = n . We will denote by 2 E the set of all subsets of E . It has cardinality | E | = 2 | E | = 2 n , as P nk =0 (cid:0) nk (cid:1) = 2 n . The subsets of 2 E are called f amilies and will be denoted by script capital letters, i.e. B . Thefamilies can be regarded as elements of the set of subsets of 2 E , 2 E . We will call the subsets of 2 E collections and denote them by boldface capital letters, i.e. B .An antichain (also called incomparable family ) in E is a family B of subsets of E such that, for all X, Y ∈ B ,from X ⊆ Y it follows that X = Y .We will also need the notion of a minimal and maximal subset B of E with respect to a certain property prop that holds for B . This is achieved by ordering the subsets by set inclusion, and regarding a subset B to bemaximal (minimal) in B iff ∄ B ′ ∈ B such that B ⊂ B ′ ( B ⊃ B ′ ) and also prop holds for B ′ . In this section we introduce the concept of a matroid. The reason for using underlined quantities as in [3] willbecome clear in the next section, where we introduce signed subsets of a set.We start with the definition of a matroid in terms of its circuits. In the vector picture a circuit can be thoughtof as a minimal linearly dependent set of vectors.
Definition 3.1 (Matroid from Circuits [41].)
A family C of subsets of a set E is called the set of circuitsof a matroid M = ( E, C ) on E if(C0) Non-emptiness: ∅ / ∈ C (C2) Incomparability: if C ⊆ C then C = C ∀ C , C ∈ C ( C is an antichain)(C3) Elimination: For all C , C ∈ C with C = C , and ∀ e ∈ E , ∃ C ∈ C such that C ⊆ ( C ∪ C ) \{ e } . Moreover we have the following definitions:
Definition 3.2 (Bases, Circuits, Hyperplanes and Cocircuits of a matroid M [41].) (i) Bases B := { B ⊆ E : B is maximal , ∄ C ∈ C , B ⊇ C }B is a family of maximal subsets B of E which contain no circuit C of M asa subset. B is called a family of bases or basic family of M and its sets B arecalled bases of M .(ii) Circuits C := { C ⊆ E : C is minimal , ∄ B ∈ B , C ⊆ B } Conversely to the definition of bases, the circuits C of M can be seen as afamily of minimal subsets C of E which are not contained in a basis B .(iii) Hyperplanes H := { H ⊆ E : H is maximal , ∄ B ∈ B , H ⊇ B } The family H of hyperplanes of M is given by the maximal subsets H of E which contain no basis of M as a subset.(iv) Cocircuits C ∗ = { C ∗ ⊆ E : E \ C ∗ ∈ H} The family C ∗ consists of all subsets C ∗ of E whose complement is a hyper-plane. C ∗ is called the family of cocircuits (also called bonds in the literature)of M . We will limit our presentation here to the finite case. However, there exists work on the infinite case e.g. [44].
6e can equivalently give the definition of a matroid in terms of its bases.
Definition 3.3 (Matroid from Bases [41].)
Let E be a finite set. For a basic family B of subsets of E thefollowing axioms hold:(b1) Nontriviality: B 6 = ∅ .(b2) Incomparability: B is an antichain of subsets in E (b3) Basis-exchange axiom: ∀ B , B ∈ B and for every b ∈ B \ B , ∃ b ∈ B \ B such that (cid:0) B \{ b } (cid:1) ∪ { b } ∈ B . Note that this implies (cid:0) B \{ b } (cid:1) ∪ { b } ∈ B as well.A pair M = ( E, B ) is called a finite matroid on the ground set E . According to [41] we will denote the collection of all basic families of E by B ( E ). In this notion of a basis onecan introduce a map χ : 2 E → { , } indicating whether a given subset B ⊆ E is a basis (a member of the basicfamily B ( M ) ∈ B ( E )) or not: χ B ( B ) = B ∈ B (b3) any two bases B , B ∈ B of a matroid M = ( E, B ) have the same cardinality, | B | = | B | =: r . The cardinality r is often called the rank of the matroid M . Finally we would like to introducethe notion of basic (co-)circuits of a matroid. Definition 3.4 (Basic (fundamental) (co-)circuits of an element with respect to a basis [3].)
Let M =( E, B ) be a matroid on E and B ∈ B be a basis of M . Let e ∈ E \ B . Then there is a unique circuit C ( e, B ) of M contained in { e } ∪ B . The circuit C ( e, B ) is called the basic (or fundamental) circuit of e with respect to B .Dually if e ∈ B then there is a unique cocircuit C ∗ ( e, B ) of M which is disjoint from the hyperplane ( B \ e ) ∈ H ,that is C ∗ ∩ ( B \ e ) = ∅ or equivalently E \ C ∗ = ( B \ e ) ∈ H . The cocircuit C ∗ ( e, B ) is called the basic (orfundamental) cocircuit of e with respect to B . Notice that the opposite does not hold in general: Given a (co-)circuit, there can be several bases from whichthis (co-)circuit can be obtained as a fundamental (co-)circuit by adding an additional element.
Lemma 3.1 (Intersection of fundamental circuits and cocircuits [3].)
Let M be a matroid. Given anycircuit C ∈ C ( M ) and e, f ∈ C , e = f , then there is a cocircuit C ∗ ∈ C ∗ ( M ) , such that C ∩ C ∗ = { e, f } . To see this, let B be a basis of M , containing C \ f . Clearly f / ∈ B , as otherwise C would be contained in B .By construction the basic cocircuit C ∗ ( e, B ) is such that C ∩ C ∗ = { e, f } . Finally we have a natural notion of duality between matroids [41].
Definition 3.5 (Dual Matroid.)
Let M = ( E, B ) be a matroid. A family B ⋆ ∈ B ( E ) with B ⋆ := { B ⋆ ⊆ E : ∃ B ∈ B , B ⋆ = E \ B } is called a dual basic family and gives rise to the matroid M ⋆ = ( E, B ⋆ ) which is called the dual matroid to M = ( E, B ) . It follows that the circuits C of M = ( E, B ) are the cocircuits C ⋆ of M ⋆ = ( E, B ⋆ ) and vice versa. To see this,notice that by definition 3.5 the map “ ⋆ ” is bijective. That is, ∀ B ∈ B there is a unique B ⋆ ∈ B ⋆ and vice versa, ∀ B ⋆ ∈ B ⋆ there is a unique B ∈ B such that ( B ⋆ ) ⋆ = B . Then, recalling the definition of C in terms of B fromdefinition 3.2 (ii) C := { C ⊆ E : C is minimal , ∄ B ∈ B , C ⊆ B }
7e can formally dualize it and write C ⋆ := { C ⋆ ⊆ E : C ⋆ minimal , ∄ B ⋆ ∈ B ⋆ , C ⋆ ⊆ B ⋆ } = { C ⋆ ⊆ E : C ⋆ minimal , ∄ B ∈ B , C ⋆ ⊆ E \ B } = { C ⋆ ⊆ E : E \ C ⋆ maximal , ∄ B ∈ B , E \ C ⋆ ⊇ B } = { C ⋆ ⊆ E : E \ C ⋆ ∈ H} = C ∗ . (3.1)Here, from the first to the second line we have used definition 3.5 and bijectivity of “ ⋆ ” in order to replaceconditions of the form “ ∄ B ⋆ ∈ B ⋆ ” by “ ∄ B ∈ B ”. From the second to the third line we have used the factfrom elementary set theory that for two subsets X, Y ⊆ E if X ⊃ E \ Y then also E \ X ⊂ E \{ E \ Y } which isequivalent to E \ X ⊂ Y . From the third to the fourth line we use definition 3.2 (iii) and finally from the fourthto the fifth line we use definition 3.2 (iv) .Hence it is obvious that the circuits of M are the cocircuits of M ⋆ and the cocircuits of M are the circuitsof M ⋆ . Therefore we will drop the symbol “ ⋆ ” in what follows, and write instead the common symbol “ ∗ ” fordual objects, such as circuits C and cocircuits C ∗ , bases B and dual bases B ∗ , matroids M and dual matroids M ∗ etc. Example.
Consider the set E = { e , e , e , e , e } of 5 vectors in R shown in figure 1. It is given by thecolumn vectors of the matrix A = . For every 3-element subset B ⊂ E we consider the map χ B ( B ) = B = 00 else . We will abbreviate the elements e k by its label k for k = 1 . . . ,
5, such that we can write E = { , , , , } . We e e e e e Figure 1: Vector example for a matroid.have the non-uniform rank 3 vector matroid M = ( E, C ) = ( E, B ) where the sets C of circuits and respectivelythe set B of bases are given by C = (cid:8) { , , , } , { , , } , { , , } (cid:9) B = (cid:8) { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } , { , , } (cid:9) Moreover we have the set of H of hyperplanes and the set C ∗ of cocircuits and dual bases B ∗ H = (cid:8) { , , } , { , , } , { , } , { , } , { , } , { , } (cid:9) C ∗ = (cid:8) { , } , { , } , { , , } , { , , } , { , , } , { , , } (cid:9) B ∗ = (cid:8) { , } , { , } , { , } , { , } , { , } , { , } , { , } , { , } (cid:9) , which can be used in order to define the rank 2 dual matroid M ∗ = ( E, C ∗ ) = ( E, B ∗ ). This can be extended to all other subsets B ′ of cardinality different from 3 by setting χ B ( B ′ ) = 0. For this example this means that we have colinear triples of vectors in E . See section 3.3 for the definition of uniform. .2 Oriented Matroids The concept of a rank r matroid M = ( E, B ) can be extended to the case of an oriented matroid M = ( E, B ).Notice however that not every matroid can be extended to an oriented matroid .For this we need to extend the definitions of the previous section. We label the elements of the ground set E with positive integers 1 , , . . . n = | E | , and will often refer to n -tuples of subsets of E in round brackets ( · ) (forwhich the ordering of the elements is important). The elements of a sorted set (such as a basis) are understoodto be tuples in which the element labels are listed in increasing order.Secondly we would like to extend the concept of circuits to signed circuits. For this we need the definition of signed subsets of E . Definition 3.6 (Signed Subset.)
See [3] p. 101 and following.A signed subset X of E is a set X ⊂ E together with a partition ( X + , X − ) of X into two distinguished subsets. X + is called the set of positive elements of X , X − is called the set of negative elements of X , X + ∩ X − = ∅ .The underlying set X = X + ∪ X − is then called the support of X . Let X, Y be signed subsets, F be an unsignedsubset of E . Then the following properties hold.(i) Equality: X = Y means X + = Y + and X − = Y − .(ii) Restriction: X is called a restriction of Y if X + ⊆ Y + and X − ⊆ Y − Y = X | F is called the restriction of X to F if Y + = X + ∩ F and Y − = X − ∩ F .(iii) Composition: X ◦ Y is the signed set defined by ( X ◦ Y ) ± = X ± ∪ ( Y ± \ X ∓ ) . Notice that thisoperation is associative, but not commutative in general.(iv) Positivity: X is called positive/negative if X − = ∅ / X + = ∅ . X is called empty if X − = X + = ∅ .(v) Opposite: − X with ( − X ) ± = X ∓ is called the opposite of X . The signed set − F X is obtainedfrom X by sign reversal or reorientation on F by ( − F X ) ± = ( X ± \ F ) ∪ ( X ∓ ∩ F )For two signed sets X, Y we write e ∈ X if e ∈ X . We denote the cardinality of X by | X | = | X | . X \ Y denotesthe restriction of X to X \ Y . Definition 3.7 (Signed Subset. Signature.)
A signed set X can also be viewed as a set X ⊂ E togetherwith a mapping sgn X : X → {− , } such that X ± = (cid:8) e ∈ X : sgn X ( e ) = ± (cid:9) . The mapping sgn X is calledthe signature of X . Definition 3.8 (Signed Subset. Extended Signature.)
The signature of X can be extended to all of E bydefining the extended signature map sgn X : E → {− , , } | E | . Here sgn X ( e ) = ± if e ∈ X ± ⊂ E , sgn X ( e ) = 0 if e ∈ E \ X . X can then be identified with an element in {− , , +1 } | E | ≡ { + , , −} | E | and we will shortly write X ∈{− , , +1 } | E | . Moreover by S ⊆ {− , , +1 } | E | we will denote a family S of signed subsets of E . For the n -element ‘ground tuple’ E = E n = (1 , , . . . , n ) we may denote X as a sign vector X ∈ {− , , +1 } n . Finallywe give a notion of orthogonality of signed subsets. Definition 3.9 (Orthogonality of Signed Subsets.)
Two signed subsets X = ( X + , X − ) , Y = ( Y + , Y − ) , X, Y ⊂ E are said to be orthogonal X ⊥ Y if X ∩ Y = ∅ or the restrictions of X and Y to their intersectionare neither equal nor opposite.That is, there are elements e, f ∈ X ∩ Y such that sgn X ( e ) sgn Y ( e ) = − sgn X ( f ) sgn Y ( f ) . For a discussion of criteria for extensibility of a matroid to an oriented matroid we refer to [3]. This is motivated from the sign properties of the scalar product of two real vectors. Consider a finite set E = { , . . . , n } withcardinality | E | = n . Let u = (cid:0) u , . . . , u n (cid:1) , v = (cid:0) v , . . . , v n (cid:1) ∈ R n be orthogonal with respect to the Euclidean inner product (cid:10) · , · (cid:11) of R n . Define X = ( X + , X − ), where X ± := { i : u i ≷ } , and Y = ( Y + , Y − ), where Y ± := { i : v i ≷ } . From (cid:10) u, v (cid:11) = n X k =1 u k · v k = 0it follows that the non-zero terms in the sum cannot all have the same sign, that is if there are non-zero components u k , v k , thentheir signs have to be different for at least one k . Hence there exists at least one k , contained in X + and Y − or X − and Y + respectively. According to definition 3.9 we then say that the signed sets X and Y are orthogonal, X ⊥ Y . Definition 3.10 (Oriented Matroid from Signed Circuits.)
A family C of signed subsets of a set E iscalled the set of signed circuits of an oriented matroid M = ( E, C ) on E if(C0) Non-emptiness: ∅ / ∈ C (C1) Symmetry: C = −C , that is for every C ∈ C also its opposite − C ∈ C .(C2) Incomparability: if C ⊆ C then either C = C or C = − C ∀ C , C ∈ C (C3) Elimination: For all C , C ∈ C with C = − C , if e ∈ C +1 ∩ C − ∃ C ∈ C such that C ± ⊆ ( C ± ∪ C ± ) \{ e } . Now we would like to define the oriented matroid M in terms of a family B of bases of the underlying rank r matroid M . An element B ∈ B can be written as a (sorted) r -tuple B = ( b , b , . . . , b r ), where each element b k stands for an element of E . In particular there exists a permutation π , which brings all elements intolexicographic order according to E , such that b < b < . . . < b r . Then B is called an ordered basis. Definition 3.11 (Oriented Matroid from Oriented Bases of a Set.)
Let M = ( E, B ) be a finite rank r matroid over the ground set E . An oriented matroid M = ( E, B ) on E is given by the bases B of the underlyingmatroid M together with a mapping χ B : E r ∋ B → {− , , +1 } with χ B ( B ) = ± iff B ∈ B elsecalled the basis orientation or chirotope. The rank r of M equals the rank of M and for χ B it holds that(B1) χ B is alternating: for B = ( b , . . . , b r ) and B ′ = ( b π (1) , . . . , b π ( r ) ) consisting of the same ele-ments as B but in different order, related to the order in B by a permutation π we have χ B ( B ) = sgn( π ) · χ B ( B ′ ) .(PV) Pivoting property: for any two sorted bases B, B ′ ∈ B ( M ) with B = ( y, b , . . . , b r ) and B ′ = ( z, b , . . . , b r ) , y = z we have χ B ( B ) = − sgn C ( y ) · sgn C ( z ) · χ B ( B ′ ) where C is one of either of the two (opposite) signed circuits of M whosesupport is contained in the set { y, z, b , . . . , b r } . We would like to remark that in general χ B is only fixed up to an overall sign: if χ B is a basis orientation for M ( E, B ), then also − χ B is a basis orientation. This must be kept in mind in subsequent computations. Seealso section 3.2.2. Signed Cocircuits.
With definition 3.9 we can define the set C ∗ of signed cocircuits of the rank r orientedmatroid M as follows: C ∗ := { C ∗ = ( C ∗ + , C ∗− ) : C ∗ ⊆ E , C ∗ ⊥ C ∀ C ∈ C (in the sense of definition 3.9) } . (3.2)Alternatively to this definition we can equip the set C ∗ of cocircuits of the underlying matroid M with a sig-nature as follows [3]. Each support C ∗ is the complement of a hyperplane H ∈ H as in definition 3.2, that is H = E \ C ∗ . By construction H = { h , . . . , h r − } is a maximal subset of E not containing a basis B ∈ B ( M ).Then the signature of C ∗ can be constructed as ( C ∗ ) ± := { e ∈ E \ H : χ B ( h , . . . , h r − , e ) = ± } . In section3.5 a geometric interpretation for C ∗ is given.Additionally we have the following definition. Here it makes no difference if e.g. e < h r − , since this would only affect the sign of ± M is non-uniform, it might happen that H contains more than r − H which contains r − H . efinition 3.12 (Basic (fundamental) signed (co-)circuits of an element wrt. an oriented basis.) Let M = ( E, B ) be a given oriented matroid of rank r with underlying matroid M . Let C = ( e, B ) be afundamental circuit of M as given in definition 3.4. Then the signed circuit C ∈ C supported on C and having sgn C ( e ) = +1 is called the basic or fundamental circuit of e with repsect to B ∈ B . Dually, if e ∈ B (as indefinition 3.4) let C ∗ = C ∗ ( e, B ) be the unique cocircuit of M disjoint from the hyperplane B \ e . Then thecocircuit C ∗ ∈ C ∗ supported on C ∗ and positive on e is called the basic or fundamental cocircuit of e with respectto B ∈ B . Given any
B, B ′ ∈ B with B = ( y, b , . . . , b r ), B ′ = ( z, b , . . . , b r ) and y = z we may construct the basic circuit C := C ( z, B ) = C ( y, B ′ ) = { y, z, b , . . . , b r } . Moreover using B ∗ , B ′∗ ∈ B ∗ with B ∗ = E \ B and B ′∗ = E \ B ′ we may construct the basic cocircuit C ∗ := C ∗ ( z, B ) = E \{ B \ z } = { E \ B } ∪ { z } = B ∗ ∪ { z } = B ′∗ ∪ { y } = { E \ B ′ } ∪ { y } = E \{ B ′ \ y } = C ∗ ( y, B ′ ) . Here once again we see very nicely that the basic cocircuits with respect to the basic family B can be regardedas basic circuits with respect to the basic family B ∗ . Clearly we have C ∩ C ∗ = { y, z } . Now choose one of thetwo oppositely signed circuits ± C ∈ C and ± C ∗ ∈ C ∗ such that for example sgn C ( y ) = sgn C ∗ ( y ) = +1. As C ⊥ C ∗ by construction we get from definition 3.9 a chirotope χ B ∗ ( B ∗ ) for any B ∗ ∈ B ∗ from χ B ( B ) · χ B ( B ′ ) = − sgn C ( y ) · sgn C ( z ) = sgn C ∗ ( y ) · sgn C ∗ ( z ) = − χ B ∗ ( B ∗ ) · χ B ∗ ( B ′∗ ) . (3.3)In the last step we have used the pivoting property of definition 3.11 in its dualized form . Note that thisidentity is independent of the choice of C, C ∗ above, as it only uses the relative signs of y, z with respect to C, C ∗ . Using (3.3) we may define the dual matroid chirotope χ B ∗ : E n − r → {− , , +1 } as follows. Considerthe lexicographically sorted E = (1 , , . . . , n ) and an arbitrary sorted ( n − r )-tuple T ∗ = ( t ∗ , . . . , t ∗ n − r ) ofelements in E . Write T = ( t , . . . , t r ) for some permutation of E \ T ∗ . Clearly, there is a permutation π mapping( T ∗ , T ) := ( t ∗ , . . . , t ∗ n − r , t , . . . , t r ) to (1 , , . . . , n ) with parity sgn π =: sgn (cid:0) ( T ∗ , T ) (cid:1) . Now we can define χ B ∗ : E n − r → {− , , +1 } T ∗ = ( t ∗ , . . . , t ∗ n − r ) χ B ∗ ( T ∗ ) := χ B ( T ) · sgn (cid:0) ( T ∗ , T ) (cid:1) . (3.4)Note that this definition is compatible with (3.3), and directly relates the values of chirotope and dual chi-rotope, as follows. Choose B = ( y, b , . . . , b r ), B ′ = ( z, b , . . . , b r ) and B ∗ , B ′∗ ∈ B ∗ with B ∗ = E \ B =( z, b ∗ , . . . , b ∗ ( n − r ) ), B ′∗ = E \ B ′ = ( y, b ∗ , . . . , b ∗ ( n − r ) ) and C, C ∗ as above. Clearly χ B ∗ ( B ∗ ) ( )= χ B ( B ) · sgn( z, b ∗ , . . . , b ∗ ( n − r ) , y, b , . . . , b r ) ( PV )= − sgn C ( y ) · sgn C ( z ) · χ B ( B ′ ) · sgn( z, b ∗ , . . . , b ∗ ( n − r ) , y, b , . . . , b r ) ( )= sgn C ∗ ( y ) · sgn C ∗ ( z ) · χ B ( B ′ ) · sgn( z, b ∗ , . . . , b ∗ ( n − r ) , y, b , . . . , b r ) z ↔ y = − sgn C ∗ ( y ) · sgn C ∗ ( z ) · χ B ( B ′ ) · sgn( y, b ∗ , . . . , b ∗ ( n − r ) , z, b , . . . , b r ) ( )= − sgn C ∗ ( y ) · sgn C ∗ ( z ) · χ B ∗ ( B ′∗ ) . (3.5) Definition 3.13 (Dual Oriented Matroid.)
Let M = ( E, B ) be an oriented rank r matroid with chirotope χ B over the ordered ground set E = (1 , . . . , n ) . A family B ∗ ∈ B ( E ) with B ∗ := { E \ B : B ∈ B} together with the map χ B ∗ : E n − r ∋ B ∗ → {− , , +1 } for which B ∗ = E \ B , B ∈ E r χ B ∗ ( B ∗ ) = χ B ( B ) · sgn(( B ∗ , B )) = ± iff B ∗ ∈ B ∗ which is equivalent to B ∈ B iff B ∗ / ∈ B ∗ which is equivalent to B / ∈ B This can always be done without loss of generality, because ± C ∈ C and ± C ∗ ∈ C ∗ by construction. Explicitly given in definition 3.13. ith sgn(( B ∗ , B )) defined as above gives rise to an oriented matroid M ∗ = ( E, B ∗ ) , called the dual orientedmatroid to M = ( E, B ) , of rank ( n − r ) . In particular it holds for χ B ∗ that(B1) χ B ∗ is alternating: for all B ∗ = ( b ∗ , . . . , b ∗ n − r ) and B ′∗ = ( b ∗ π (1) , . . . , b ∗ π ( n − r ) ) consisting of thesame elements as B ∗ but in different order, related to the order in B ∗ bya permutation π , we have χ B ∗ ( B ∗ ) = sgn( π ) · χ B ∗ ( B ′∗ ) .(PV*) Dual Pivotingproperty: for any B ∗ , B ′∗ ∈ B ∗ with B ∗ = ( y, b ∗ , . . . , b ∗ ( n − r ) ) and B ′ = ( z, b ∗ , . . . , b ∗ ( n − r ) ) where y = z we have χ B ∗ ( B ∗ ) = − sgn C ∗ ( y ) · sgn C ∗ ( z ) · χ B ∗ ( B ′∗ ) , where C ∗ is one of the twooppositely signed cocircuits of M whose support is contained in the set { y, z, b ∗ , . . . , b ∗ ( n − r ) } . Note that again it holds that ( M ∗ ) ∗ = M . The signed cocircuits of M are the signed circuits of M ∗ and thesigned circuits of M are the signed cocircuits of M ∗ .Notice that ( P V ) and (
P V ∗ ) are equivalent for a map χ B if M is an oriented matroid due to (3.3) [3].In practice the identity in equation (3.3) might often give an efficient way to compute the dual chirotope χ B ∗ from the chirotope χ B . Example continued.
For the vector example at the end of section 3.1 we have E = (cid:0) , , , , (cid:1) and theaccording set of sorted bases B . The set C of signed circuits is given by C = {± C , ± C , ± C } with C C C C + k { , , } { , } { , } C − k { } { } { } . For the chirotope we consider for every 3 element subset B of E the map χ B ( B ) = sgn(det B ) iff det B = 00 otherwise . which leads to the following basis orientation : B (1 , ,
3) (1 , ,
4) (1 , ,
5) (1 , ,
4) (1 , ,
5) (1 , ,
5) (2 , ,
4) (2 , ,
5) (2 , ,
5) (3 , , χ B ( B ) + + 0 − − − + + + 0 Here we follow [3] p. 132 and following.The equivalence of defining an oriented matroid M of rank r over a ground set E in terms of its signed circuits C , as in definition 3.10, or in terms of its oriented basis B , as in theorem 3.11, can be made explicit by using theso-called basis graph BG M of the underlying matroid M , which will be constructed in the sequel. Choose afamily B of bases of M . Associate to each basis B ∈ B a vertex of BG M . Now link any two bases B, B ′ ∈ B byan edge if they differ by exactly one element, for example B = ( b , b , . . . , b r ), B ′ = ( b ′ , b , . . . , b r ) and b = b ′ .By construction BG M is connected.Now let M = ( E, C ) be given. Fix an ordering of the ground set E such that E = (1 , , . . . , n ). Sort theelements of each basis B ∈ B according to that ordering. Now assume we have two sorted bases B, B ′ ∈ B with B = ( b , . . . , b r ) and B ′ = ( b ′ , . . . , b ′ r ) such that B ∆ B ′ := ( B ∪ B ′ ) \ ( B ∩ B ′ ) = { b i , b ′ j } . Let C ∈ C be one of If χ B is a chirotope, then − χ B is also, depending of our notion of ‘positive’ orientation. C ( b ′ j , B ) whose support is contained in the set { b , . . . , b r , b ′ j } ⊇ C ( b ′ j , B ).Then by (B1) of definition 3.11 we have: χ B ( b i , b , . . . , b i − , b i +1 , . . . , b r ) = ( − i − χ B ( B ) χ B ( b ′ j , b ′ , . . . , b ′ j − , b ′ j +1 , . . . , b ′ r ) = ( − j − χ B ( B ′ ) . (3.6)Moreover by construction ( b , . . . , b i − , b i +1 , . . . , b r ) ≡ ( b ′ , . . . , b ′ j − , b ′ j +1 , . . . , b ′ r ) and we can define η ( B, B ′ ) := χ B ( B ) · χ B ( B ′ ) = ( − i + j · χ B ( b i , b , . . . , b i − , b i +1 , . . . , b r ) · χ B ( b ′ j , b ′ , . . . , b ′ j − , b ′ j +1 , . . . , b ′ r )= ( − i + j · sgn C ( b i ) · sgn C ( b ′ j ) (3.7)where we have used (PV) of definition 3.11 in the last line. Hence, starting from an arbitrarily chosen B ∈ B ,we can construct the so-called signed basis graph SBG M by attaching η ( B, B ′ ) to every edge of BG M . Clearly SBG M depends on the ordering chosen on the ground set E .A chirotope χ : E r → {− , , +1 } can be constructed from SBG M as follows. Choose an arbitrary B ∈ B . Set χ ( B ) = 1. Then for any B ∈ B we have χ ( B ) = k Y i =1 η ( B i − , B i )where B , B , B , . . . , B k = B is an arbitrary path from B to B contained in SBG M . It can be shown thatthis definition is consistent, that is χ ( B ) is independent of the choice of a path, see [3] p. 132 and following.Notice that this construction can easily be applied to cocircuits and the dual oriented matroid M ∗ . Moreoverhaving computed the chirotope χ B of M = ( E, C ), one can read off the chirotope for the dual matroid by usingidentity (3.3). Let us complete our introduction to oriented matroids by giving definitions which frequently occur in theliterature [3]. Let the oriented matroid M = ( E, C ) be given with its set C of signed circuits. Vectors and covectors.
A composition C ◦ . . . ◦ C k of signed circuits in the sense of (iii) of definition 3.6 iscalled a vector of M . The set of all vectors of M is denoted by V . Accordingly a composition of signed cocircuitsof M is called a covector of M . The set of all covectors of M is denoted by L . Note that by construction vectorsand covectors of M are signed subsets of E . In particular any v ∈ V and any l ∈ L determines an extendedsignature in the sense of definition 3.8. Instead of giving M in terms of its signed circuits C or cocircuits C ∗ , M can also be determined in terms of its vectors V or covectors L [3]. Loops and coloops.
Moreover an element e ∈ E is called a loop of M if the signed set ( { e } , ∅ ) ∈ C . Thesubset of loops contained in E is denoted by E o . If e / ∈ C ∀ C ∈ C then e is called a coloop of M . Parallel elements.
Two elements e, f ∈ E \ E o are called parallel e k f if sgn l ( e ) = sgn l ( f ) or sgn l ( e ) = − sgn l ( f ) for all covectors l ∈ L . Simple oriented matroid.
An oriented matroid M is called simple if it does not contain loops or parallelelements. Uniform oriented matroid.
An oriented matroid M = ( E, B ) of rank r is called uniform if every r -elementsubset X ⊆ E is contained in the family of bases B , that is χ B ( X ) = 0 ∀ X ⊂ E such that | X | = r . That is, the fundamental (basic) signed circuit or its opposite, as given by definition 3.12. See section 3.4 for an example in which it is a proper subset. In the picture of a vector configuration a loop corresponds to a zero vector. .4 Oriented Matroids from Directed Graphs In this section we will show how the abstract framework of oriented matroids naturally arises from directedgraphs. This will be done by reviewing an example from [3] in great detail.PSfrag replacements e e e e e e v v v v Figure 2: Undirected graph γ . PSfrag replacements e e e e e e v v v v (cid:9) { , , , } Figure 3: Directed graph γ .Consider the graph γ of figure 2 with edge set E = E ( γ ) = { e , . . . , e } and vertex set V ( γ ) = { v , . . . , v } .We notice that γ contains simple (undirected) cycles X , for example { , , , } . The set of all such cycles isdenoted by C . Then C defines the circuits of a matroid M = ( E, C ).Now consider the directed graph γ of figure 3 now with a set E ( γ ) = { e , . . . , e } of oriented edges and vertexset V ( γ ) = { v , . . . , v } . Consider the cycle X = { , , , } . If we introduce an anti-clockwise orientation forthis cycle as indicated by (cid:9) in figure 3, then it contains positive elements X + = { , } and negative elements X − = { , } . Hence X = ( X + , X − ) is a signed circuit or in the wording of definition 3.6. X = ( X + , X − ) is asigned set with support X = X + ∪ X − . The set of all signed circuits obtained in this way from the orientedgraph γ is denoted by C . This defines the oriented matroid M = ( E, C ). We sometimes refer to an orientedmatroid which arises from a directed graph in this way as a graphic oriented matroid .We may define the extended signature sgn X introduced in definition 3.8 with respect to the chosen orientation (cid:9) ,to write the signed circuit X as an element of { + , , −} | E ( γ ) | : X = { sgn X ( e ) , . . . , sgn X ( e ) } = {− , + , , , − , + } or shorter (denoting e k as simply k ) X = 1256. In this notation the set C can be written as a matrix, whereeach circuit is a row or the negative of a row [3]: C = + − − − − + − + − − + 0 − + 0 +0 + − − + 0 ← C = 124 ← C = 135 ← C = 236 ← C = 456 ← C = 1256 ← C = 1346 ← C = 2345It is instructive to check that C = {± C , . . . , ± C } indeed fulfills the oriented matroid circuit axioms of defini-tion 3.10. In particular ( C
2) holds.The bases B for M , M are given by the set of spanning trees T ( γ ) of γ [41]. For a connected graph (orcomponent) γ the spanning tree T ( γ ) is given by a collection of edges which uniquely connects any two vertices v , v of γ by a path. If γ contains | V ( γ ) | vertices, then T ( γ ) consists of | V ( γ ) | − M γ is | V ( γ ) | − γ is not connected but consists of N connected components γ = { γ , γ , . . . , γ N } , then B is given by thecollection of trees T ( γ ) := { T ( γ ) , T ( γ ) , . . . , T ( γ N ) } , which is also called the forest [45] of γ .Hence in our example for γ in figure 3 we find that M γ has rank 3. By (i) in definition 3.2 of bases as maximalsubsets of E containing no circuit we find that all triples except C , C , C , C of E ( γ ) = { , , , , , } arebases. The fact that we have 3-element circuits indicates that M γ is non-uniform.
14n order to compute the according chirotope, we use the signed basis graph construction (3.7) of section 3.2.2: Asthe chirotope is only given up to an overall sign, we choose χ B (1 , ,
3) = − χ B ( B ′ ) = χ B ( B ) · ( − i + j · sgn C ( b i ) · sgn C ( b ′ j ) : B = ( b , b , b ) B ′ = ( b ′ , b ′ , b ′ ) B ∆ B ′ = { b i , b ′ j } i j B ∪ B ′ ⊇ C sgn C ( b i ) sgn C ( b ′ j ) χ B ( B ) χ B ( B ′ )(1 , ,
3) (1 , , { , } { , , , } C − − − , ,
3) (1 , , { , } { , , , } C − − − , ,
3) (1 , , { , } { , , , } C − − , ,
3) (1 , , { , } { , , , } C +1 +1 − − , ,
3) (2 , , { , } { , , , } C +1 +1 − , ,
3) (2 , , { , } { , , , } C +1 +1 − , ,
4) (1 , , { , } { , , , } C − − , ,
6) (1 , , { , } { , , , } C − − − , ,
6) (1 , , { , } { , , , } C − − − , ,
5) (2 , , { , } { , , , } C − − − , ,
5) (2 , , { , } { , , , } C +1 − − − , ,
5) (2 , , { , } { , , , } C − − − − , ,
4) (3 , , { , } { , , , } C +1 +1 +1 − , ,
4) (3 , , { , } { , , , } C +1 +1 +1 − , ,
5) (3 , , { , } { , , , } C − − − − χ B ( B ) one can choose any B ′ ∈ B which differs from B in one element andthe according signed circuit contained in B ∪ B ′ . For instance for B = (123) one could take B ′ = (125)instead with B ∆ B ′ = { , } , ❀ i = 3, j = 3. Then one simply uses the according circuit C ∈ C withsupport C ⊆ B ∪ B ′ = { } and computes equivalently χ B (123) = ( − · sgn C (3) · sgn C (5) · χ B (125) =( − · ( − · (+1) · ( −
1) = − E and decide whetherthey are a basis element ( • ) or not ( • ). The initial choice for χ B (1 , ,
3) = − χ B Basis Triple χ B Basis Triple χ B Basis Triple χ B • − •
124 0 • − • − •
134 +1 •
234 +1 •
135 0 •
235 +1 • − •
236 0 • − • − • − • − • − • − • − • − • − •
456 0 (3.8)Table 1: Chirotope data of the graphic oriented matroid encoded in figure 3, computed by the signed basisgraph method.We will now compute the basis chirotope χ B ∗ of the dual basis as defined in definition 3.13. B ∗ := { E \ B : B ∈ B} χ B ∗ ( B ∗ ) = χ B ( B ) · sgn (cid:0) ( B ∗ , B ) (cid:1) =: χ B ( B ) · sgn (cid:0) π ( b ∗ , b ∗ , b ∗ , b , b , b ) (cid:1) where we have written B = ( b , b , b ) and B ∗ = ( b ∗ , b ∗ , b ∗ ) and the defined lexicographically sorted ground setis E = (1 , , , , , (cid:0) π ( · ) (cid:1) denotes the parity of the permutation π in order to bring the set inthe argument into lexicographic order. The result is given in table 2. B χ B ( B ) B ∗ sgn(( B ∗ , B )) χ B ∗ ( B ∗ )123 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − B χ B ( B ) B ∗ χ B ∗ ( B ∗ )124 0 356 0135 0 246 0236 0 145 0456 0 123 0Table 2: Dual chirotope data for the oriented matroid given by figure 3.Using χ B ∗ from table 2 we can now compute the signed cocircuits C ∗ as signed circuits of the dual matroid M ∗ = ( E, B ∗ ). As E = (1 , , , , ,
6) consists of 6 elements we know that M ∗ has rank r ∗ = 6 − r = 3. That is,all possible cocircuits should be supported within subsets having at most 4 elements. Now consider all (cid:0) (cid:1) = 15four-element subsets of E . 1234 1235 1236 1245 1246 1256 1345 1346 1356 14562345 2346 2356 24563456We simply compute the circuit signature induced by χ ∗B for every tuple, using again the signed basis graphconstruction (3.7) in its dualized version (dual bases and cocircuits as circuits of the dual matroid).16rom the dual non-bases given in table 2 we already see that C ∗ := { , , } ⊂ { , , , } , { , , , } , { , , , } C ∗ := { , , } ⊂ { , , , } , { , , , } , { , , , } C ∗ := { , , } ⊂ { , , , } , { , , , } , { , , , } C ∗ := { , , } ⊂ { , , , } , { , , , } , { , , , } Hence we have to additionally analyze the cocircuits C ∗ , C ∗ , C ∗ whose support is equal to { , , , } , { , , , } , { , , , } respectively. As an example, let us compute the signed sets for ± C ∗ . First we again choosesgn C ∗ (1) = −
1. In order to compute the signs for the elements 2 , ∈ C ∗ we have to choose pairs B ∗ , ( B ∗ ) ′ of dual bases, such that C ∗ ⊆ (cid:0) B ∗ ∪ ( B ∗ ) ′ (cid:1) and B ∗ ∆( B ∗ ) ′ = { , } , { , } , for example we may choose B ∗ = (1 , , , ( B ∗ ) ′ = (2 , ,
6) and B ∗ = (1 , , , ( B ∗ ) ′ = (2 , , C ∗ (2) = sgn C ∗ (1) · ( − · χ B ∗ (1 , , · χ B ∗ (2 , ,
6) = ( − · ( − · ( − · (+1) = − C ∗ (3) = sgn C ∗ (1) · ( − · χ B ∗ (1 , , · χ B ∗ (2 , ,
6) = ( − · (+1) · (+1) · (+1) = − C ∗ = 123. One can check that C ∗ ∈ C ∗ is orthogonal toall C k ∈ C as by definition 3.2. We get: C ∗ ∩ C k = { a, b } sgn C k ( a ) sgn C k ( b ) sgn C ∗ ( a ) sgn C ∗ ( b ) C = 124 { , } + − − − C = 135 { , } + − − − C = 236 { , } + − − − C = 456 ∅ C = 1256 { , } + − − − C = 1346 { , } + − − − C = 2345 { , } + − − − (3.9)and see that the orthogonality condition is fulfilled. Similarly one can proceed with the remaining cocircuits,where we use as before B ∗ ∆( B ∗ ) ′ := ( B ∗ ∪ ( B ∗ ) ′ ) \ ( B ∗ ∩ ( B ∗ ) ′ ) = { b i , b ′ j } , B ∗ = ( b , b , b ) , ( B ∗ ) ′ = ( b ′ , b ′ , b ′ ) As these 4-element subsets are extensions of dual bases by one element, and we already have computed all cocircuits with fewerthan 4 elements, the remaining cocircuits must be supported on 4 elements. C ∗ ⊆ (cid:8) B ∗ ∪ ( B ∗ ) ′ (cid:9) : C ∗ choose { b i , b ′ j } ( b , b , b ) ( b ′ , b ′ , b ′ ) i j χ B ∗ ( B ∗ ) χ B ∗ (( B ∗ ) ′ ) sgn C ∗ ( b j ) { , , } sgn C ∗ (1) = − { , } (1 , ,
5) (2 , ,
5) 1 2 − { , } (1 , ,
4) (2 , ,
5) 1 2 +1 +1 +1 { , , } sgn C ∗ (2) = − { , } (2 , ,
6) (3 , ,
6) 1 2 +1 +1 − { , } (2 , ,
4) (3 , ,
6) 1 3 +1 +1 +1 { , , } sgn C ∗ (3) = − { , } (2 , ,
6) (2 , ,
6) 2 2 +1 − − { , } (3 , ,
5) (3 , ,
6) 1 3 − − { , , , } sgn C ∗ (1) = − { , } (1 , ,
5) (2 , ,
6) 1 1 +1 − − { , } (1 , ,
6) (2 , ,
6) 1 2 +1 − { , } (1 , ,
5) (2 , ,
6) 1 3 − − { , , , } sgn C ∗ (1) = − { , } (1 , ,
6) (3 , ,
6) 1 1 − − { , } (1 , ,
6) (3 , ,
6) 1 2 − { , } (1 , ,
4) (3 , ,
6) 1 3 − − { , , , } sgn C ∗ (2) = − { , } (2 , ,
4) (3 , ,
5) 1 1 +1 − − { , } (2 , ,
5) (3 , ,
5) 1 2 − − − { , } (2 , ,
4) (3 , ,
5) 1 3 +1 − − C ∗ = 123 C ∗ = 145 C ∗ = 246 C ∗ = 356 C ∗ = 1256 C ∗ = 1346 C ∗ = 2345 (3.10)Certainly it is also possible to directly use the fact that cocircuits are orthogonal to all circuits. Writingsgn C k ( a ) , sgn C k ( b ) as in (3.9) for every pair { a, b } contained in the set C of all circuits, one can compare thesedata for all elements of the cocircuits and obtain their partition into positive and negative parts. Consider a finite set E = { v , . . . , v n } of n ≥ r non-zero vectors, spanning the r -dimensional vector space R r .We can express the minimal linear dependencies among the elements of E by the solutions to n X i =1 λ i v i = 0 with λ i ∈ R . A solution is given by elements of the form ~λ = ( λ , . . . , λ n ) ∈ R n having at least two non-zero components.For each such solution ~λ we define the set C = { i : λ i = 0 } , called a circuit. The set of all C then builds upthe set of all circuits C of the (unoriented) matroid M = ( E, C ). By construction M has rank r .In order to obtain an oriented matroid, consider for each solution ~λ the signed set C = ( C + , C − ) with C + = { i : λ i > } and C − = { i : λ i < } , giving a signed circuit. We denote the set of all C by C . This defines anoriented matroid M = ( E, C ), also called a vector oriented matroid .18e may also introduce the extended signature sgn C for all C ∈ C . Then the signed circuit C can be writtenas an element in { + , , −} n as C = (sgn C ( λ ) , . . . , sgn C ( λ n )). Notice that the linear dependencies encoded ineach circuit C are only given up to an overall scalar. Hence each element C ∈ C gives rise to two elements C = ( C + , C − ) and − C = ( C − , C + ) in C .It is also straightforward to see that the set B of bases of M = ( E, C ) is given by all subsets B of E whichcontain r linearly independent vectors. The natural notion of a basis orientation χ B ( B ), B = ( b , . . . , b r ), canbe written in terms of an ( r × r )-matrix with column vectors ( v b , . . . , v b r ) by χ B ( B ) = χ B ( b , . . . , b r ) = sgn (cid:0) det( v b , . . . , v b r ) (cid:1) =: sgn (cid:0) [ b , . . . , b r ] (cid:1) . (3.11)Here and in what follows we write [ b , . . . , b r ] to abbreviate determinant expressions like det( v b , . . . , v b r ). Itmay be checked that this notion of χ B is consistent with definition 3.11 by the determinant properties.Moreover the set C ∗ of signed cocircuits of M = ( E, C ) gets the following geometric interpretation: Consideran ( r − { h , . . . , h r − } of E such that H = { v h , . . . , v h r − } spans an ( r − R n into two half spaces H + , H − . Equip H with an orientation and choose the labels H + , H − suchthat H + is the positive and H − the negative half space with respect to the chosen orientation. Then a signedcocircuit C ∗ of M with E \ C ∗ = H is given by C ∗ = ( C ∗ + , C ∗− ) where C ∗± is the set of vectors in E containedin the according half space H ± . By construction C ∗ contains ± C ∗ , since the orientation of H can be chosenarbitrarily without loss of generality. Grassmann Pl¨ucker identities and determinants [3].
Given any two sorted sets B, B ′ ⊂ E where B = ( b , . . . , b r ), B ′ = ( b ′ , . . . , b ′ r ), the product [ b , . . . , b r ] · [ b ′ , . . . , b ′ r ] can be expressed as[ b , . . . , b r ] · [ b ′ , . . . , b ′ r ] = r X k =1 [ b ′ k , b , . . . , b r ] · [ b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r ] , (3.12)one special example being the Laplace expansion for determinants . In general expressions like (3.12) are called Grassmann Pl¨ucker relations for determinants . Notice that the difference between the left hand side and righthand side of (3.12) gives an antisymmetric linear expression in the ( r + 1) arguments b , b ′ , b ′ , . . . , b ′ r , that isan ( r + 1)-form on R r . This vanishes by construction [3].Obviously (3.12) implies [ b , . . . , b r ] · [ b ′ , . . . , b ′ r ] ≥ b ′ k , b , . . . , b r ] · [ b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r ] ≥ k = 1 . . . r . With the identification of signs of determinants and chirotopes given in (3.11), this gives rise to Definition 3.14 (Grassmann Pl¨ucker relations for chirotopes [3].)
Let the oriented matroid M = ( E, B ) be given with the chirotope χ B giving a basis orientation of B . Let B, B ′ ⊂ E be given with B = ( b , . . . , b r ) , B ′ = ( b ′ , . . . , b ′ r ) . Then(B2) For all B, B ′ such that χ B ( b ′ k , b , . . . , b r ) · χ B ( b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r ) ≥ ∀ k = 1 . . . r it holds that χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) ≥ or equivalently(B2’) For all B, B ′ such that χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) = 0 ∃ k ∈ { , . . . , r } such that χ B ( b ′ k , b , . . . , b r ) · χ B ( b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r ) = χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) . The equivalence of ( B
2) and ( B ′ ) can be seen as follows [3]: For r = 1 it is trivially fulfilled. For r ≥ ǫ k := χ B ( b ′ k , b , . . . , b r ) · χ B ( b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r ). Observe that under the permutation b ′ ↔ b ′ the sign tuple ( ǫ , ǫ , . . . , ǫ r ) changes into ( − ǫ , − ǫ , . . . , − ǫ r ) as well as χ B ( b , . . . , b r ) · χ B ( b ′ , b ′ , . . . , b ′ r ) = − χ B ( b , . . . , b r ) · χ B ( b ′ , b ′ , . . . , b ′ r ). Not necessarily
B, B ′ ∈ B . Let
B, B ′ ∈ B and ( b , . . . , b r ) be the coordinate vectors, that is b = (1 , , . . . , t , b = (0 , , . . . , t , . . . , b r = (0 , , . . . , t ( t stands for transposed). Let ( b ′ , . . . , b ′ r ) be another basis. Certainly, [ b , . . . , b r ] corresponds to the determinant of the unit( r × r )-matrix. Inserting this, (3.12) corresponds to the Laplace expansion of the determinant of the matrix whose column vectorsare given by ( b ′ , . . . , b ′ r ). B
2) holds and χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) >
0. Then there must be a k such that ǫ k >
0, asotherwise (3.12) would give a contradiction. By permuting b ′ ↔ b ′ χ B ( b , . . . , b r ) · χ B ( b ′ , b ′ , . . . , b ′ r ) < ǫ k change their sign also ǫ k <
0. Hence ( B ′ ) holds.Conversely assume ( B ′ ) holds. Then we know that whenever χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) = 0 there is a k ∈ { , . . . , r } such that ǫ k = χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) = 0. Now if ǫ k ≥ ∀ k then it must hold that χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) ≥
0. By applying the permutation argument again it also holds for ǫ k ≤ ∀ k that χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) <
0. The case χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) = 0 is contained in both argu-mentations.The question arising here is whether ( B B ′ ) arise only in case of a vector realization of the orientedmatroid M = ( E, B ). The answer is given in the following Lemma 3.2 (Grassmann-Pl¨ucker Relations and Basis Orientation [3].)
Let the oriented matroid M be given in terms of the ground set E and its set of cocircuits C ∗ , M = ( E, C ∗ ) with basis orientation χ B .Then its chirotope χ B satisfies ( B ′ ) . To see this [3], consider
B, B ′ ⊂ E with B = ( b , . . . , b r ), B ′ = ( b ′ , . . . , b ′ r ), such that ǫ := χ B ( b , . . . , b r ) · χ B ( b ′ , . . . , b ′ r ) ∈ {− , +1 } . If b ∈ B ′ then ( B ′ ) is trivially satisfied.If b / ∈ B ′ consider the basic circuit C = ( b , B ′ ) and the basic cocircuit C ∗ = ( b , B ) defined as in def-inition 3.4. By construction b ∈ C ∩ C ∗ . By orthogonality of C, C ∗ there is a k ∈ { , . . . , r } such thatsgn C ( b ) sgn C ( b ′ k ) = − sgn C ∗ ( b ) sgn C ∗ ( b ′ k ). Now M is an oriented matroid and hence ( P V ∗ ) and ( P V ) aresatisfied by χ B . Therefore it holds that χ B ( b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r ) = − sgn C ( b ) · sgn C ( b ′ k ) · χ B ( b ′ , . . . , b ′ k − , b ′ k , b ′ k +1 , . . . , b ′ r ) χ B ( b ′ k , b , . . . , b r ) = sgn C ∗ ( b ) · sgn C ∗ ( b ′ k ) · χ B ( b , b , . . . , b r ) ❀ ǫ = χ B ( b ′ , . . . , b ′ r ) · χ B ( b , . . . , b r ) = χ B ( b ′ k , b , . . . , b r ) · χ B ( b ′ , . . . , b ′ k − , b , b ′ k +1 , . . . , b ′ r )However the existence of k ∈ { , . . . , r } such that the last line holds is precisely the statement of ( B ′ ).We see that ( B , ( B ′ ) of definition 3.14 is not a consequence of the realization of M as a vector configu-ration. Rather it is a general consequence from the combinatorial definition 3.11. As an example [3], consider the vectors E = ( v , . . . , v ) in R , given by the columns of the matrix. A = From A we can read off the set of minimal linear dependencies among the vectors in E . For instance we have v − v + v − v = 0, that is ~λ = (1 , − , , , , −
1) and X = { + , − , , , + , −} . This can be compared tothe graph example in the previous section. Here we also obtain the same list of signed circuits as in the graphexample above. The column vectors of the matrix A contain the set B of bases of the oriented matroid M obtained in the previous section. There are two additional realizations of oriented matroids, related to vector configurations, which we introducefor completeness [3, 43].Consider the rank r vector oriented matroid M = ( E, C ) as defined at the beginning of section 3.5. Eachelement v i of ground set E = { v , . . . , v n } of M can be used to define an ( r − H i through the origin in R r as H i = { x ∈ R r : (cid:10) v i , x (cid:11) = 0 } , where (cid:10) · , · (cid:11) denotes the usual Euclidean innerproduct on R r . That is, v i is the normal vector to H i . Obviously the orientation of v i can be used to definean orientation of H i . Then we may define the positive side of H i by H + i := { x ∈ R r : (cid:10) v i , x (cid:11) > } and thenegative side H − i := { x ∈ R r : (cid:10) v i , x (cid:11) < } . 20n this way E corresponds to an oriented arrangement A = { H , . . . , H n } of n ( r − A gives rise to a decomposition of R r into r -dimensional cells W . The interior of each such cell isuniquely described by its relative position with respect to the hyperplanes contained in A . For each H i ∈ A we can say if W is on the positive or negative side of H i . That is W is uniquely described by an element of { +1 , − } n .Obviously the oriented vector matroid M = ( E, B ) can be equivalently encoded in the arrangement A . To spec-ify M we need to specify the unique cell W ≡ (+1 , . . . , +1) situated on the positive side of every hyperplane.We may alternatively consider the intersection of A with the unit sphere S r − = { x ∈ R r : k x k = 1 } aroundthe origin of R r . This gives rise to an arrangement S = { S , . . . , S n } of n unit ( r − S i on S r − defined by S i := S r − ∩ H i . Equivalently we define S ± i := S r − ∩ H ± i . This is depicted in figure 4. The spherearrangement S = { S , . . . , S n } together with the specification of a positive and negative side S ± i for every S i ∈ S is called a signed sphere arrangement.PSfrag replacements S + i S − i S i k v i k v i Figure 4: S r − split into two open hemispheres S + i , S − i by an r − S i , induced by the element v i of theground set E .Now recall the rank r vector oriented matroid M = ( E, C ) from the beginning of section 3.5. For a signedcircuit C ∈ C it holds that [ v i ∈ C S sgn C ( v i ) i = S r − where S sgn C ( v i ) i = S ± i and S ± i denotes the closure S ± i = S ± i ∪ S i We will see in section 3.7 how sphere arrangements can be used for deciding the so-called realizability problemof oriented matroids.
In sections 3.5 and 3.6 we have outlined how oriented matroids can arise from different geometric setupscorresponding to vector configurations. This section will be concerned with the opposite point of view: Givenan oriented rank r matroid M = ( E, B ) defined as in section 3.2, when can it be represented as a vectorconfiguration? This question is called the realizability problem for oriented matroids. It is of particular relevanceif one wants to compute all possible vector oriented matroids of rank r constructable over an abstract groundset E . For instance one would like to compute all classes of diffeomorphic embeddings of a graph vertex asdiscussed in section 4.1.The main statement [3, 43] here is that reorientation equivalence classes ( ≡ equivalence classes under rela-belling of the elements of the ground set E and a possible change of orientation for each element e ∈ E ) oforiented matroids correspond to signed arrangements of modified arrangements of spheres called signed pseu-dosphere arrangements on S r − ⊂ R r . Notice that the definition of H ± i is slightly different in [3] and [43]. In the former, H ± i contains H i and is thus closed, whereasin the latter H i is defined as an open half space of R r . We regard the notation of [43] as more convenient, as it avoids difficultiesin assigning a sign to points x ∈ H i ⊂ R r , and separates H i from H ± i . In general a circuit C ∈ C corresponds to a minimal systemof closed hemispheres that cover the whole unit sphere S r − . See section 3.8 for the definition of reorientation. r − S i as depicted in figure 4. A subset S i of S r − is called a pseudosphere if S = h ( S i )for some homeomorphism h : S r − → S r − . Using h we can also define the open subspaces S ± i := h ( S ± i ), asdepicted in figure 5. The arrangement S = { S , . . . , S } of pseudospheres S i ⊂ S r − ⊂ R r together with afixed choice of positive side S + i and negative side S − i for each S i is called a signed pseudosphere arrangement.PSfrag replacements S + i S − i S i Figure 5: S r − split into two half spaces S + i , S − i by an ( r − S i homeomorphic to the ( r − S i induced by the matroid element v i as shown in figure 4.Now conversely let a signed pseudosphere arrangement S = { S , . . . , S n } ⊂ S r − ⊂ R r be given. Choose anarbitrary n -element set E = { , . . . , n } as a ground set and associate to every i ∈ E a S i ∈ S . Define C ( S ) asthe set of sign vectors C ∈ { + , − , } | E | with support C = { e ∈ E : sgn C ( e ) = 0 } , which satisfy (i) [ e ∈ C S sgn C ( e ) e = S r − where S sgn C ( e ) e := S ± e and S ± e denotes the closure S ± e = S ± e ∪ S e (ii) The support C is minimal with respect to (i) , that is C contains only minimal subsets of E such that theunion of the closed half spaces S ± e covers S r − .Then we get the following Theorem 3.1 (Topological Representation Theorem.)
The following conditions are equivalent [3]:(1) If S = ( S e ) e ∈ E is a signed arrangement of pseudeospheres in S r − , then C ( S ) is the family of circuitsof a rank r simple oriented matroid on E .(2) If ( E, C ) is a rank r simple oriented matroid, then there exists a signed arrangement of pseudospheres S in S r − such that C = C ( S ) .(3) C ( S ) = C ( S ′ ) for two signed arrangements S and S ′ in S r − iff S ′ = h ( S ) for some self-homeomorphism h of S r − . It follows that there is a one-to-one correspondence between (equivalence classes of) pseudosphere arrangementsin S r − and (reorientation classes of) simple rank r oriented matroids. That is, a pseudosphere arrangementspecifies an oriented matroid only up to reorientation and permutation (relabeling of the ground set).Moreover the stretchability of the arrangement (existence of a self-homeomorphism on S r − transforming eachpseudosphere S i into a proper sphere S i ) is equivalent to the realizability of the oriented matroid as a vectorconfiguration. Note that the problem of deciding stretchability can be encoded in a system of equalities andinequalities of polynomial expressions. Finding a solution to this system is equivalent to deciding the realizabilityproblem for an oriented matroid given by a pseudosphere arrangement [3]. For rank r = 3 it turns out thatfrom n ≥ r = 3 results for such a counting are shown in table 3. Notice that our notation distinguishes between pseudospheres and spheres, in contrast to [3]. Again notice the difference in notation between [3] and [43] as described in footnote 31. As in the case of spheres S i weunderstand the S ± i as open subpaces in the sense of [43]. Moreover we understand C ∈ { + , − , } | E | as coming from an extendedsignature map, assigning a sign to every e ∈ E in the sense of definition 3.8. emark. Notice that the term realizability is used in two different contexts throughout this work. In themathematics literature such as [3] realizability of a given matroid aims at answering the question whether agiven set C of oriented circuits of a matroid can be realized by a vector configuration. In the context of [4, 5]the term realizability is directed towards the question whether a given set ~ǫ of sign factors in the definition ofthe volume operator ˆ V v at a vertex v in the vertex set of a graph corresponds to the chirotope of an orientedmatroid of rank 3 which can be realized (in the mathematical sense) as a vector configuration. Either questionof realizability not only decides whether a given family of oriented sets respectively sign factors represents anoriented matroid, but it also can be used to count the number of realizable oriented matroids of a given rank r over an n -element set E . Finally we would like to define certain operations which can be performed on oriented matroids. Let again M = ( E, C ) = ( E, B ) be an oriented rank r matroid over the ground set E . Then the following operations aredefined [3]. Reorientation.
According to definition 3.6 one can define a reorientation − F C =: e C of a signed set C on F ⊆ E by ( e C ) ± = ( C ± \ F ) ∪ ( C ∓ ∩ F ), that is(a) sgn e C ( e ) := ( − | F ∩{ e }| · sgn C ( e ) and(b) − F χ B ( e , . . . , e r ) = e χ B ( e , . . . , e r ) := χ B ( e , . . . , e r ) · ( − | F ∩{ e ,...,e r }| . Mutation.
Let M furthermore be a uniform oriented matroid. Then every r -tuple B = ( b , . . . , b r ) is abasis. An r -tuple B yields a mutation of M if the mapping B χ B given by B χ B ( B ) := − χ B ( B ) if B = Bχ B ( B ) otherwise (3.13)defines another oriented matroid, that is B χ B satisfies the Grassmann Pl¨ucker relations (B2) / (B2’) of definition3.14. Deletion.
If a subset A ⊂ E is removed from E then the remaining matroid M\ A over the ground set E \ A is defined by its set of signed circuits C\ A := { C ∈ C : C ∩ A = ∅} . (3.14)Suppose M\ A has rank s < r . Choose an arbitrary basis B ∈ B . Now take the ( r − s ) tuple B ∩ A =:( a , . . . , a r − s ) ∈ A . Then the chirotope of χ B\ A can be constructed (up to an overall sign) as follows: χ B \ A : ( E \ A ) s → {− , , +1 } ( e , . . . , e s ) χ B ( e , . . . , e s , a , . . . , a r − s ) (3.15)By construction χ B \ A has all properties of a chirotope. Moreover the above definition specifies χ B \ A uniquelyup to a sign [3], independent of the choice of B ∈ B . Example.
As an example consider a ground set E = ( e , e , e ) of 3 vectors e , e , e in R . Let a family B of bases be given by B = ( B , B ) with B = ( e , e ), B = ( e , e ), assume e , e to be co-linear, such that C = { C } with C = { e , e } . The resulting oriented matroid M = ( E, B ) then has rank r = 2. Now consider A = { e } . Clearly M\ A has rank s = 1 with bases B\ A = { e , e } . in order to construct χ B\ A , choose B ∈ B with B ∩ A = ( e ) =: ( a ). Applying (3.15) then gives χ B\ A ( e ) = χ B ( e , a ) and χ B\ A ( e ) = χ B ( e , a ). See section 3.3 for the definition. Oriented Matroids in Loop Quantum Gravity
Having outlined the construction of LQG in section 2, and oriented matroids in the previous section, we willnow demonstrate how oriented matroids naturally enter into LQG. As we have seen, both the local geometricproperties of a graph γ as well as its global topological (connectedness) properties can be described by orientedmatroids. As described in section 2, the measure µ on the kinematical Hilbert space H is sensitive only to supp( l ( γ )) of the graph γ underlying a spin network function [47] in H , but not to other properties of γ such as relativeorientations of edges at the vertices.In contrast to µ , the flux and a composite thereof, namely the volume operator [1], are sensitive to relativeorientations of edges. These relations are preserved by analytic diffeomorphisms. Even more, this sensitivity iscrucial for a consistent formulation of the theory [48] and the constraint operators [12]. This is based on thefact that the spectrum of the volume operator due to [1] is characterized by signature factors resulting fromthe embedding of γ in Σ as shown in [4, 5]. They encode the relative orientation of all tangent vectors of N v edges intersecting at a vertex v of γ . The classification of all possible signature factor combinations turns outto be intimately related to a central question of oriented matroid theory [3, 2], namely whether a given orientedmatroid of rank 3 can be realized as a configuration of N v ≥ R . Within LQG, the operator corresponding to the volume of a region R in three dimensional Riemannian spaceplays a prominent role for the implementation of the scalar (Hamilton) constraint operator on the quantumlevel. In [4, 5], the spectral properties of the volume operator according to [1, 49] were analyzed on H Gauss ,the gauge invariant subspace of the kinematical Hilbert space H of LQG. Starting from the classical volumeexpression rewritten in terms of Ashtekar variables V ( R ) = Z R d x p det q ( x ) = Z R d x r(cid:12)(cid:12)(cid:12) ǫ ijk ǫ abc E ai ( x ) E bj ( x ) E ck ( x ) (cid:12)(cid:12)(cid:12) (4.1)one can work out the action of V ( R ) on a cylindrical function f γ : G n ∋ ( h e , . . . , h e n ) → f γ ( h e , . . . , h e n ),which has support on the n copies of G = SU (2), each labelled by one of the edges ( e , . . . , e n ) contained inthe edge set E ( γ ) of a graph γ embedded into three dimensional Riemannian space. Notice that the graph γ underlying the cylindrical function f γ is taken to be adapted to the flux operator E i ( S ). That is, the elementsin E ( γ ), which intersect S , are subdivided, such that their subdivisions either end or start at S . With theseassumptions the action of E ( S ) on f γ is explicitly given as follows [50]: (cid:2) E i ( S ) f γ (cid:3) ( h e , . . . , h e n ) = 12 X e ∈ E ( γ ) X AB ǫ ( e, S ) (cid:16) δ e ∩ S,b ( e ) τ i h e + δ e ∩ S,f ( e ) h e τ i (cid:17) AB ∂f γ ( h e , . . . , h e n ) ∂ ( h e ) AB = 14 X e ∈ E ( γ ) ǫ ( e, S ) h(cid:0) δ e ∩ S,b ( e ) R ie + δ e ∩ S,f ( e ) L ie (cid:1) f γ i(cid:0) h e , . . . , h e n (cid:1) . (4.2)Here we denote by [ R ie f γ ]( h e , . . . , h e n ) := ddt f γ (cid:0) h e , . . . , e t τ i h e , . . . , h e n (cid:1)(cid:12)(cid:12) t =0 and respectively by[ L ie f γ ]( h e , . . . , h e n ) := ddt f γ (cid:0) h e , . . . , h e e t τ i , . . . , h e n (cid:1)(cid:12)(cid:12) t =0 the action of right / left invariant vector fields onthe copy of SU (2) labeled by e ∈ E ( γ ). By τ i we represent a basis of the Lie algebra su (2), given by τ i = − i σ i l ( γ ) is defined in section 2. Here ǫ abc , ǫ ijk denote the antisymmetric symbol ( ǫ = 1 = − ǫ etc.) and we use Einstein’s sum convention. Moreoverdet q ( x ) denotes the determinant of the spatial metric on the Cauchy surface Σ. E ai ( x ) is an su (2)-valued vector density, called adensitized triad. It holds that E ai ( x ) E bj ( x ) δ ij = det q ( x ) q ab ( x ). Here δ ij is the Cartan-Killing metric of su (2), which is just theEuclidean metric. Moreover q ab ( x ) are the components of the inverse spatial metric in the tangent space T x Σ at the point x ∈ Σ,with respect to a given basis (e.g. a coordinate basis ∂ a ). Let S ⊂ Σ be an orientable two-dimensional surface in Σ. Then the flux E i ( S ) is defined as the densitized triad E ci ( x ) integratedover the 2-surface S , that is E i ( S ) = R S du dv ǫ abc ∂x a ( u,v ) ∂u ∂x b ( u,v ) ∂v E ci ( x ( u, v )) where x : ( u, v ) x ( u, v ) denotes the embedding ofthe surface S into Σ parametrized by the pair ( u, v ). σ i being the according Pauli matrix . Moreover δ e ∩ S,b ( e ) = 1 if the intersection e ∩ S is the (b)eginningpoint of e , that is e is outgoing from S and zero otherwise. Accordingly δ e ∩ S,f ( e ) = 1 if the intersection e ∩ S is the (f)inal point of e , that is e is incoming to S and zero otherwise. Also in (4.2) each e ∈ E ( γ ) labelsa particular copy of SU (2). The sum over A, B can be taken in defining a representation of SU (2), but inprinciple any representation could be chosen . The orientation factor ǫ ( e, S ) indicates the relative orientationof the edge tangent at the intersection point and the chosen surface orientation ~n S of S .PSfrag replacements ~n S S x ˙ e ( x ) e ǫ ( S, e ) = +1 x = b ( e )PSfrag replacements ~n S S x ˙ e ( x ) e ǫ ( S, e ) = − x = f ( e )PSfrag replacements ~n S S x ˙ e ( x ) e ǫ ( S, e ) = +1 x = f ( e )PSfrag replacements ~n S S x ˙ e ( x ) e ǫ ( S, e ) = − x = b ( e )Figure 6: The four distinct configurations of the surface S and an edge e . The intersection point of S and e isdenoted by x . If S ∩ e = e or e ∩ S = ∅ then ǫ ( S, e ) = 0.Moreover if we set e f γ ( . . . , h, . . . ) := f γ ( . . . , h − , . . . ) then it holds that[ R ie e f γ ]( . . . , h e , . . . ) = ddt f γ (cid:0) . . . , ( e t τ i h e ) − , . . . (cid:1)(cid:12)(cid:12) t =0 = ddt f γ (cid:0) . . . , h − e e − t τ i , . . . (cid:1)(cid:12)(cid:12) t =0 = − [ L ie f γ ]( . . . , h − e , . . . )(4.3)by the exchange property of left / right invariant vector fields. Now consider a cylindrical function f e ( h e ),where e is given as in the first case of figure 6. Moreover e − = e , l ( e ) = l ( e ) where e is given as in thesecond case of figure 6. Then (cid:2) E i ( S ) f e (cid:3) ( h e ) = 14 (cid:2) R ie f e (cid:3)(cid:0) h e (cid:1) = 14 (cid:2) L ie f e (cid:3)(cid:0) h − e (cid:1) = 14 (cid:2) L ie f e (cid:3)(cid:0) h e − (cid:1) = 14 (cid:2) L ie f e (cid:3)(cid:0) h e (cid:1) (4.4)where we have used the transformation property of the edge holonomy under edge reorientation e → e − , thatis h e − = h e − . Notice that ǫ ( S, e ) = − ǫ ( S, e − ) = − ǫ ( S, e ) compensates the minus sign between the actionof R ie and L ie . A similar statement holds for case 3 and 4 in figure 6. That is, the action (4.4) of the fluxoperator on a cylindrical function f γ only depends on the support l ( γ ), and not on the actual direction of edgescontained in E ( γ ).In contrast to this, situation 1 and 3 in figure 6 have identical relative orientations ǫ ( S, e ) = ǫ ( S, e ) = 1, but (cid:2) E i ( S ) f e (cid:3) ( h e ) = 14 (cid:2) R ie f e (cid:3)(cid:0) h e (cid:1) and (cid:2) E i ( S ) f e (cid:3) ( h e ) = 14 (cid:2) L ie f e (cid:3)(cid:0) h e (cid:1) (4.5)because x = b ( e ) = f ( e ). Strictly speaking, (4.5) describes the action of the flux on two different cylindricalfunctions f e , f e , which have different supports l ( e ) = l ( e ).Now we are going to discuss the implication of this on the volume operator. In its final form, the actionof the volume operator on a cylindrical function f γ is given by [1, 49]:ˆ V ( R ) γ f γ ( · ) = Z R d x \ q det ( q ( x )) γ f γ ( · ) = Z R d x ˆ V ( x ) γ f γ ( · ) (4.6)where the “ b ” symbolizes the operator corresponding to the classical expression, R ⊆ Σ denotes a region inΣ, and ˆ V ( x ) γ = ℓ P X v ∈ V ( γ ) δ ( x, v ) ˆ V v,γ . (4.7) σ = σ = − ii σ = − with (cid:2) σ i , σ j (cid:3) = 2 i ǫ ijk σ k This is a quantization ambiguity. ℓ P denotes the Planck length and δ ( x, v ) is Dirac’s delta distribution. The action of ˆ V ( R ) γ decomposesinto a local action at all vertices v in the vertex set V ( γ ) with valence N v ˆ V v,γ = s(cid:12)(cid:12)(cid:12) Z X I 8. Thisconfirms the remarkable applicability of the Monte Carlo method to this problem. Having the oriented matroiddata at hand, we are now sure that we had detected all uniform oriented matroids of rank 3 over a ground setof up to N v = 8 elements. In addition, one can write down an analytic expression which gives a lower bound to the total number ~ǫ ( N v )of realizable ~ǫ sign factors for N v vectors [52]. For N v ≥ 3, it is ~ǫ ( N v ) ≥ ( N v − N v − N v − Y s =1 ( s − s + 7 s + 2) . (4.12)Note that this expression is able to exactly reproduce ~ǫ ( N v ) for N v up to 6. A derivation of this bound willappear in [53]. N v ~ǫ ( N v )(lower bound by(4.12) ) ~ǫ ( N v )(sprinkling) ~ǫ permutationequivalenceclasses ~σ confi-gurations ≥ 747 735 880 28 287 1359 84 486 888 837 120 ? ? 4 381Table 3: Sign factor combinatorics for 3–9-valent non-coplanar vertices embedded in R .As can be seen from (4.8), the volume operator evaluated at a vertex v is not only sensitive to the local edgetangent vector configuration encoded in the ǫ ( I, J, K ) factors, but also on whether v is the beginning or finalpoint of the N v edges intersecting at v . To see this, notice that the reorientation of an edge, say e L , outgoingfrom v , flips the edge tangent ˙ e L ( v ), and hence inverts all signs { ǫ ( IJL ) } containing the label L . However,at the same time, the action of [ R ie L f γ ]( . . . , h e L , . . . ) changes to [ L ie L f γ ]( . . . , h − e L , . . . ) = − [ R ie L e f γ ]( . . . , h e L , . . . )according to (4.3). This compensates the sign inversion of all ǫ ( IJL ) similarly to (4.4). This situation is depictedin figure 7 and figure 8. There e is reoriented, and hence the orientation of the edge tangent ˙ e ( v ) is flippedcompared to ˙ e ( v ). Although the configuration of sign factors in figures 7 and 8 differs accordingly, the volumespectrum at v and v is identical. One can compute the permutation equivalence classes from the reorientation equivalences classes as follows. Choose a repre-sentative ~ǫ Reor of each reorientation equivalence class, and apply all 2 N v reorientations to it. For each such reorientation applyeach of the N v ! permutations of the elements, and record the canonical representative of the resulting chirotope. The set of suchcanonical representatives gives the set of permutation equivalence classes of chirotopes. To be completely correct we were not able to perform the sprinkling of the 8-vertex in [4, 5], because the enormous numberof chirotopes overwhelmed the system memory of any machines we had available. We performed the sprinkling with N v = 8 morerecently, by computing the canonical representative of each chirotope as it arose, thus having to save only 28 287 chirotopes inplace of more than 10 . 27o rephrase this in terms of oriented matroids: If an element in the graphic matroid constructed from γ isreoriented, this induces a reorientation of the according element in the local vector oriented matroid at v .Because of the described sign compensation, the volume spectrum is thus invariant under reorientations of thegraphic oriented matroid of γ .In contrast to this, figure 9 seems to have the same sign configuration ~ǫ induced from the edge tangents at v aswe find at v . However, the support (tracing of e ) of the underlying graph γ is different in figure 8 and figure9. Hence by (4.5) the signs ~ǫ of (4.8) containing e are effectively opposite at v and v , and it follows that thevolume spectra at v and v are different. This is equivalent to stating that ˆ V ( R ) is evaluated on spin networkfunctions supported on non-diffeomorphic graphs.It follows that the spectrum of the volume operator is invariant under relabellings (permutation) of edges adja-cent to v . It is furthermore invariant under reorientations of the adjacent edges, which induce reorientationson the vector oriented matroid encoded by the ~ǫ factors (figures 7 and 8). However it is not invariant under anisolated reorientation of the oriented matroid encoded in the ~ǫ factors alone (figure 9). Hence different volumespectra at v are characterized by permutation equivalence classes of ~ǫ configurations.PSfrag replacements e e e e e e ˙ e ˙ e ˙ e ˙ e ˙ e ˙ e v Figure 7: Vertex v all edges out-going. PSfrag replacements e e e e e e ˙ e ˙ e ˙ e ˙ e ˙ e ˙ e v Figure 8: Vertex v generatedfrom v by orientation reversal ofedge e . PSfrag replacements e e e e e e ˙ e ˙ e ˙ e ˙ e ˙ e ˙ e v Figure 9: Vertex v with identi-cal edge tangent orientation as v but with all edges outgoing.A more general discussion, also including non-uniform oriented matroids (with coplanar edges), using methodsdeveloped in [55], will be presented in a forthcoming paper [53]. We have seen in section 3.4 that connectedness properties of (directed) graphs can be described by (oriented)matroids. Although originally introduced for planar graphs [41], the definition of graphic oriented matroidsin terms of its signed circuits depends only on the topology of the graph (adjacency of vertices, orientation ofedges), and not on any embedding. Therefore the framework of graphical oriented matroids is general enoughto be applied to any abstract graph, and in particular to the graphs underlying (gauge invariant) spin networkfunctions in LQG .Given the fact that the construction of LQG as outlined in section 2 is based on partially ordered sets of graphs,it is natural to ask if we can reformulate this construction in terms of oriented matroids. This seems feasible,because it makes sense to speak of the restriction of an (oriented) matroid, if its ground set E is restricted. Thiscan be directly seen from the deletion property in section 3.8. Hence, similarly to the partial order among thelabel set L of section 2, the according graphic (oriented) matroids will be equipped with a partial order givenby set inclusion of their ground sets. This will be further investigated in [58]. By the definition of the graphicoriented matroid in terms of circuits it seems obvious that the oriented matroid formalism should be applied atthe gauge invariant level H Gauss . See also section 6 for a discussion. In [49, 54, 5] the details for computing the matrix elements ˆ q IJK in (4.10) of the volume operator with respect to a basis ofgauge invariant spin network functions is given. This basis is formulated in terms of so-called recoupling schemes. Permuting edgelabels amounts to changing the recoupling order, which is a unitary basis transformation. This can be seen from the fact that thetransformation matrix between two recoupling schemes can be written in terms of Clebsch-Gordan coefficients which are unitary.Details will be given in [53]. The possibility of winding numbers between edges certainly needs to be further investigated. See also [56, 57] and section 6. 28n order to demonstrate the possibly new and interesting features of such a reformulation, we would like to givethe following example on H Gauss :If the Gauss constraint, and hence gauge invariance, is imposed on spin network functions, then the fundamentalgroup of its underlying graph γ becomes relevant . Taking into account the properties of the measure µ asdescribed in section 2, a gauge invariant spin network function f γ over γ is equivalent to a gauge variant spinnetwork function over another graph e γ ⊂ γ , the remainder of γ after the retraction to a spanning tree T ( γ ) ofedges [59]. This also holds if additionally invariance under graph automorphisms is imposed [47]. The circuitstructure of a directed graph can in turn be used to define an oriented matroid [3, 2]. By definition, the orientedmatroid M γ constructed from γ has the set of all spanning trees of γ as its basic family B ( M γ ). Then eachpossible edge set E ( e γ ) of e γ obtained by removal of a B ∈ B ( M γ ) corresponds to an element B ∗ of the dual basis B ∗ by definition 3.13. Hence we can reformulate the findings of [59] by saying that gauge invariant informationof f γ can be encoded in cylindrical functions f e γ , which have support only on B ∗ , the family of bases of the dualoriented matroid M ∗ γ . In [4, 5] an extensive numerical analysis of the spectrum of the volume operator was presented. For valences N v = 4 − 7, and spins up to some j max ( N v ), for every chirotope which arose from the Monte Carlo sprinklingdiscussed in section 4.1.2, we presented graphs and histograms which detail various aspects of the collection ofeigenvalues of the volume operator for vertices within this range of parameters.Casting the analysis into the context of oriented matroids now allows us to better organize the collection ofeigenvalues, in particular to take advantage of the permutation symmetry of the operator, as touched upon insection 4.1.2. Instead of computing the volume spectrum for all realizable chirotopes, we can restrict attentionto the canonical representatives of each permutation equivalence class. Since the volume operator dependsupon these chirotopes through the sigma values of (4.11) (a sigma configuration ~σ ), we compute this for eachcanonical representative of a permutation equivalence class.When counting eigenvalues to form histograms, we follow the same practice as in [4, 5], regarding each eigenvaluefrom each chirotope as distinct. Thus we associate a redundancy with each permutation equivalence class ofchirotopes, equal to the number of members in that class. We then add these redundancies for each chirotopewhich yields the same sigma configuration, to get a redundancy for each sigma configuration. The numbers ofeigenvalues reported in histograms are then weighted by (half of) these redundancies.The analysis in [4, 5] considers only ‘sorted’ spin values j ≤ . . . ≤ j N v , because any other assignment of spinsto the edges is related to one of these by a permutation. However, this same permutation will also transformthe chirotope ~ǫ to another ~ǫ ′ . Thus the separation of the eigenvalues into those arising from a given chirotope ~ǫ , while at the same time considering only sorted spin values j ≤ . . . ≤ j N v , is ‘misleading’, in the sense thatthe eigenvalues which arise from the same chirotope ~ǫ , but with non-sorted spin values j , . . . , j N v , are foundinstead under a different chirotope ~ǫ ′ which is related to ~ǫ by a permutation which brings the spins into a sortedordering j ≤ . . . ≤ j N v .A more ‘correct’ presentation of the eigenvalue data would combine all eigenvalues which arise from verticeswhich are related to each other by a permutation. We achieve this by working only with the canonical rep-resentative of each permutation equivalence class of the chirotopes, but allow all values of spins on the edges j , . . . , j N v , not just those which are sorted j ≤ . . . ≤ j N v .The resulting spectra are presented below. Remarkably it appears that the property found in [4, 5], that different Consider a cylindrical function f γ ( h e , . . . , h e N ) defined over the digraph γ with edge set E ( γ ) = { e , . . . , e N } and vertexset V ( γ ) = { b ( e K ) , f ( e K ) } e K ∈ E ( γ ) . In the context of LQG, gauge invariance of f γ denotes invariance under generalized gaugetransformations G , which consist of all mappings g : Σ ∋ x g ( x ) ∈ SU (2) (no continuity assumption). Applying g ∈ G , aholonomy h e K transforms as h e K g h ( g ) e K := g (cid:0) b ( e K ) (cid:1) h e K g (cid:0) f ( e K ) (cid:1) − . To see this, choose a spanning tree T ( γ ) = { e , . . . , e t } ⊆ E ( γ ) of γ , with cardinality t . Denote e E := E \ T ( γ ) = { e e , . . . , e e N − t } .Consider the cylindrical function f γ ( h e e , . . . , h e e N − t , h e , . . . , h e t ) as before. Given any ( h e e , . . . , h e e N − t , h e , . . . , h e t ) ∈ SU (2) N ,we can choose a g ∈ G , such that h ( g ) e K = SU (2) for every e K ∈ T ( γ ) . This implies that the gauge invariant information of f γ canbe encoded in f e γ , where e E = E ( e γ ). This half stems back to the overall sign symmetry of chirotopes. In our code we consider only chirotopes which assign +1 tothe basis 123. However it is interesting to note that a chirotope C will in general not lie in the same permutation equivalence classas its negative − C . However we have carefully checked that nevertheless the true redundancies as defined here are exactly doubleof those that arise by ignoring all chirotopes with basis 123 = -1, in spite of the separation of C from − C by permutations. σ -configurations have a different behavior of the smallest non-zero eigenvalue (increases, decreases, or remainsconstant as j max is increased) is preserved among the permutation equivalence classes for the 5-vertex. Therewe find precisely four permutation equivalence classes and the corresponding four different spectral propertiesas j max is increased: identically zero spectrum, increasing/decreasing smallest non-zero eigenvalue, and constantsmallest non-zero eigenvalue. At the six vertex we also observe all four such behaviors, however at the 7-vertexthe increasing smallest eigenvalue sequences seem to vanish. This may affect the notion of a semiclassical limitfor the volume operator as discussed in section 6.Our improved understanding of the action of permutations on the whole spin network vertex, including edgespins and orientation of the embedded edge tangents (as encoded in the chirotope) together, has revealed someerrors in our previous analysis of [4, 5].One such error is in the handling of the overall sign symmetry of chirotopes (that if χ B is a valid chirotope(equivalently ~ǫ a valid sign configuration), then so − χ B (equivalently − ~ǫ ). There we made a mistake in mappingthis symmetry to the sigma configurations, and thus some of the sigma configurations we used in our analysiswere incorrect. This affects some of the counting of sigma configurations which yield the different behaviorsof the smallest non-zero eigenvalues, and some of the eigenvalues themselves. However none of the qualitativeresults are affected.We were also able to utilize the permutation symmetry of the volume operator as a cross check on our code,which at the level of sigma configurations is extremely subtle. In using this we discovered a bug which affectedall our results on the 7-vertex. The corrected eigenvalues are shown in section 5.4.In addition there was a bug in our code with respect to the handling of the sigma configuration for the cubic6-vertex. Corrected eigenvalues are shown in section 5.3.The results from the computations shown below come from running a modified version of the code developedfor references [4, 5], which is written as a ‘thorn’ within the Cactus high performance computing framework[60], on the ‘whale’ cluster of the SHARCNET grid in Southern Ontario. The computations for the 7-vertexran efficiently on 225 cores. In figure 10 we show histograms of the three non-trivial permutation equivalence classes of the 5-vertex, for j max up to . Each is distinct yet shows similar behavior. Note the small lip at zero for the purple ~σ = (2 , , , j max up to 25/2. Note that the lip atzero is not really visible at this scale. Figure 12 zooms in on the small eigenvalue region. There we can discernthe lip just beginning to show at j max = 11.Figures 13 and 14 show the smallest and largest non-zero eigenvalues of the 5-vertex, for each ~σ and j max . Forthese ~σ -index 0 corresponds to ~σ = (2 , , , ~σ = (2 , , , ~σ = (2 , , , ~σ -index 0 increasing, 1 constant, and 2 decreasing with j max . The largest eigenvalues behave similarlyfor each non-trivial ~σ . Histograms for the 6-vertex are displayed in figures 15, 16, and 17, for j max up to 13/2. Here there are 39distinct sigma configurations. The beginnings of a lip at zero in the spectrum are just visible for several sigmaconfigurations ~σ , as seen in figure 16.Figures 18 and 19 depict the smallest and largest non-zero eigenvalues for each ~σ , as a function of maximumspin. Most of the sigma configurations yield smallest non-zero eigenvalues which decrease with j max , howevera number of them remain constant (such as the rather degenerate ~σ = (2000000000), in which only four of thesix spins play any role beyond their effect in determining the recoupling basis), and one ( ~σ = (0222222220))leads to increasing smallest non-zero eigenvalues. It can also be noted, in particular in figure 18, that in generalthere are several sigma configurations which yield the same exact sequence of smallest non-zero eigenvalues.The largest eigenvalues all appear to increase by a power law, as expected from (4.9). There is a fourth ~σ at the 5-vertex, which is all zeros, and hence leads to only zero eigenvalues. There will be such a ~σ forevery valence. Zero eigenvalues are suppressed on all plots. PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 10: Histograms for each sigma configuration ~σ at the 5-vertex, up to j max = 25 / 2. From bottom to topat volume 100, the blue is for ~σ = ( σ , σ , σ , σ ) = (2 , , , ~σ = (2 , , , ~σ = (2 , , , PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 11: Histograms for the overall generic 5-vertex, up to every j max ≤ / 2. (By ‘generic’ we mean excludingco-planar edges, which in a sense form a set of measure zero.) Each histogram has 512 bins.31 PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 12: Histograms for the overall 5-vertex, up to every j max ≤ / 2, zoomed to the small eigenvalue region. PSfrag replacements λ min j max ~σ -indexFigure 13: Smallest non-zero eigenvalues λ min at the 5-vertex. 0 0.5 1 1.5 2 0 5 10 15 20 25 1 10 100 1000 PSfrag replacements λ max j max ~σ -indexFigure 14: Largest eigenvalues λ max of the 5-vertex.32 PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 15: Histograms for each sigma configuration ~σ at the 6-vertex, up to j max = 13 / 2. Two histograms have128 bins, all others have 512. PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 16: Histograms for each sigma configuration ~σ at the 6-vertex, up to j max = 13 / 2, zoomed to the smalleigenvalue region. 33 PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 17: Histograms for the overall (generic) 6-vertex, up to every j max ≤ / 2. All have 512 bins. PSfrag replacements λ min j max ~σ -indexFigure 18: Smallest non-zero eigenvalues λ min at the6-vertex. 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 1 10 100 1000 PSfrag replacements λ max j max ~σ -indexFigure 19: Largest eigenvalues λ max of the 6-vertex.34 .3 Cubic 6-Vertex The cubic 6-vertex arises from a ‘tilation’ of R by cubes. I.e. it is the network of lines formed by restrictingthe Cartesian coordinates x, y ∈ Z , with z ∈ R , and likewise for x, z ∈ Z , y ∈ R , and y, z ∈ Z , x ∈ R . The edgetangents at such a vertex are of course coplanar — this is the one exception to our rule of excluding coplanaredges. The resulting chirotope and sigma configuration are detailed in [5].In figure 20 we present histograms for the volume eigenvalues of the cubic 6-vertex, for maximam spins j max upto 5. Figure 21 displays the minimum and maximum non-zero eigenvalues, as a function of j max . One can see PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 20: Histograms of volume eigenvalues for the cubic 6-vertex, one for each value of j max up to 5.that the cubic 6-vertex is one of the rare ones for which the smallest non-zero eigenvalue increases with spin. Histograms for the gauge invariant 7-vertex are shown in figures 22, 23, and 24, for j max up to 7/2. There are673 permutation equivalence classes of sigma configurations for the 7-vertex. Here there seem to be roughlytwo sorts of sigma configurations — those that lead to extremely jagged histograms in figure 22 (we believethat these come from degenerate ~σ , which have many zeros, and thus can be effectively of lower valence), andthe others which all have more-or-less the same smooth shape. j max = 7 / j max and ~σ at the 7-vertex. Here there are no ~σ forwhich the smallest non-zero eigenvalue increases with j max , though there are several for which this smallest eigen-value is constant, such as ~σ = (22002000002000000000), which is one of the ~σ which yields the largest smallestnon-zero eigenvalues, and the ‘4-vertex-like’ ~σ = (20000000000000000000). ~σ = (00000000022000000000) is anexample of a 7-vertex sigma configuration with decreasing smallest non-zero eigenvalue. The largest eigenvaluesbehave as usual.It is interesting to note that the sigma configuration ~σ in which the only non-zero sigma is σ (1 , , 3) = +2appears to arise at every valence, as the sigma configuration corresponding to the chirotope ǫ (1 , , 3) = +1, ǫ ( I, J, K ) = − I, J, K . This chirotope appears as the canonical representative of anequivalence class at every valence, because it corresponds to the integer 1 in our numbering scheme (andpresumably because chirotopes ‘0’ and ‘1’ cannot be transformed into each other by a permutation, at anyvalence). It leads to a smallest non-zero eigenvalue of √ 12 for every j max .35 PSfrag replacements λ min λ max j max e x t r e m a l e i g e n v a l u e Figure 21: Smallest and largest non-zero volume eigenvalues of the cubic 6-vertex, as a function of j max ≤ PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 22: Histograms for each sigma configuration ~σ at the 7-vertex, up to j max = 7 / PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 23: Histograms for each sigma configuration ~σ at the 7-vertex, up to j max = 7 / 2, zoomed to the smalleigenvalue region. PSfrag replacements e i g e n v a l u e s volume in ℓ Figure 24: Histograms for the overall (generic) 7-vertex, up to every j max ≤ / PSfrag replacements λ min j max ~σ -indexFigure 25: Smallest non-zero eigenvalues λ min at 7-vertex. 0 100 200 300 400 500 600 700 2 3 4 5 6 7 1 10 100 PSfrag replacements λ max j max ~σ -indexFigure 26: Largest eigenvalues λ max at 7-vertex. As we have demonstrated, the framework of oriented matroids can capture both local embedding properties ofgraphs as well as their global connectedness properties.Originally developed for describing planar graphs, the framework of oriented matroids is general enough to beapplied to graphs embedded in three dimensional Riemannian space, as is the case for LQG. Although theexample from [3], revisited in detail in section 3.4, corresponds to a planar graph and uses a global orientation,the computation of the set C of signed circuits of the according oriented matroid does not rely on this fact.Because the set C of signed circuits as well as the chirotope χ B are symmetric with respect to a global signreversal , one can just take any vertex v contained in a circuit C as a starting point and then follow C alongeither direction until one comes back to v . The relative orientation of edges to the chosen direction thendetermines C ± .In the outlined treatment, we certainly neglect the global knotting of edges (e.g. one edge could wind aroundanother) of graphs embedded in three dimensional space [56]. If one wants to include these properties, one has tothink about a possible extension of the oriented matroid framework for embedded digraphs. On the other hand,the knotting properties are not seen by any quantum operator corresponding to classical geometric quantities,unless they are interpreted as e.g. matter excitations as suggested e.g. in [56, 57]. Note that knotting also becomesirrelevant if automorphisms of graphs are considered [47]. Therefore we prefer to postpone considerations ofknotting of edges until the connection between oriented matroids and the framework of LQG is worked out inmore detail.Regarding the local embedding of graph vertices, the occurrence of so-called moduli parameters was discussedin [61], which seems to prevent H diff from having a countable label set. However the solution suggested thereerases all local geometric information of the vertex embedding, encoded in sign factors. In contrast, we thinkthat this information is physically relevant and should be kept, as it is crucial for the implementation of theHamilton constraint operator [12] and a consistent formulation of LQG [48]. A solution to this issue is suggestedin [40], giving H diff a countable basis, while keeping the information on the local embedding.The level at which the graphic oriented matroids can be introduced into the LQG formalism has to be furtheranalyzed. At the level of H , one can have graphs underlying spin network functions, which have “open ends”,that is edges which have a 1-valent vertex as their beginning or final point. This difficulty could be dealt with bytreating such “open ends” as effective re-tracings, however as one finally has to solve the Gauss constraint, onecan directly start at the level of gauge invariant spin network functions in H Gauss , because there every edge ofa graph has at least a two-valent vertex as beginning and final point. From the combinatorics side introducinggraphic oriented matroids directly at the level of H diff seems to be the most natural approach.We have seen in section 5 that the spectral properties of the volume operator are closely related to the orientedmatroid resulting from the local embedding of the vertices of a graph. Interestingly, it is not only sensitive to That is if χ B is a chirotope, then so is − χ B . Also ± C ∈ C for all signed circuits C . Similar statements hold for the dualoriented matroid. j max at a vertex is increased. We find, in agreement with [4, 5],increasing, constant, and decreasing smallest non-zero eigenvalue sequences. In the latter two cases even verylarge spins can contribute microscopically small non-zero eigenvalues, and hence one cannot say that the limit j max → ∞ produces only large volume eigenvalues. This is different from the area operator [62], for whichlarge spins imply large eigenvalues. If one wants to keep this property one might regard this as a “physical”preference for vertices whose smallest non-zero eigenvalue grows with j max , such as the 4-vertex, cubic 6-vertex,or the (permutation equivalence class of) 6-vertex with ~σ = (0222222220).As a consequence we would like to point out that high valent vertices do not necessarily overcount the volumein a semiclassical analysis as presented in [20]. That is, high valence does not automatically shift the volumespectrum towards larger eigenvalues, as one might naively expect from the sum structure in (4.10). This is dueto, for example, the presence of vertex embeddings which only have a small number of non-zero signs σ ( I, J, K ),as defined in (4.11). In particular the number of triples ( I, J, K ) with σ ( I, J, K ) = 0 can be independent of thenumber of edges attached to the vertex. One example for such a vertex embedding is given by ~σ = (2 , , , . . . , j max and valence N v .Given this fact, it will be instructive to analyze if one can find embedded higher valent vertices which alsoresemble the correct semiclassical limit in the sense of [20]. Approximating the volume of a spatial regionby semiclassical coherent state techniques introduced in [63] might be a subtle mixture of the embedding ofthe graph supporting the coherent state, as well as choosing its topology, in particular the number of verticesand their connectedness in that region. Here the unified description of these properties in terms of orientedmatroids might give us a better understanding of the possible choices one has to make in the construction ofthe complexifier coherent states used in [20]. In this work we have demonstrated a connection between the LQG framework and the field of oriented matroids.For this we have described the combinatorial properties of LQG in section 2, and introduced matroids andoriented matroids in section 3 in an abstract way, without referring to any particular realization. In sections 3.4and 3.5 we showed how this abstract framework can be applied in order to describe global (connectedness) andlocal (geometry of vertex embedding) graph properties by oriented matroids. We have also briefly discussedthe issue of realizability of oriented matroids in terms of pseudosphere arrangements in section 3.7, that is thequestion of when an abstract oriented matroid of rank r gives rise to a configuration of vectors in R r . Theobvious connection of oriented matroids to LQG is then discussed in section 4.We see several possible benefits in this approach. First, it provides one unified framework to describe local andglobal graph combinatorics, in an abstract way. That is, in this setup we do not need to refer to the underlyingmanifold into which graphs are embedded. In addition the oriented matroid concept introduces the possibilityof a dual description of oriented graphs in terms of vector configurations, and vice versa.Secondly, it gives explicit ways to classify oriented matroids, and e.g. to generate representatives of equivalenceclasses of vector configurations under permutation. We have already taken advantage of this in section 5, wherewe revisit the results of [4, 5]. Using our insights from oriented matroids we are able to drastically simplify thepresentation, and to confirm the vertex combinatorics used in [4, 5] by direct computation from reorientationclasses of oriented matroids.The present work will serve as a starting point to further explore this subject.The constructions outlined in section 3.8, e.g. reorientation and deletion, can be used in order to construct ananalogy to the projective limit of the graph poset in LQG, using only oriented matroids. Here the constructionof matroids for infinite graphs [44] will become relevant. This question will be analyzed in [58]. We have seen39n section 4.2 gauge invariance of a cylindrical function can be decribed by the selection of a dual basis ofthe underlying graph. This points to a connection between oriented matroids and recoupling theory of SU (2)representations. This is an interesting perspective for a possible extension of the oriented matroid frameworkby techniques from the graphical calculus of angular momentum, as for example given in [64]. For LQG it willalso be crucial to describe the action of graph changing operators (holonomies) by modifications to the graphicoriented matroid. If this turns out to be possible, then it becomes feasible to cast the action of the Hamilton[12], and respectively Master constraint operator [39, 40], in LQG into the oriented matroid framework. Thedifficulty of finding eigenstates could then be re-formulated in terms of oriented matroids.In section 3.2 an abstract notion of orthogonality between signed subsets is given. An interesting question is ifthis notion can be related to the inner product on H diff , the diffeomorphism invariant sector of LQG, where theinner product is given by a rigging map construction [38, 30]. Also, in the context of [47], it will be instructiveto see if the outstanding problem of giving a basis for the automorphism invariant sector of LQG can be tackledusing an oriented matroid labelling.Besides these exciting conceptual perspectives, we have already seen that oriented matroids can be used inorder to perform explicit computations in LQG. We expect the spectral analysis of the volume operator to becompleted in [53], where also non-uniform oriented matroids, that is coplanar triples of tangent vectors, willbe included using techniques from [55]. In this context it should also be possible to extend the analysis of [20]to all realizable vertex embeddings. Additionally the issue of taking the semiclassical limit, as discussed at theend of section 6, can be addressed in this setup.Moreover it is now possible to classify diffeomorphic graphs , and to compute the number of diffeomorphismequivalence classes. For this the issue of realizability in section 3.7 becomes important. 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