Geometric Bijections for Regular Matroids, Zonotopes, and Ehrhart Theory
GGEOMETRIC BIJECTIONS FOR REGULAR MATROIDS,ZONOTOPES, AND EHRHART THEORY
SPENCER BACKMAN, MATTHEW BAKER, CHI HO YUEN
Abstract.
Let M be a regular matroid . The Jacobian group
Jac( M ) of M isa finite abelian group whose cardinality is equal to the number of bases of M .This group generalizes the definition of the Jacobian group (also known as thecritical group or sandpile group) Jac( G ) of a graph G (in which case bases ofthe corresponding regular matroid are spanning trees of G ).There are many explicit combinatorial bijections in the literature betweenthe Jacobian group of a graph Jac( G ) and spanning trees. However, most ofthe known bijections use vertices of G in some essential way and are inherently“non-matroidal”. In this paper, we construct a family of explicit and easy-to-describe bijections between the Jacobian group of a regular matroid M andbases of M , many instances of which are new even in the case of graphs. Wefirst describe our family of bijections in a purely combinatorial way in terms oforientations; more specifically, we prove that the Jacobian group of M admits acanonical simply transitive action on the set G ( M ) of circuit-cocircuit reversalclasses of M , and then define a family of combinatorial bijections β σ,σ ∗ between G ( M ) and bases of M . (Here σ (resp. σ ∗ ) is an acyclic signature of the set ofcircuits (resp. cocircuits) of M .) We then give a geometric interpretation ofeach such map β = β σ,σ ∗ in terms of zonotopal subdivisions which is used toverify that β is indeed a bijection.Finally, we give a combinatorial interpretation of lattice points in the zono-tope Z ; by passing to dilations we obtain a new derivation of Stanley’s formulalinking the Ehrhart polynomial of Z to the Tutte polynomial of M . Introduction
The main bijection in the case of graphs.
Let G be a connected finitegraph. The Jacobian group
Jac( G ) of G (also called the sandpile group, criticalgroup, etc.) is a finite abelian group canonically associated to G whose cardi-nality equals the number of spanning trees of G . Since in most cases there isno distinguished spanning tree to correspond to the identity element, there isno canonical bijection between Jac( G ) and the set T ( G ) of spanning trees of G .However, many constructions of combinatorial bijections starting with some fixedadditional data are known. We mention, for example: the Cori–Le Borgne bijec-tions that use an ordering of the edges as well as a fixed vertex [14], Perkinson, Date : April 24, 2019. a r X i v : . [ m a t h . C O ] A p r SPENCER BACKMAN, MATTHEW BAKER, CHI HO YUEN
Yang and Yu’s bijections that use an ordering of the vertices [30], and Bernardi’sbijections that use a cyclic ordering of the edges incident to each vertex [9].In this paper we describe a new family of combinatorial bijections betweenJac( G ) and T ( G ). Our bijections are very simple to state, though proving thatthey are indeed bijections is not so simple. Another feature is that our bijectionsare formulated in a “purely matroidal” way, and in particular they generalizefrom graphs to regular matroids . We will first state the main result of this paperin the language of graphs, and then give the generalization to regular matroids.What we will in fact do is establish a family of bijections between T ( G ) andthe set G ( G ) of cycle-cocycle equivalence classes of orientations of G . The latterwas introduced by Gioan [21, 22] and by definition is the set of equivalenceclasses of orientations of G with respect to the equivalence relation generated bydirected cycle reversals and directed cut reversals. We will write [ O ] to denotethe equivalence class containing an orientation O . G ( G ) is known to be a torsorfor Jac( G ) in a canonical way (i.e., there is a canonical simply transitive groupaction of Jac( G ) on G ( G )) [4]. By fixing a class in G ( G ) to correspond to theidentity element of Jac( G ), we then obtain a bijection between Jac( G ) and T ( G ).To state our main bijection for graphs, let C ( G ) (resp. C ∗ ( G )) denote theset of simple cycles (resp. minimal cuts, i.e., bonds) of G , and define a cyclesignature (resp. cut signature ) on G to be a choice, for each C ∈ C ( G ) (resp. C ∈ C ∗ ( G )), of an orientation of C . By fixing an reference orientation for eachedge, we can identify directed cycles (resp. directed cuts) with elements of Z E ( G ) .Now we call a cycle signature σ (resp. cut signature σ ∗ ) acyclic if whenever a C are nonnegative reals with (cid:88) C ∈C ( G ) a C σ ( C ) = 0in Z E ( G ) (resp. (cid:80) C ∈C ∗ ( G ) a C σ ∗ ( C ) = 0) we have a C = 0 for all C . Example 1.1.1.
Fix a total order e < · · · < e m and a reference orientation O of E ( G ), and orient each simple cycle C compatibly with the reference ori-entation of the smallest element in C . This gives an acyclic signature of C ( G ).Indeed, suppose the signature is not acyclic and take some nontrivial expression (cid:80) C ∈C ( G ) a C σ ( C ) = 0. Let e be the minimum element appearing in some cycle inthe support of this expression. Then the element e must be appear with differentorientations in at least two different cycles, and thus one of these cycles is notoriented according to σ , a contradiction. One can, in an analogous way, use O to define an acyclic signature of C ∗ ( G ).Recall that if T is a spanning tree of G and e (cid:54)∈ T (resp. e ∈ T ), there is aunique cycle C ( T, e ) (resp. cut C ∗ ( T, e )) contained in T ∪ { e } (resp. containing T \{ e } ), called the fundamental cycle (resp. fundamental cut ) associated to T ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 3 and e . With this notation in place, we can now state our main bijection in thecase of graphs: Theorem 1.1.2.
Let G be a connected finite graph, and fix acyclic signatures σ and σ ∗ of C ( G ) and C ∗ ( G ), respectively. Given a spanning tree T ∈ T ( G ), let O ( T ) be the orientation of G in which we orient each e (cid:54)∈ T according to its orien-tation in σ ( C ( T, e )) and each e ∈ T according to its orientation in σ ∗ ( C ∗ ( T, e )).Then the map T (cid:55)→ [ O ( T )] is a bijection between T ( G ) and G ( G ).The bijection in Theorem 1.1.2 appears to be new even in the special casewhere σ and σ ∗ are defined as in Example 1.1.1. Example 1.1.3.
Suppose that G is a plane graph and define σ by orienting eachsimple cycle of G counterclockwise. Similarly, define σ ∗ by orienting each simplecycle of the dual graph G ∗ clockwise and composing with the natural bijectionbetween oriented cuts of G and oriented cycles of G ∗ . By [38, Theorem 15], thesimply transitive action of Jac( G ) on T ( G ) afforded by Theorem 1.1.2 in thiscase coincides with the “Bernardi torsor” defined in [8] and a posteriori with the“rotor-routing torsor” defined in [12, 13]. In particular, we get a new “geometric”proof of the bijectivity of the Bernardi map.This example is in fact a special case of Example 1.1.1. Indeed, let q ∗ be thevertex of G ∗ corresponding to the unbounded face of G , and fix a spanning tree T ∗ of G ∗ . Let O ∗ be any orientation of G ∗ in which the edges of T ∗ are orientedaway from q ∗ , and fix any total order on E ( G ∗ ) in which every edge of the rootedtree T ∗ has a larger label than its ancestors while being smaller than all the edgesoutside T ∗ . Using the natural bijection between oriented edges of G and of G ∗ ,this gives an orientation O of G and a total order < on E ( G ). Then the cyclesignature σ associated to ( O , < ) by the rule in Example 1.1.1 will orient everysimple cycle of G counterclockwise.1.2. Generalization to regular matroids.
As mentioned previously, an inter-esting feature of the bijection given by Theorem 1.1.2 is that it admits a directgeneralization to regular matroids .Regular matroids are a particularly well-behaved and widely studied class ofmatroids which contain graphic (and co-graphic) matroids as a special case. Moreprecisely, a regular matroid can be thought of as an equivalence class of totallyunimodular integer matrices. See § G is a graph, one can associate a regular matroid M ( G ) to G by taking the(modified) adjacency matrix of G . By a theorem of Whitney, the equivalenceclass of A determines the graph G up to “2-isomorphism” (and in particulardetermines G up to isomorphism if G is assumed to be 3-connected). If e and e ∗ are dual edges of G and G ∗ , respectively, then given an orientation for e ∗ weorient e by rotating the orientation of e ∗ clockwise locally near the crossing of e and e ∗ . SPENCER BACKMAN, MATTHEW BAKER, CHI HO YUEN
Let M be a regular matroid. One can define the set C ( M ) of signed circuits of M (resp. the set C ∗ ( M ) of signed cocircuits of M ) in a way which generalizesthe corresponding objects when M = M ( G ). Similarly, one has a set B ( M ) of bases of M , generalizing the notion of spanning tree for graphs, and a set G ( M )of cycle-cocycle equivalence classes generalizing the corresponding set for graphs.In Section 4.3 of his Ph.D. thesis, Criel Merino defined the critical group (whichwe will call the Jacobian ) Jac( M ) of M , generalizing the critical group of a graph.By results of Merino and Gioan, the cardinalities of Jac( M ), B ( M ), and G ( M )all coincide .Generalizing the known case of graphs [4], we prove: Theorem 1.2.1. G ( M ) is canonically a torsor for Jac( M ).In view of this result, in order to construct a bijection between elements ofJac( M ) and bases of M , it suffices to give a bijection between B ( M ) and G ( M ):if we fix an arbitrary element of G ( M ) (e.g. by fixing a reference orientation of M ), our torsor induces a bijection between Jac( M ) and G ( M ). One can gen-eralize the notion of acyclic signature and fundamental cycles (resp. cuts) in astraightforward way from graphs to regular matroids. Theorem 1.1.2 then admitsthe following generalization to regular matroids: Theorem 1.2.2.
Let M be a regular matroid, and fix acyclic signatures σ and σ ∗ of C ( M ) and C ∗ ( M ), respectively. Given a basis B ∈ B ( M ), let O ( B ) be theorientation of M in which we orient each e (cid:54)∈ B according to its orientation in σ ( C ( B, e )) and each e ∈ B according to its orientation in σ ∗ ( C ∗ ( B, e )). Thenthe map B (cid:55)→ [ O ( B )] gives a bijection β : B ( M ) → G ( M ).Most known combinatorial bijections between elements of Jac( G ) and spanningtrees of a graph G do not readily extend to the case of regular matroids, as theyuse vertices of the graph in an essential way. The only other work we are awareof giving explicit bijections between elements of Jac( M ) and bases of a regularmatroid M are the papers of Gioan and Gioan–Las Vergnas [20, 23] and theas-yet unpublished recent work of Shokrieh [33]. Our family of combinatorialbijections appears to be quite different from those of Gioan–Las Vergnas. The fact that these cardinalities are equal is essentially a translation of the natural extensionof Kirchhoff’s Matrix-Tree theorem to regular matroids [27],[28, Theorem 4.3.2]. A “volumeproof” of the Matrix-Tree theorem for regular matroids based on zonotopal subdivisions is givenin [16]. These authors do not consider the problem of giving explicit combinatorial bijectionsbetween bases of M and the Jacobian group. Technically speaking, Gioan and Las Vergnas do not produce a bijection between bases andelements of Jac( M ); they produce a bijection between B ( M ) and X ( M ; σ, σ ∗ ), where σ and σ ∗ are determined by a total order on the edges and a reference orientation as in Example 1.1.1;see § X ( M ; σ, σ ∗ ). ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 5
Brief overview of the proof of the main combinatorial bijections.
Although the statement of Theorem 1.1.2 and its generalization Theorem 1.2.2to regular matroids M are completely combinatorial, we do not know any simplecombinatorial proof. Our proof involves the geometry of a zonotopal subdivisionassociated to a matrix A representing M .Concretely, fix a totally unimodular r × m matrix A representing M , where r is the rank of A , and the columns of A are indexed by a set E of cardinality m .Denote by V ∗ ⊆ R E the row space of A and by π V ∗ the orthogonal projectionfrom R E to V ∗ . Let u e ∈ R E be the standard coordinate vector corresponding to e ∈ E . The column zonotope Z A ⊂ R r (resp. row zonotope (cid:102) Z A ⊂ R E ) associatedto A is defined to be the Minkowski sum of the columns of A (resp. the Minkowskisum of the orthogonal projections π V ∗ ( u e ) for e ∈ E ). The linear transformation L : v (cid:55)→ Av gives an isomorphism from V ∗ to R r taking (cid:102) Z A to Z A and preservinglattice points (cf. Lemma 3.5.1). The reason that we introduce two versions ofessentially the same object is mostly for the sake of notational convenience.An orientation O of M is a function E → {− , } . An orientation O is compatible with a signed circuit C of M if O ( e ) = C ( e ) for all e in the supportof C .If O is an orientation and C is a signed circuit compatible with O , we canperform a circuit reversal taking O to the orientation O (cid:48) defined by O (cid:48) ( e ) = −O ( e ) if e is in the support of C and O (cid:48) ( e ) = O ( e ) otherwise.Let σ be an acyclic signature of C ( M ). We say that O is σ -compatible ifevery signed circuit C of M compatible with O is oriented according to σ . ByProposition 4.1.4, every circuit-reversal equivalence class of orientations containsa unique σ -compatible orientation.The connection between σ -compatible orientations and the zonotopes definedabove is given by the following result. For the statement, given an orientation O of M and e ∈ E , define w e ∈ R r to be 0 if O ( e ) = − e th columnof A if O ( e ) = 1. Define ψ ( O ) ∈ Z A by(1.1) ψ ( O ) := (cid:88) e ∈ E w e ∈ Z A . Then the map ψ induces a bijection between circuit-reversal classes of orientationsof M and lattice points of the zonotope Z A (cf. Proposition 4.1.3).Fix a reference orientation O of M . Each acyclic signature σ of C ( M ) givesrise to a subdivision of Z A into smaller zonotopes Z ( B ), one for each basis B of M , in the following way.Let B be a basis of M . For each e (cid:54)∈ B , define v e ∈ V ∗ to be 0 if the referenceorientation of e coincides with the orientation of e in σ ( C ( B, e )), and to be the
SPENCER BACKMAN, MATTHEW BAKER, CHI HO YUEN e th column of A otherwise. Define Z ( B ) := (cid:88) e ∈ B [0 , A e ] + (cid:88) e (cid:54)∈ B v e ⊆ Z A ⊂ R r . By Proposition 3.4.1, the collection of Z ( B )’s gives a zonotopal subdivision (also known in the literature as a tiling ) Σ of Z A . Similarly, via the map L , thevarious (cid:93) Z ( B ) := L − ( Z ( B ))’s give a zonotopal subdivision (cid:101) Σ of (cid:102) Z A .We now explain briefly how these results are used to prove Theorem 1.2.2.Let σ, σ ∗ be acyclic signatures of C ( M ) and C ∗ ( M ), respectively. An orienta-tion is called ( σ, σ ∗ ) -compatible if it is both σ -compatible and σ ∗ -compatible, andwe denote the set of such orientations by X ( M ; σ, σ ∗ ). Theorem 1.3.1.
Let ˆ β be the map which sends a basis B to the orientation O ( B ) defined in Theorem 1.2.2. Let χ be the map which sends an orientation O to its circuit-cocircuit reversal class [ O ], so that β = χ ◦ ˆ β .(1) The image of ˆ β is contained in X ( M ; σ, σ ∗ ), and ˆ β gives a bijection be-tween B ( M ) and X ( M ; σ, σ ∗ ).(2) The map χ restricted to X ( M ; σ, σ ∗ ) induces a bijection between X ( M ; σ, σ ∗ )and G ( M ). Remark 1.3.2.
The proofs of Theorem 1.2.1 and Theorem 1.2.2 do not assume a priori that | B ( M ) | = |X ( M ; σ, σ ∗ ) | = |G ( M ) | = | Jac( M ) | for a regular matroid M , thus our work provides an independent proof of these equalities. Furthermore,we will show in Theorem 1.4.1 below that the equality | B ( M ) | = |X ( M ; σ, σ ∗ ) | continues to hold under the weaker assumption that M is realizable over R .By Lemma 3.1.1, we may choose a vector w (cid:48) ∈ V ∗ which is compatible with σ ∗ , in the sense that w (cid:48) · σ ∗ ( C ) > C of M . Note that thezonotopal subdivision (cid:101) Σ of (cid:102) Z A depends only on σ (and the reference orientation O ) and the vector w (cid:48) depends only on σ ∗ .The following theorem shows that the combinatorially defined map ˆ β : B ( M ) →X ( M ) can be interpreted geometrically as first identifying a basis with a maximalcell in our zonotopal subdivision and then applying a “shifting map”. Theorem 1.3.3. (1) Let B be a basis of M . For all sufficiently small (cid:15) > (cid:93) Z ( B ) under the map v (cid:55)→ v + (cid:15)w (cid:48) contains a unique latticepoint (cid:102) z B of (cid:102) Z A , which corresponds to a unique ( σ, σ ∗ )-compatible discreteorientation O (cid:48) B .(2) The map φ which takes each basis B to the orientation O (cid:48) B coincides withthe map ˆ β appearing in the statement of Theorem 1.2.2, and hence ˆ β gives a bijection between B ( M ) and X ( M ; σ, σ ∗ ). ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 7
Theorem 1.2.2 is a simple consequence of Theorem 1.3.1 and Theorem 1.3.3.
Example 1.3.4.
Let G be a graph, and fix a vertex q of G . In [1], the authorsprove that the break divisors of G are the divisors associated to q -connectedorientations offset by a chip at q . In other words (in the notation of [1, Lemma3.3]), a divisor D is a break divisor if and only if D = ( q )+ ν O for some q -connectedorientation O . They also show that break divisors of the corresponding metricgraph Γ induce a canonical subdivision of the g -dimensional torus Pic g (Γ) intoparalleletopes indexed by spanning trees of G , with the vertices of the subdivisioncorresponding to the break divisors of G . By applying a small generic shift tothe vertices, this yields a family of “geometric bijections” between break divisorsand spanning trees (cf. [1, Remark 4.26]).We claim that the geometric bijections defined in [1] can be thought of asspecial cases of the bijections afforded by Theorem 1.2.2. By [38, Theorem 10],each such geometric bijection gives rise in a natural way to an acyclic orientation σ of the cycles of G . On the other hand, we can use the recipe described inExample 1.1.3 to produce a cut signature σ ∗ such that every cut is oriented awayfrom q . Given a spanning tree T , the orientation O T associated to the pair ( σ, σ ∗ )by Theorem 1.2.2 will have the property that every edge e in T (considered asa tree rooted at q ) is oriented away from q , and therefore O T is q -connected[3, Section 3]. Let D T = ν O T + ( q ) be the corresponding break divisor. Then T (cid:55)→ D T will be the geometric bijection we started with.1.4. A partial extension to matroids realizable over R . Although theequality | B ( M ) | = |G ( M ) | = | Jac( M ) | does not hold for general oriented ma-troids (indeed, Jac( M ) is not even well-defined in the general case), the notions ofacyclic circuit/cocircuit signatures and ( σ, σ ∗ )-compatible orientations continueto make sense whenever M is realizable over R . Furthermore, the geometric setupused to prove Theorem 1.3.3, as well as the first half of Theorem 1.3.1, does notrequire M to be regular but only realizable. Therefore we have the followingresult, which will be proved in § Theorem 1.4.1.
Let M be an oriented matroid which is realizable over R , andlet σ, σ ∗ be acyclic signatures of C ( M ) , C ∗ ( M ), respectively. Then the map ˆ β : B ( M ) → X ( M ; σ, σ ∗ ) is a bijection.An ingredient in the proof of Theorem 1.4.1 is a continuous analogue of orien-tations (which we will refer to as discrete orientations whenever there is a risk ofconfusion), hence continuous analogues of acyclic signatures and circuit-cocircuitreversal systems. Using these notions, we can provide a combinatorial interpreta-tion of all points of the zonotope Z A (not just the lattice points), thereby givingan alternate description of the zonotopal subdivision Σ (resp. (cid:101) Σ) of Z A (resp. (cid:102) Z A ), which was defined above. SPENCER BACKMAN, MATTHEW BAKER, CHI HO YUEN
Random sampling of bases.
As in [7], any computable bijection betweenbases and elements of Jac( M ) gives rise an algorithm for randomly samplingbases of M . The idea is simple: it is easy to uniformly sample random elementsfrom Jac( M ), and applying the bijection yields a random spanning tree.In order to make this into a practical method, one needs efficient algorithmsfor computing the basis associated to an element of Jac( M ). In § M ) to B ( M ) has a strong combinatorial flavor, ourinverse algorithm uses ideas from linear programming.We also remark that there are other algorithms for sampling random bases ofa regular matroid, such as the random walk based method by Dyer and Frieze[18] (whose analysis also makes use of zonotopes). Since our algorithm requiressolving multiple linear programs, its runtime is probably slower than some otherknown algorithms. However, our method can generate an exact uniform distri-bution using an information theoretical minimum amount of randomness, cf. thediscussion by Lipton [26].1.6. Connections to Ehrhart theory and the Tutte polynomial.
Everymatroid M of rank r has an associated Tutte polynomial T M ( x, y ), and everylattice polytope P (e.g. the zonotope Z A ) has an associated Ehrhart polynomial E P ( q ) which counts the number of lattice points in positive integer dilates of P . Using the relationship between Z A and σ -compatible (discrete or continuous)orientations of M , we obtain a new proof of the following identity originally dueto Stanley:(1.2) E Z ( q ) = q r T M (1 + 1 /q, . The proof involves defining a “dilation” qM of M for each positive integer q , withassociated zonotope qZ A .We also describe a direct bijective proof (without appealing to Ehrhart reci-procity) of the fact that the number of interior lattice points in qZ A is q r T M (1 − /q, . Related literature.
The study of zonotopal tilings, i.e. tilings of a zono-tope by smaller zonotopes, is a classical topic in the theory of oriented matroidsfirst initiated by Shephard [32]. The central theorem in this area is the Bohne-Dress Theorem [11, 17], which states that the poset of zonotopal tilings orderedby refinement is isomorphic to the poset of single-element lifts of the associatedoriented matroid. For instance, it can be shown that the subdivisions we considerin the paper correspond to precisely the generic, realizable single-element lifts ofrealizable oriented matroids; we will not further elaborate on this connection asit does not play a significant role in this paper.
ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 9 Background
Regular matroids.
In this section we recall the definition of regular ma-troids and related objects. We assume that the reader is familiar with the basictheory of matroids; some standard references include the book on matroids byOxley [29] and the book on oriented matroids by Bj¨orner et al. [10].An r × m matrix A of rank r with integer entries is called totally unimodular if every k × k submatrix has determinant in { , ± } for all 1 ≤ k ≤ r . A matroidis called regular if it is representable over Q by a totally unimodular matrix.The following lemma (see [35]) is the key fact used to show that various defini-tions in the subject are independent of the choice of a totally unimodular matrix A representing M : Lemma 2.1.1. If A, A (cid:48) are totally unimodular r × m matrices representing M ,one can transform A into A (cid:48) by multiplying on the left by an r × r unimodularmatrix U , then permuting columns or multiplying columns by − M is a regular matroid of rank r on E and A is any r × m totally unimodularmatrix representing M over Q , we define Λ A ( M ) := ker( A ) ∩ Z E . By Lemma 2.1.1,the isometry class of this lattice depends only on M , and not on the choice ofthe matrix A . It is denoted by Λ( M ) and called the circuit lattice of M .Similarly, we define Λ ∗ A ( M ) to be the intersection of the row space of A with Z E , or equivalently the Z -span of the rows of A . The isometry class of this latticealso depends only on M . It is denoted by Λ ∗ ( M ) and called the cocircuit lattice of M . (For proofs of all these statements, see [28, § § M ) is defined to be the determinant group of Λ( M ),i.e., Jac( M ) = Λ( M ) / Λ( M ) where Λ is the dual lattice of Λ, i.e.,Λ = { x ∈ Λ ⊗ Q : (cid:104) x, y (cid:105) ∈ Z ∀ y ∈ Λ } . There are canonical isomorphisms(2.1) Λ( M ) / Λ( M ) ∼ = Λ ∗ ( M ) / Λ ∗ ( M ) ∼ = Z E Λ A ( M ) ⊕ Λ ∗ A ( M )for every totally unimodular matrix A representing M (cf. [2, Lemma 1 of § M ) is equal to the number of bases of the matroid M (cf. [28,Theorem 4.3.2]). Moreover, we have | Jac( M ) | = | det( A T A ) | (cf. [24, p.317]),and in fact Jac( M ) can naturally be identified with the cokernel of A T A : Proposition 2.1.2.
The map Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) → coker( AA T ) given by [ γ ] (cid:55)→ [ Aγ ]is well-defined and is an isomorphism. Proof.
The map is well-defined because A (Λ A ( M ) ⊕ Λ ∗ A ( M )) = A (Λ ∗ A ( M )) = A (Col Z A T ) = Col Z AA T , the equality also shows the map is injective. It is surjective because Ax = b has a solution in Z E for every b ∈ Z r , using theunimodularity of A . (cid:3) We now discuss regular matroids from an oriented matroid point of view. By[10, Corollary 7.9.4], every oriented matroid structure on a regular matroid isrealizable by some totally unimodular matrix, hence any two such structuresdiffer by reorientations.Let C ( M ) (resp. C ∗ ( M )) be the set of circuits (resp. cocircuits) of M . Let A be any r × m totally unimodular matrix representing M over Q . An element α ∈ Λ A ( M ) (resp. Λ ∗ A ( M )) is called a signed circuit (resp. signed cocircuit ) of M if α (cid:54) = 0, all coordinates of α are in { , ± } , and the support of α is a circuit(resp. cocircuit) of M . We let C A ( M ) (resp. C ∗ A ( M )) denote the set of signedcircuits (resp. signed cocircuits) of M . The notion of signed circuit (resp. signedcocircuit) is in fact intrinsic to M , independent of the choice of A , and thusit makes sense to speak of C ( M ) and C ∗ ( M ) as subsets of Λ( M ) and Λ ∗ ( M ),respectively (cf. [35, Lemma 10 and Proposition 12] and [28, Theorem 4.3.4]).There is a natural map C ( M ) → C ( M ) taking α ∈ C A ( M ) to its support (withrespect to any choice of A ). This map induces a bijection C ( M ) / (cid:104)± (cid:105) → C ( M ),i.e, for every circuit C of M there are precisely two signed circuits ± C withsupp( C ) = C . (cf. [35, Lemma 8] and [28, Theorem 4.3.5]) The same holds forcocircuits.2.2. Equivalence classes of orientations, and signatures. An (discrete) ori-entation of a regular matroid M is a map from the ground set E of M to {− , } .An orientation O is compatible with a signed circuit C of M if O ( e ) = C ( e ) forall e in the support of C .The circuit reversal system is the equivalence relation on the set O ( M ) of allorientations of M generated by circuit reversals , in which we reverse the sign of O ( e ) for all e in (the support of) some signed circuit C compatible with O . Wecan make the same definitions for cocircuits by replacing M with its dual.The circuit-cocircuit reversal system is the equivalence relation generated byboth circuit and cocircuit reversals. It is a theorem of Gioan [22, Theorem10(v)](originally proved by a deletion-contraction argument) that the numberof circuit-cocircuit equivalence classes of orientations is equal to the number ofbases of M ; Theorem 1.2.1 gives a bijective proof of this fact.A signature of C ( M ) is a map σ : C ( M ) → C ( M ) that picks an orientationfor each (unsigned) circuit of the matroid underlying M .A signature σ of C ( M ) is called acyclic if the only solution to (cid:80) C i ∈ C ( M ) a i σ ( C i ) =0 with the a i non-negative numbers is the trivial solution where all a i are equalto zero. Signatures for C ∗ ( M ) are defined analogously. ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 11 Matroids over R and the main combinatorial bijection Throughout this section, M will denote an oriented matroid which is realizableover R . Note in particular that every regular matroid has this property.3.1. Continuous circuit reversals and the zonotope associated to a rep-resentation of M . The main goal of this section is to prove Theorem 1.4.1,which (when specialized to regular matroids) is a major component in the proofof Theorem 1.2.2.Our proof is geometric. In order to explain the basic idea, we fix once and forall a real r × m matrix A realizing M , where r is the rank of M and the columnsof A are indexed by the elements of the ground set E of M (which we sometimesregard as { , , . . . , m } ).We first briefly explain how certain important notions that we introduced forregular matroids extend more generally to matroids representable over R .For every circuit C of M , the elements in ker( A ) whose support is C , togetherwith the zero vector, form a one-dimensional subspace U C in ker( A ). Conversely,the support of any support-minimal nonzero element of ker( A ) corresponds to acircuit of M . The two rays of U C correspond to the two orientations of C . Hencewe may identify a signed circuit C with an arbitrary vector v C in the ray. Thesame holds for cocircuits if we replace ker( A ) by the row space of A .The definition of an acyclic circuit (resp. cocircuit) signature follows verbatimfrom the discussion in § (cid:80) C i ∈ C ( M ) a i v σ ( C i ) =0, which is well-defined as different choices of v σ ( C i ) differ by a positive scalar mul-tiple.As a simple consequence of Gordan’s alternative in the theory of linear pro-gramming [10, p. 478], we have the following criterion/alternative description ofan acyclic signature.
Lemma 3.1.1.
Let σ be a signature of C ( M ). Then σ is acyclic if and only ifthere exists w ∈ R E such that w · v σ ( C ) > C of M .In the situation of Lemma 3.1.1, we say that w induces σ . By the orthogonalityof vectors representing signed circuits and cocircuits, given any pair of acyclicsignatures σ, σ ∗ of C ( M ) and C ∗ ( M ), respectively, there exists w ∈ R E thatinduces both σ and σ ∗ .We state an elementary lemma for realizable oriented matroids. In abstractoriented matroid terms, it is to say that every signed vector of an oriented matroidis a conformal composition of signed circuits. Lemma 3.1.2. [39, Lemma 6.7] Let u ∈ R E be a vector in ker( A ). Then u canbe written as a sum of signed circuits (cid:80) v C where the support of each C is inside the support of u , and for each e in the support of C , the signs of e in C and u agree.A continuous orientation O of M is a function E → [ − , e -thcoordinate of O is the value O ( e ). If O ( e ) ∈ {− , } for all e ∈ M , we say that O is a discrete orientation .A continuous orientation O is compatible with a signed circuit C of M if O ( e ) (cid:54) = − sign( C ( e )) for all e in the support of C . Given a continuous orientation O compatible with a signed circuit C , a continuous circuit reversal with respectto C replaces O by a new continuous orientation O − (cid:15) v C for some (cid:15) >
0. (Inparticular, we require (cid:15) to be small enough so that (
O − (cid:15) v C )( e ) ∈ [ − ,
1] for all e ∈ E .)The continuous circuit reversal system is the equivalence relation on the set CO ( M ) of all continuous orientations of M generated by all possible continuouscircuit reversals. We can make the same definitions for cocircuits by replacing M with its dual.Next we define the (column) zonotope Z A associated to A to be the Minkowskisum of the columns of A (thought of as line segments in R r ), i.e., Z A = { m (cid:88) i =1 c i v i : 0 ≤ c i ≤ } where v , . . . , v r are the columns of A . Remark 3.1.3.
When M = M ( G ) is a graphic matroid, it is usually more con-venient to take A to be the full adjacency matrix of G , rather than a modifiedadjacency matrix with one row removed, when defining the corresponding zono-tope. This has the advantage of producing a canonically defined object, and sinceall of these different zonotopes are isomorphic, there is little harm in doing this.There are several important connections between the zonotope Z A and equiva-lence classes of orientations of M . For the statement, given α ∈ [ − ,
1] we denoteby ˆ α the real number ( α + 1) ∈ [0 , ψ : O ( M ) → Z A be the map takingan orientation O (thought of as an element of [ − , E ) to(3.1) ψ ( O ) := m (cid:88) i =1 (cid:91) O ( e i ) v i ∈ Z A . Proposition 3.1.4.
The map ψ gives a bijection between continuous circuit-reversal classes of continuous orientations of M and points of the zonotope Z A . Some authors consider variations on this zonotope, e.g. (cid:80) mi =1 [ − v i , v i ], (cid:80) mi =1 [ − v i / , v i / (cid:80) mi =1 [ v − i , v + i ], where v − and v + are the negative and positive parts of v , respectively. ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 13
Proof.
By definition, ψ sends every continuous orientation to some point in Z A ,and ψ is surjective. By the orthogonality of circuits and cocircuits, two continuousorientations in the same circuit-reversal class map to the same point of Z A , soit remains to show the converse. Suppose ψ ( O ) = ψ ( O (cid:48) ). By Lemma 3.1.2, O − O (cid:48) can be written as a sum of signed circuits in which each signed circuitis compatible with O , and O can be transformed to O (cid:48) via the correspondingcontinuous circuit reversals in any order. (cid:3) Distinguished orientations within each equivalence class.
If we fixan acyclic signature σ of C ( M ), there is a natural way to pick out a distinguishedcontinuous orientation from each continuous circuit reversal class.Define a continuous orientation O to be σ -compatible if every signed circuit C of M compatible with O is oriented according to σ . Proposition 3.2.1.
Let σ be an acyclic signature of C ( M ). Then each contin-uous circuit-reversal class M contains a unique σ -compatible continuous orienta-tion. Proof.
By Lemma 3.1.1, there exists w ∈ R E such that w · v σ ( C ) > C of M . Consider the function P ( O (cid:48) ) := w · O (cid:48) . If − σ ( C ) is compatiblewith O for some circuit C , then performing a continuous circuit reversal with − σ ( C ) strictly increases the value of P , so every maximizer of P inside a class (ifexists) must be σ -compatible. The set of continuous orientations in a continuouscircuit reversal class is the fiber of ψ over a point in Z A , which is a closed subsetof the hypercube, so such maximizer must exist as P is continuous.For uniqueness, suppose there are two distinct σ -compatible continuous ori-entations O , O (cid:48) in a continuous circuit-reversal class. By Lemma 3.1.2, O canbe transformed to O (cid:48) via a series of continuous circuit-reversals in which eachsigned circuit involved is compatible with O , hence agrees with σ . If the lastsigned circuit involved in the series of reversals is C , then − C is a signed circuitcompatible with O (cid:48) , hence it agrees with σ as well, which is a contradiction. (cid:3) Remark 3.2.2.
By interpreting σ -compatible orientations as maximizers of thelinear function P , it is easy to see that the map µ : Z A → CO ( M ), which takes apoint z of Z A to the unique σ -compatible continuous orientation in the continuouscircuit-reversal class corresponding to z , is a continuous section to the map ψ .Such a point of view is closely related to the classical theory of zonotopal tilings.We will call orientations that are compatible with both σ and σ ∗ ( σ, σ ∗ ) -compatible orientations .The set of discrete ( σ, σ ∗ )-compatible orientations will be denoted by X ( M ; σ, σ ∗ ).In §
4, we will establish an analogue of Proposition 3.2.1 for discrete orientationsof regular matroids.
Bi-orientations and bases.
Let O be a continuous orientation of M . Wecall an element e ∈ E bi-oriented with respect to O if O ( e ) ∈ ( − , orient any bi-oriented element e in a σ -compatible continuousorientation O , i.e., we set O ( e ) equal to either 1 or −
1, the new continuousorientation is still σ -compatible. Proposition 3.3.1.
Let σ be an acyclic signature of C ( M ).(1) If O is a σ -compatible continuous orientation then the set of e ∈ E whichare bi-oriented with respect to O is independent in M .(2) If B is a basis for M and b : B → ( − ,
1) is any function, there is a unique σ -compatible continuous orientation O = O ( B, b ) such that O ( e ) = b ( e )for all e ∈ B and O ( e ) ∈ {± } for all e (cid:54)∈ B . Proof.
For the first part, suppose the set S of bi-oriented elements in a continuousorientation O is not independent. Then S contains some circuit C , and O iscompatible with both orientations of C , so O is not σ -compatible.For the uniqueness assertion in (2), note that each element not in B must beoriented in agreement with the orientation of its fundamental circuit given by σ , as for otherwise the fundamental circuit will be compatible with − σ . Suchunique choice of orientations outside B , together with b itself, gives a continuousorientation O .Now we claim that such O is σ -compatible. If not, then O is compatible with − σ ( C ) for some circuit C . We choose C such that | C \ B | (cid:54) = 0 is minimum amongall such circuits. Pick any e ∈ C \ B and let C (cid:48) be its fundamental circuit withrespect to B . Then O is compatible with σ ( C (cid:48) ) by construction. Pick suitable v − σ ( C ) , v σ ( C (cid:48) ) such that they agree on the e -th coordinate. Using Lemma 3.1.2, wewrite v − σ ( C ) − v σ ( C (cid:48) ) = (cid:80) D v D with D ’s being signed circuits conformal with theleft hand side, hence they do not contain e . Since v σ ( C ) + v σ ( C (cid:48) ) + (cid:80) D v D = 0,at least one such D is oriented opposite to σ by acyclicity. Then such D iscompatible with O : each element of D is either in B (which is bi-oriented in O ),or from C and oriented as in − σ ( C ) (which is compatible with O ). However, D \ B ⊂ ( C \ B ) \ e , contradicting the minimality of C . (cid:3) Polyhedral subdivision of the zonotope.
Let σ be an acyclic signatureof C ( M ). For each basis B of M , let CO ◦ ( B ) be the set of σ -compatible continu-ous orientations of the form O ( B, b ) as b ranges over all possible b : B → ( − , Z ◦ ( B ) = ψ ( CO ◦ ( B )) be the projection of CO ◦ ( B ) to Z A , and let Z ( B ) bethe topological closure of Z ◦ ( B ) in Z A . Finally, let CO ( B ) = µ ( Z ( B )) be theclosure of CO ◦ ( B ) in CO ( M ). Proposition 3.4.1. (1) The union of Z ( B ) over all bases B of M is equal to Z A , and if B, B (cid:48) are distinct bases then Z ◦ ( B ) and Z ◦ ( B (cid:48) ) are disjoint. ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 15 (2) The collection of Z ( B ) as B varies over all bases B for M gives a poly-hedral subdivision of Z A whose vertices (i.e., 0-cells) correspond via ψ tothe σ -compatible discrete orientations of M . Proof.
The only non-trivial part of (1) is the first half. By Proposition 3.1.4 and3.2.1, every point of Z A is of the form ψ ( O ) for some σ -compatible continuousorientation O . Hence by Proposition 3.3.1, it suffices to show that if the set ˆ B of bi-oriented elements in O do not form a basis, then we can bi-orient someelement in O while maintaining σ -compatiblility; by induction, we will end upwith a bi-oriented basis B , which implies that ψ ( O ) is a limit point of Z ◦ ( B ).Suppose that for every e (cid:54)∈ ˆ B such that ˆ B ∪ { e } is independent in M , bi-orienting e in O will cause the new continuous orientation O e to no longer be σ -compatible. Then every such O e is compatible with − σ ( C e ) for some circuit C e containing e . Pick, among all such elements e and circuits C e , the pair thatmaximizes w · v σ ( C e ) , where we always choose the normalized v σ ( C e ) whose e -thcoordinate is σ ( C e )( e ). The circuit C e must contain another element f (cid:54)∈ ˆ B suchthat ˆ B ∪ { f } is independent in M , so there exists some circuit C f containing f such that O f is compatible with − σ ( C f ). The signs of σ ( C e ) and σ ( C f ) over f are different, so we can choose a suitable positive multiple v f of v σ ( C f ) such thatthe f -th coordinates of v f and v − σ ( C e ) are equal.By Lemma 3.1.2, v − σ ( C e ) − v f can be written as a sum (cid:80) ki =1 v C i of signedcircuits. Each such signed circuit C i that does not contain e must be compatiblewith O (hence σ ), while those signed circuits that contain e would at least becompatible with O e . Since w · ( (cid:80) ki =1 v C i ) = w · ( v − σ ( C e ) − v f ) <
0, some C i is notcompatible with σ (hence O ), thus they contain e . In particular, the sign of the e -th coordinate of v − σ ( C e ) − v f agrees with − σ ( C e ). But as the signs of σ ( C e )( e )and σ ( C f )( e ) are different, the absolute value of the e -th coordinate of v f is atmost the absolute value of the e -th coordinate of v − σ ( C e ) , which is 1.Without loss of generality, the circuits containing e are C , C , . . . , C j . Wechoose v C i ’s so that their e -th coordinates equal − σ ( C e )( e ), and rewrite v − σ ( C e ) − v f as (cid:80) ki =1 λ i v C i for some λ i >
0. By comparing e -th coordinate, (cid:80) ji =1 λ i ≤ w · ( j (cid:88) i =1 λ i v C i ) = w · (cid:32) v − σ ( C e ) − v f − k (cid:88) i = j +1 λ i v C i (cid:33) < − w · v σ ( C e ) < , i.e., there exists some C i with i ≤ j that is compatible with O e , disagrees with σ , and w · v σ ( C i ) > w · v σ ( C e ) , contradicting our choice of C e .For (2), Z ◦ ( B ) can be identified, up to an affine linear transformation of fullrank, with the open paralleletope (0 , B , where the e -th coordinate of ψ ( O ) ∈ Z ◦ ( B ) is the value (cid:91) O ( e ). Thus Z ( B ) can be identified with the paralleletope [0 , B in an analogous manner, and restricting to a face of Z ( B ) of codimension i can be described as orienting i elements in B . This gives a combinatorialdescription of each face as the combinatorial type of any σ -compatible orientationin its relative interior, and conversely, every combinatorial type of σ -compatibleorientation determines a unique face. Hence if the relative interiors of two facesintersect, then the two faces must be equal, showing that the collection of Z ( B )’sgives a polyhedral subdivision of Z A . (cid:3) Remark 3.4.2.
We give a description of the incidence relation between cells inthe polyhedral subdivision; we will not give a proof as we will not make use ofit. Let B be a basis, and let F e be a facet of Z ( B ) corresponding to orientingsome e ∈ B . Let O be a continuous orientation obtained from orienting e in anycontinuous orientation of the form O ( B, b ). Then either (1) C ∗ ( B, e ) is a positivecocircuit in O , in which case F e lies on the boundary of Z A , or (2) there exists aunique element f ∈ C ∗ ( B, e ) \ e such that the orientation obtained by reversing f in O is also σ -compatible, in which case F e is a facet of Z (( B \ { e } ) ∪ { f } ).
1" 2"3"Edge"order" Reference"orienta2on"
Figure 1.
The subdivision of the zonotope associated to K as de-scribed in Proposition 3.4.1 using σ induced by the total order andreference orientation on the right as described in Example 1.1.1.The red edges are bi-oriented.3.5. Geometric interpretation of the combinatorial map.
Let σ, σ ∗ beacyclic signatures of C ( M ) and C ∗ ( M ), respectively. By Lemma 3.1.1, there ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 17 exists w ∈ R E that induces both σ and σ ∗ . Our next goal is to show that thecombinatorially defined basis-to-orientation map ˆ β (whose definition depends on σ and σ ∗ ) can be interpreted geometrically as a “shifting map”.To present the calculation in our proof more clearly, for the rest of §
3, we willwork in the cocircuit space V ∗ ( M ) of M , which is the R -span of C ∗ ( M ) (and isequal to the row space of A ). Let π V ∗ ( M ) be the orthogonal projection from R E onto V ∗ ( M ) and let { u e : e ∈ E } be the standard basis for R E . Consider the(row) zonotope (cid:102) Z A := { (cid:80) e ∈ E c e π V ∗ ( u e ) : 0 ≤ c e ≤ } ⊂ V ∗ ( M ). The followinglemma shows that (cid:102) Z A and the previously defined zonotope Z A are essentiallyequal: Lemma 3.5.1.
The map L : v (cid:55)→ Av is a lattice points preserving isomorphismbetween V ∗ ( M ) and R r taking (cid:102) Z A to Z A . Proof.
Since AA T has full rank, V ∗ ( M ) = { A T z : z ∈ R r } is isomorphic to R r via L . By simple linear algebra, we have L ( π V ∗ ( u e )) = L ( A T ( AA T ) − Au e ) = A e .Thus L ( (cid:80) e ∈ E c e π V ∗ ( u e )) = (cid:80) e ∈ E c e A e and L ( (cid:102) Z A ) = Z A . L preserves latticepoints because A is totally unimodular. (cid:3) In particular, the subdivision of Z A constructed in § (cid:102) Z A . We denote by (cid:93) Z ( B ) the cell L − ( Z ( B )) in (cid:102) Z A .The key to defining the shifting map is the following lemma: Lemma 3.5.2. If w (cid:48) is the orthogonal projection of w onto V ∗ ( M ), then for allsufficiently small (cid:15) > (cid:93) Z ( B ) under the map v (cid:55)→ v + (cid:15)w (cid:48) containsa unique point corresponding (via ψ ) to a σ -compatible discrete orientation O B . Proof.
By Proposition 3.4.1, the vertices of each (cid:93) Z ( B ) correspond to σ -compatiblediscrete orientations. It therefore suffices to prove that w (cid:48) does not lie in theaffine span of any facet of (cid:93) Z ( B ). The affine span of a facet of (cid:93) Z ( B ) is spannedby directions of the form π V ∗ ( e ) for e ∈ ˆ B where ˆ B (cid:40) B . Since | ˆ B | < r , there isa cocircuit K of M avoiding ˆ B . Any direction v := (cid:80) e ∈ ˆ B λ e π V ∗ ( e ) in the spansatisfies (cid:104) v, v σ ∗ ( K ) (cid:105) = (cid:80) e ∈ ˆ B λ e (cid:104) e, v σ ∗ ( K ) (cid:105) = 0. On the other hand, since w induces σ ∗ , (cid:104) w (cid:48) , v σ ∗ ( K ) (cid:105) = (cid:104) w, v σ ∗ ( K ) (cid:105) > (cid:3) We define φ to be the map that takes a basis B to the orientation O B definedin Lemma 3.5.2. Theorem 3.5.3.
The map φ coincides with the previously defined map ˆ β . Proof.
Let B be a basis. Then φ ( B ) can be obtained by orienting each (bi-oriented) basis element from a continuous σ -compatible orientation in the interiorof (cid:93) Z ( B ) (which is of the form O ( B, b )), so by the greedy procedure described in
Proposition 3.3.1, the elements outside B are oriented according to their funda-mental circuits, hence φ ( B ) agrees with ˆ β ( B ) outside B .For elements inside B , we work with the basis { π V ∗ ( u e ) : e ∈ B } for V ∗ ( M )and write w (cid:48) = (cid:80) e ∈ B w e π V ∗ ( u e ). Identifying (cid:93) Z ( B ) with [0 , B and the vertices of (cid:93) Z ( B ) with { , } B . If a vertex v is identified with ( s e : e ∈ B ), then it correspondsto a σ -compatible discrete orientation where each element e ∈ B is oriented inagreement with (resp. opposite to) its reference orientation when s e = 1 (resp. s e = 0). The cell (cid:93) Z ( B ) will contain v after shifting if and only if the sign patternof the s e ’s agrees with the sign pattern of the w e ’s, i.e., if and only if s e = 1precisely when w e > f ∈ B , and let K be the fundamental cocircuit of f with respect to B . Bya calculation similar to the above,0 < (cid:104) w (cid:48) , v σ ∗ ( K ) (cid:105) = (cid:88) e ∈ B w e (cid:104) u e , v σ ∗ ( K ) (cid:105) = w f (cid:104) u f , v σ ∗ ( K ) (cid:105) , as f is the unique element in B ∩ K . If w f >
0, then (cid:104) u f , v σ ∗ ( K ) (cid:105) > f agrees with σ ∗ ( K ), i.e., the orientation of f in φ ( B ) isthe same as the reference orientation of f . From the last paragraph, f is orientedaccording to its reference orientation in ˆ β ( B ) as well, because w f >
0. A similaranalysis in the case where w f < φ ( B )( f ) = ˆ β ( B )( f ). (cid:3) Proposition 3.5.4.
Let B be a basis. Then ˆ β ( B ) is ( σ, σ ∗ )-compatible. Proof.
Since φ ( B ) is σ -compatible, ˆ β ( B ) is also σ -compatible by Theorem 3.5.3.And since the procedure described in Theorem 1.2.2 is symmetric with respect tocircuits and cocircuits, a dual argument shows that ˆ β ( B ) is σ ∗ -compatible. (cid:3) Theorem 3.5.5 (Theorem 1.4.1) . The map ˆ β : B ( M ) → X ( M ) is a bijection. Proof. ˆ β = φ is injective for the simple geometric reason that a vertex can onlybe contained in the interior of at most one cell (cid:93) Z ( B ) after shifting. To prove thesurjectivity part, we need to show that for every ( σ, σ ∗ )-compatible orientation O , there exists a continuous orientation O (cid:48) such that the displacement from O (cid:48) to O , interpreted as points of (cid:102) Z A , is π V ∗ ( w ) (here we assume w is sufficiently short).For simplicity, we negate suitable columns of A in order to assume without loss ofgenerality that O ≡ , and we modify w accordingly. For such to be determined O (cid:48) , denote by f e ≥ (cid:91) O ( e ) = 1 and (cid:91) O (cid:48) ( e ). By simple linearalgebra, our condition on O (cid:48) in terms of displacement becomes A f = Aw , hence O (cid:48) exists if and only if the linear program(3.2) min { T f : A f = Aw, f ≥ } ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 19 is feasible. But the σ ∗ -compatible condition implies “if z T A ≥
0, then ( z T A ) w ≥ z such that z T A ≥ , z T ( Aw ) < f ≥ A f = Aw . (cid:3) Figure 2.
An example of the bijection for K using the pair ( σ, σ ∗ )induced by the total order and reference orientation from Figure 1.3.6. Computability of the inverse map.
We now describe an inverse algo-rithm which furnishes an inverse to the map φ , and hence to ˆ β . Again we assumethe inputted ( σ, σ ∗ )-compatible discrete orientation O is equal to for simplic-ity. Suppose O was shifted into the cell (cid:93) Z ( B ) after moving by a displacement of − π V ∗ ( w ). By solving the linear program (3.2) we obtain a continuous orientation O (cid:48) (resp. f ) in the cell Z ( B ). Therefore it remains to find the σ -compatiblecontinuous orientation O (cid:48)(cid:48) equivalent to O (cid:48) , and the desired basis B will then bethe set of bi-oriented elements in O (cid:48)(cid:48) .To do so, we solve the linear program(3.3) max { w T y : A y = 0 , f e − ≤ y e ≤ f e , ∀ e } . Let ˜ y be an optimal solution. Consider the continuous orientation O (cid:48)(cid:48) := O (cid:48) + 2˜ y ,we claim this is the continuous orientation we are looking for. The conditionsin the linear program guarantee that O (cid:48)(cid:48) is a valid continuous orientation circuitreversal equivalent to O (cid:48)(cid:48) , and it is σ -compatible: indeed, if O (cid:48)(cid:48) is compatible with some − σ ( C ), then one can easily check that ˜ y + δ v σ ( C ) is also a feasiblesolution for sufficiently small δ >
0, contradicting the optimality of ˜ y .Since linear programming admits a polynomial-time algorithm [31], the linearprogram (3.3), together with the dual version of it, imply the following: Proposition 3.6.1.
There is a polynomial-time algorithm to compute the unique( σ, σ ∗ )-compatible continuous orientation circuit-cocircuit equivalent to a givencontinuous orientation.Summarizing the discussion, we have the following theorem: Theorem 3.6.2.
There is a polynomial-time algorithm to compute the inverseof ˆ β .4. The discrete circuit-cocircuit reversal system for a regularmatroid and its Jacobian
We now return to the setting of regular matroids. Throughout this section, M will denote a regular matroid on E and A will be a totally unimodular matrixrepresenting M . We will investigate the (original) discrete version of circuit(-cocircuit) reversal system which was introduced by Gioan [21, 22], and showthat the σ -compatible discrete orientations also give distinguished representativesfor this system. Moreover, we will show that discrete circuit-reversal classescorrespond to lattice points of the zonotope Z A (which by Proposition 3.4.1 areprecisely the vertices of the zonotopal subdivision Σ). Finally, we show that thediscrete circuit-cocircuit reversal system is canonically a torsor for Jac( M ).4.1. The discrete circuit-cocircuit reversal system.
For totally unimodularmatrices, we have the following integral version of Lemma 3.1.2:
Lemma 4.1.1.
Let u ∈ Λ A ( M ). Then u can be written as an integral sum ofsigned circuits (as elements of Λ A ( M )) (cid:80) λ C C with λ C >
0, such that each C isconformal to u . In particular, if u is a { , ± } -vector, then the λ C ’s are 1 andthe C ’s are disjoint. Proof.
Without loss of generality, we may assume u ≥
0. We first pick a signedcircuit C conformal to u as in the statement of Lemma 3.1.2. By total unimodu-larity, v C can be chosen as a { , } -vector, and we choose λ C to be the maximumnumber such that u − λ C C ≥
0. In such case λ C must be an integer and thesupport of u − λ C C ∈ Λ A ( M ) is strictly contained in the support of C . Proceedby induction to obtain the desired decomposition. The second assertion followseasily from the first. (cid:3) A discrete orientation O of M is a function E → {− , } . A discrete orientation O is compatible with a signed circuit C of M if O ( e ) (cid:54) = − C ( e ) for all e in thesupport of C . ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 21 If O is a discrete orientation and C is a signed circuit compatible with O , wecan perform a (discrete) circuit reversal taking O to the orientation O (cid:48) definedby O (cid:48) ( e ) = −O ( e ) if e is in the support of C and O (cid:48) ( e ) = O ( e ) otherwise. The discrete circuit reversal system is the equivalence relation on the set CO ( M ) ofall discrete orientations of M generated by all possible discrete circuit reversals.We can make the same definitions for cocircuits by replacing M with its dual.We first state a basic fact about orientations (which is true more general fororiented matroids)[10, Corollary 3.4.6]: Proposition 4.1.2.
Given an orientation O of M and e ∈ E , exactly one of thefollowing holds:(1) There is a signed circuit C of M with e ∈ supp( C ) such that O ( f ) = C ( f )for every f in the support of C . In this case we say that e belongs to the circuit part of O .(2) There is a signed cocircuit C ∗ of M with e ∈ supp( C ∗ ) such that O ( f ) = C ∗ ( f ) for every f in the support of C ∗ . In this case we say that e belongsto the cocircuit part of O . Proposition 4.1.3.
The map ψ from § M modulo discrete circuit reversals and lattice points of Z A . Proof.
As the columns of A are integral, ψ takes an orientation of M to a latticepoint of Z A ; conversely, for any lattice point y ∈ Z A , A ˆ α = y, ≤ ˆ α i ≤ ∀ i has a solution ˆ α , which can be chosen to be integral by the total unimodularityof A , hence it corresponds to an orientation. Thus the image of ψ is preciselythe set of lattice points of Z A . By the orthogonality of circuits and cocircuits,two orientations in the same circuit-reversal class map to the same point of Z A .Conversely, suppose ψ ( O ) = ψ ( O (cid:48) ). By Lemma 4.1.1, O − O (cid:48) can be written asa sum of disjoint signed circuits in which each signed circuit is compatible with O , and O can be transformed to O (cid:48) via the corresponding circuit-reversals in anyorder. (cid:3) Proposition 4.1.4.
Each discrete circuit-reversal class of discrete orientationsof M contains a unique σ -compatible discrete orientation. Proof.
The uniqueness assertion follows from Lemma 4.1.1 and a similar argumentas in Proposition 3.2.1. For existence, start with any orientation O in the classand reverse some signed circuit C compatible with O but not compatible with σ . We claim that the process will eventually stop. Indeed, suppose not: sincethe number of discrete orientations of M is finite, the orientation will withoutloss of generality return to O after reversing some signed circuits C , . . . , C k inthat order (the circuits might not be distinct). Then − C − · · · − C k = 0, whichmeans that σ ( C ) + · · · + σ ( C k ) = 0, contradicting the acyclicity of σ . (cid:3) Corollary 4.1.5.
The lattice points of Z A are exactly the vertices of the subdi-vision Σ. Proof.
This follows from Propositions 4.1.3, 4.1.4 and 3.4.1. (cid:3) ("1,"1,2)'(1,"1,0)' (1,"2,1)' (0,"2,2)'(0,0,0)' ("1,0,1)'(0,"1,1)'
Figure 3.
The zonotope associated to K and the circuit reversalclasses associated to its lattice points by the map ψ from Proposi-tion 3.1.4. Taking the acyclic signature σ from Figure 1, the cyclein blue is σ -compatible, while the cycle in red is not. (Note that weare using the full adjacency matrix of K to define the zonotope,cf. Remark 3.1.3.)Let χ : X ( M ) → G ( M ) be the map which associates to each ( σ, σ ∗ )-compatibleorientation the discrete circuit-cocircuit reversal class which it represents. Theorem 4.1.6 (Part (2) of Theorem 1.3.1) . The map χ is a bijection. Proof.
This follows directly from Propositions 4.1.4 and 4.1.2 (cid:3)
Corollary 4.1.7 (Theorem 1.2.2) . The map β : B ( M ) → G ( M ) given by B (cid:55)→ [ O ( B )] is a bijection. Proof.
The map ˆ β : B (cid:55)→ O ( B ) is a bijection between B ( M ) and X ( M ; σ, σ ∗ ) byTheorem 3.5.5. Now compose this map with χ . (cid:3) ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 23
The circuit-cocircuit reversal system as a
Jac( M ) -torsor. In this sec-tion we will define a natural action of Jac( M ) on the set G ( M ) of circuit-cocircuitequivalence classes of orientations of M and prove that the action is simply tran-sitive. We will also discuss an efficient algorithm for computing this action, alongwith an application to randomly sampling bases of M .4.3. Definition of the action.
Recall from (2.1) that Jac( M ) can be identifiedwith Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) . Note that such group is generated by [ −→ e ] , e ∈ E (here we usean overhead arrow to emphasize that we are keeping track of orientations).The group action Jac( M ) (cid:8) G ( M ) is defined by linearly extending the fol-lowing action of each generator [ −→ e ] on circuit-cocircuit reversal classes: pick anorientation O from the class so that e is oriented as −→ e in O , reverse the orienta-tion of e in O to obtain O (cid:48) , and set [ −→ e ] · [ O ] = [ O (cid:48) ]. This action generalizes theone defined in terms of path reversals by the first author in the graphical case [4,Section 5]. [-f+2e]• [-f+e]• [-f]• [-f+e]• [-f]• [ ] e f [ ] [ ] [ ] [ ] [ ] Figure 4.
Example of the torsor. Here the reference orientationsof e, f are the same as the orientation we begin with.Our main goal for the rest of this section will be to prove:
Theorem 4.3.1.
The group action (cid:8) is well-defined and simply transitive.
Remark 4.3.2.
For ease of exposition, in the rest of this section we will use theterm positive (co)circuit (with respect to an orientation O ) to denote a signed(co)circuit that is compatible with O . Furthermore, given an orientation O and asubset X ⊂ E , we denote by − X O the orientation obtained by reversing elements of X in O . For a (co)circuit C of O , we say that − X C is positive if C is a positive(co)circuit of − X O . Finally, we denote by χ X the { , } -characteristic vectorwhose support is X .4.4. The action is well-defined.
In order to show that the action of Jac( M )on G ( M ) is well-defined, we first show that the corresponding action (which byabuse of notation we continue to write as (cid:8) ) of Z E on G ( M ) is well-defined, thenthat the action descends to the quotient by Λ A ( M ) ⊕ Λ ∗ A ( M ). Lemma 4.4.1.
Let e ∈ E , and suppose X ⊂ E \ e is a positive cocircuit in O \ e but not in O . Then Y := X ∪ { e } is a cocircuit in O , and either Y or − e Y ispositive. Proof.
By assumption, w T A | E \ e = χ X for some w . Hence w T A = χ X + λχ { e } for some λ (cid:54) = 0. By the dual of Proposition 3.1.2, Y contains a cocircuit D .If D ∩ X = ∅ , then D = { e } , which in turn shows that X itself is a positivecocircuit. Now we must have X ⊂ D , or otherwise D ∩ X (cid:40) X would be acocircuit in M \ e . Therefore Y = D is a cocircuit, and λ = ±
1, i.e. either Y or − e Y is positive. (cid:3) Lemma 4.4.2.
Suppose e ∈ M is contained in some positive circuit of O , andthat Y is a subset of E containing e such that − e Y is a positive cocircuit. Thenany positive circuit containing e intersects Y in exactly two elements. Proof.
Let C be a positive circuit containing e . By assumption, there exists avector v such that v T A = χ Y − e − χ { e } . Then 0 = v T Aχ C = | ( Y − e ) ∩ C | − Y intersects C in e together with exactly one more element. (cid:3) Proposition 4.4.3.
For every [ O ] ∈ G ( M ) and oriented element −→ e , there exists˜ O ∈ [ O ] so that e is oriented as −→ e in ˜ O . Proof.
By Proposition 4.1.2, e is either contained in a positive circuit or cocircuit C . If e is not already oriented as −→ e in O , reverse C . (cid:3) Proposition 4.4.4.
The action of −→ e on [ O ] is independent of which orientationwe choose. Proof.
Suppose
O ∼ O (cid:48) and they agree on e , then O and O (cid:48) differ by a disjointunion of positive circuits and cocircuits which do not contain e by Lemma 3.1.2and its dual. Thus − e O ∼ − e O (cid:48) using the same reversals. (cid:3) Proposition 4.4.5.
For any −→ e , −→ f ∈ Z E and [ O ] ∈ G ( M ), −→ e · ( −→ f · [ O ]) = −→ f · ( −→ e · [ O ]). Hence it is valid to extend · linearly, and (cid:8) is indeed a group actionof Z E on G ( M ) . Proof.
The statement is tautological if −→ e = −→ f . If −→ e = −−→ f , then without lossof generality the orientation of e in O is −→ e . Let C be a positive (co)circuitcontaining e . Then −→ f · ( −→ e · [ O ]) = [ O ] = [ − C O ] = −→ e · ( −→ f · [ O ]). ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 25
Otherwise e (cid:54) = f . We may again assume that e is oriented as −→ e in O . Thestatement is easy if there exists some positive (co)circuit in O that contains f but not e , as we can reverse it and obtain an orientation in which the orientationsof e, f are already −→ e , −→ f . So without loss of generality e, f are in the circuit partof O and every positive circuit containing f also contains e ; fix any such positivecircuit C . f must be in some positive cocircuit D (cid:48) of O − e , since otherwise f is in some positive circuit of O − e , which is a positive circuit in O avoiding e .By Lemma 4.4.1, D := D (cid:48) ∪ { e } is a cocircuit in O and − e D is positive, and byLemma 4.4.2, we know that C ∩ D = { e, f } .On one hand we have −→ f · ( −→ e · [ O ]) = −→ f · [ − e O ] = −→ f · [ − ( D − e ) O ] = [ − ( D −{ e,f } ) O ].On the other hand, −→ e · ( −→ f · [ O ]) = −→ e · ( −→ f · [ − C O ]) = −→ e · [ − ( C − f ) O ] = −→ e · [ − ( C ∪ D − e ) O ] = [ − ( C ∪ D ) O ]. But C is positive in − ( C ∪ D ) O , so [ − ( C ∪ D ) O ] =[ − ( C ∪ D ) (cid:52) C O ] = [ − ( D −{ e,f } ) O ]. (cid:3) Now we know that Z E (cid:8) G ( M ) is well-defined, so we show next that this actiondescends to a group action Jac( M ) (cid:8) G ( M ). Proposition 4.4.6.
The stabilizer of the action on any [ O ] contains Λ A ( M ) ⊕ Λ ∗ A ( M ). Proof.
Let −→ C ∈ Λ A ( M ) be a signed circuit. Let F be the set of elements in C whose orientations in O are the same as in −→ C . Then −→ C · [ O ] = ( (cid:80) −→ e ∈−→ C \ F −→ e ) · [ − F O ] = ( (cid:80) −→ C \ F −→ e i ) · [ − ( C − F ) O ] = [ O ]. The proof for Λ ∗ A ( M ) is similar. (cid:3) The action is simply transitive.Proposition 4.5.1.
The group action Jac( M ) (cid:8) G ( M ) is transitive. Proof.
Given any two orientations O , O (cid:48) , let γ be the sum of the (oriented) ele-ments in O whose orientation in O (cid:48) is different, then [ γ ] · [ O ] = [ O (cid:48) ]. (cid:3) By Proposition 4.4.6 and Proposition 4.5.1, we know that Jac( M ) (cid:8) G ( M )is well-defined and transitive, and we know that | Jac( M ) | = |G ( M ) | , so theaction is automatically simple. However, it seems worthwhile to give a directproof of the simplicity of the action which does not make use of the equality | Jac( M ) | = |G ( M ) | , since this yields an independent and “bijective” proof of theequality. We begin with the following reduction. Proposition 4.5.2.
The simplicity of the group action Jac( M ) (cid:8) G ( M ) is equiv-alent to the statement that every element of the quotient group Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) contains a coset representative whose coefficients are all 1 , , − Proof.
Suppose such a set of coset representatives exists. We need to show thatwhenever [ γ ] ∈ Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) fixes some circuit-cocircuit reversal class, [ γ ] = [0]. By transitivity, [ γ ] will fix every equivalence class in such a case. Without lossof generality, the coefficients of γ are all 1 , , − F ⊂ E . Pick anorientation O in which the orientation of every element of F agrees with γ ; then[ O ] = [ γ ] · [ O ] = [ − F O ]. Therefore O ∼ − F O , meaning that F is a disjoint union ofpositive circuits and cocircuits in O , i.e., γ ∈ Λ A ( M ) ⊕ Λ ∗ A ( M ) and [ γ ] = [0]. Theproof of the other direction is omitted as it is not being used in this paper. (cid:3) Proposition 4.5.3.
Every element of Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) contains a coset representa-tive whose coefficients are all 1 , , − Proof.
We will show that there is such a representative in [ γ ] for every γ = (cid:80) e ∈ E c e e ∈ Z E by lexicographic induction on | γ | ∞ := max e ∈ E | c e | and the numberof elements e with | c e | = | γ | ∞ . The assertion is trivial if | γ | ∞ ≤
1, so suppose | γ | ∞ >
1. By choosing a suitable reference orientation we may assume that allcoefficients of γ are non-negative. Pick an element e whose coefficient c e equals | γ | ∞ and pick a positive (co)circuit C containing e . By subtracting γ C := (cid:80) f ∈ C f from γ , all positive coefficients c f with f ∈ C decrease by 1, while the zerocoefficients become −
1. Hence | γ − γ C | ∞ ≤ | γ | ∞ and the number of elements f with | c f | = | γ | ∞ strictly decreases. By our induction hypothesis, there exists arepresentative with the desired form in [ γ − γ C ] = [ γ ]. (cid:3) Corollary 4.5.4.
The group action Jac( M ) (cid:8) G ( M ) is simple.4.6. Computability of the group action.
We now show that the simply tran-sitive action of Jac( M ) on G ( M ) is efficiently computable. Proposition 4.6.1.
The action of Jac( M ) on G ( M ) can be computed in poly-nomial time, given a totally unimodular matrix A representing M . Proof.
First we show that computing the action of a generator [ −→ e ] on a circuit-cocircuit reversal class can be done in polynomial time. To see this, note that byProposition 4.4.3, it suffices to find a positive circuit/cocircuit containing a givenelement e in O . For positive circuits, this can be done by solving the integerprogram min( T v : Av = 0 , v e = 1 , ≤ v i ≤ , v i ∈ Z ) and take the supportof the minimizer (if exists), but it is actually a linear program (thus polynomialtime computable [31]) as A is totally unimodular. The cocircuit case is similar.It remains to show that it is possible to find, in polynomial time, a coset rep-resentative with small polynomial-size coefficients in each element of Jac( M ) ∼ = Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) . For the practical reason of generating random elements of Jac( M )(cf. § y ∈ Z r representing a coset of Z r Col Z ( AA T ) ,before lifting y to a vector γ ∈ Z E . Thus we describe a two-step algorithm toin fact find a representative in Z E where all coefficients belong to {− , , } (theexistence of which is guaranteed by Proposition 4.5.3), starting with an inputvector in y ∈ Z r . ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 27
In step 1, replace y by y (cid:48) := y − ( AA T ) (cid:98) ( AA T ) − y (cid:99) , where (cid:98) (cid:99) is the coordinate-wise truncation. The new vector represents the same element in Z r Col Z ( AA T ) , andit is equal to ( AA T )(( AA T ) − y − (cid:98) ( AA T ) − y (cid:99) ). Since 0 ≤ x − (cid:98) x (cid:99) < AA T is between − m and m , the absolute value of each coordinateof y (cid:48) is at most mr . To work in Z E Λ A ( M ) ⊕ Λ ∗ A ( M ) , we solve the equation Aγ = y (cid:48) ,which is a simple linear system; since A is totally unimodular, the absolute valueof each coefficient of γ is at most mr .In step 2, starting with an element γ ∈ Z E we obtained in step 1. We applythe procedure described in Proposition 4.5.3 with some modification, namelythat after choosing a positive (co)circuit C which contains an element e whosecoefficient c e is maximum in γ , we subtract (cid:98) c e (cid:99) γ C from γ . No new element withthe absolute value of its coefficients being larger than (cid:100) | γ | ∞ (cid:101) is created in eachsuch step, so after every O ( m ) steps the maximum absolute value of coefficientsis halved, and in a total of O ( m log m ) steps the maximum absolute value ofcoefficients is reduced to at most 1. We remark that step 2 by itself can yield apolynomial time algorithm if we work in Z E from the beginning. (cid:3) An algorithm for sampling bases of a regular matroid.
By mimick-ing the strategy from [7], we can now produce a polynomial-time algorithm forrandomly sampling bases of a regular matroid. The high-level strategy is:(1) Compute the Smith Normal Form of a matrix A representing M , anddecompose Jac( M ) as a direct sum of finite abelian groups.(2) Use such a decomposition to choose a random element γ ∈ Jac( M ).(3) Given a reference orientation O , compute [ O (cid:48) ] := γ · [ O ] ∈ G ( M ), where · is the group action from Theorem 1.2.1.(4) Compute the basis B corresponding to the ( σ, σ ∗ )-compatible orientationin [ O (cid:48) ], which can be found in polynomial time by Proposition 3.6.1.5. Dilations, the Ehrhart polynomial, and the Tutte polynomial
Metric graphs can either be viewed as limits of subdivisions of discrete graphsor as intrinsic objects. See, for example, Section 2 of [6]. Similarly, one can viewcontinuous orientations of regular matroids as a limit of discrete orientations oras intrinsic objects. So far in this paper we have taken the latter viewpoint,but in this section we shift towards the former. In doing so, we will see thatthe bridge between discrete and continuous orientations of regular matroids isintimately related to Ehrhart theory for unimodular zonotopes. For example,we demonstrate how this perspective allows for a new derivation of a result ofStanley which states that the Ehrhart polynomial of a unimodular zonotope isa specialization of the Tutte polynomial. Stanley’s original proof utilizes a half-open decomposition of a zonotopal tiling. In contrast, zonotopal tilings will not make an appearance in our proof, although Corollary 4.1.5 provides a connectionto Stanley’s argument.5.1.
The Ehrhart polynomial and the Tutte polynomial.
The Tutte poly-nomial T M ( x, y ) is a bivariate polynomial associated to a matroid M which en-codes a wealth of information associated to M . One of its key properties is that T M ( x, y ) is “universal” with respect to deletion and contraction, in the followingsense: Proposition 5.1.1 (see [37, Theorem 1] and [36, Theorem 2.16]) . Let M be theset of all matroids. Suppose a, b, x , y ∈ R and that f : M → R is a functionwith f ( ∅ ) = 1 and such that for every matroid M and every element e of M , f ( M ) = af ( M/e ) + bf ( G \ e ) if e is neither a loop nor a coloop f ( M ) = x f ( M \ e ) if e is a coloop f ( M ) = y f ( M/e ) if e is a loop.Then f ( M ) = a rk ( M ) b rk ( M ∗ ) T M ( x a , y b ) . Given an integer polytope P , its Ehrhart polynomial E P ( q ) counts the numberof lattice points in qP , the q -th dilate of P . The fact that such a polynomialexists for any integer polytope was proven by Ehrhart [19]. Let M be a regularmatroid represented by the totally unimodular r × m matrix A . Given a positiveinteger q , define qA to be the r × qm matrix obtained by repeating each columnof A q times consecutively. Let qM be the corresponding regular matroid. Notethat the zonotope Z qA associated to qA is just the q -th dilate qZ A .Let σ q be an acyclic signature of qM . Using the interpretation of lattice pointsof Z qA as σ q -compatible orientations of qM , we give a new proof of the followingtheorem. Theorem 5.1.2 (Stanley [34]) . Let A be a totally unimodular matrix with as-sociated zonotope Z = Z A , and let M be the corresponding regular matroid.Then E Z ( q ) = q rk( M ) T M (1 + 1 q , . Stanley’s result extends to general integer zonotopes, but a priori our proofdoes not. For a calculation of the Ehrhart polynomial of an integral zonotopeusing the language of the arithmetic Tutte polynomial, see [15]. Before givingthe proof of Theorem 5.1.2, we need a few definitions.
ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 29
By a partial orientation of a regular matroid M , we mean a function E →{− , , } , where elements mapping to 0 are called bi-oriented . Given t ∈ Z > ,a t -partial orientation of M will be a partial orientation where each bi-orientededge receives some integer weight s with 1 ≤ s ≤ t . (By convention, a 0-partialorientation of M will mean the same thing as an orientation.)Fix a reference orientation O ref on M . Setting t + 1 = q , there is a map fromorientations of qM to t -partial orientations as follows. Given e ∈ M , if all q copies of e are oriented similarly, we map them to the corresponding orientationof e in M . On the other hand, if s copies of e of are oriented in agreement with O ref and q − s copies are oriented oppositely, with 1 ≤ s ≤ t , we map this set ofedges to a bi-oriented element of weight s in M .A non-empty subset F of E is called a potential circuit in a t -partial orientation O if F is a circuit of M and there is a choice of orientation of each bi-orientedelement so that F becomes a positive circuit. We will call a t -partial orientationof M circuit connected if for each e which is the minimum element in a potentialcircuit, either e is not bi-oriented and is oriented in agreement with the referenceorientation, or e is bi-oriented and replacing it with the opposite orientation of e in O ref does not produce any potential circuits containing e . Proof. (of Theorem 5.1.2) For each positive integer q , we will define an acyclicsignature σ q on C ( qM ). By Proposition 3.1.4 and Proposition 4.1.4, it willthen suffice to prove that the number of σ q -compatible orientations of qM is q rk( M ) T M (1 + q − , σ q will come from a total orderand reference orientation of qM . We now explain how given an arbitrary σ , wecan naturally define σ q . Given e ∈ M , let e , . . . e q be the q copies of e in qM .We orient the e i in σ q similarly to e in σ , i.e., so that together they form apositive cocircuit in their induced matroid. Let e i be the list of the elements of M according to σ . Given e ik and e j(cid:96) in qM , we define σ q so that e ik < q e j(cid:96) if i < j ,or i = j and k < (cid:96) .We are attempting to count objects associated to qM using the Tutte polyno-mial of M , so we would first like to produce a bijective map from σ q -compatibleorientations of qM to certain objects associated to M alone. To do this, notethat given a σ q -compatible orientation O of qM , a reference orientation O q ref ,and a set of parallel elements e , . . . e q , there are only q + 1 possible orientationsof these elements: e . . . e k will be oriented in agreement with O q ref , for some k = 0 , , . . . , q , and e k +1 . . . e q will be oriented oppositely. (If this were not thecase, we would have a 2-element positive circuit whose minimum edge is orientedin disagreement with O q ref .) Using this observation, it is not difficult to checkthat the map defined above from orientations of qM to t -partial orientations of In the case of graphs, Hopkins and the first author would call such objects type B partial orientations , but we will suppress the term “type B” here. M (where t = q −
1) takes σ q -compatible orientations of qM bijectively to circuitconnected t -partial orientations of M .We first prove that the sets X t,M \ e and X t,M/e are the images of X t,M underdeletion and contraction, respectively. Given O ∈ X t,M \ e (the case of O ∈ X t,M/e being similar), suppose that both orientations of e produce t -partial orientationswhich are not elements of X t,M . This implies that both orientations of e producepotential circuits C and C which are not σ q -compatible. For 1 ≤ i ≤
2, wecan choose orientations of the bioriented elements in C i to produce a circuit C (cid:48) i which is not σ q -compatible. The sum C (cid:48) + C (cid:48) is in the kernel of A and does notcontain e , therefore we can apply Lemma 3.1.2 and decompose C (cid:48) + C (cid:48) into asum of directed circuits not containing e such that the signs of the elements areinherited from C (cid:48) + C (cid:48) . Let e (cid:48) be the minimum labeled element in C (cid:48) ∪ C (cid:48) . Itis possible that e (cid:48) appears in only one of the circuits C (cid:48) or C (cid:48) , otherwise it mustbe oriented similarly in both C (cid:48) and C (cid:48) as they are not σ q -compatible. Thus e (cid:48) is in the support of C (cid:48) + C (cid:48) , and there exists a circuit C containing e (cid:48) whosesupport is contained in the support of C (cid:48) + C (cid:48) . Moreover, C has size largerthan 2 as e (cid:48) was oriented similarly in C (cid:48) and C (cid:48) , thus it does not correspondto a bioriented element of O . By assumption, e (cid:48) is oriented in disagreementwith its reference orientation, therefore C is not σ q -compatible. After possiblyrebiorienting some of the elements in C , we obtain a potential circuit in O whichis not σ q -compatible. This contradicts the assumption that O ∈ X t,M \ e .Let X t,M be the set of circuit connected t -partial orientations of M (cf. Fig-ure 5.3). Let e be the largest element of M . If e is a loop, then | X t,M | = | X t,M \ e | ,and if e is a coloop, then | X t,M | = ( t + 2) | X t,M/e | . If e is neither a bridge nora loop, we claim that | X t,M | = | X t,M \ e | + ( t + 1) | X t,M/e | . Given this claim, weconclude from Proposition 5.1.1 that | X t,M | = ( t + 1) rk(M) T M ( t + 2 t + 1 ,
1) = q rk( M ) T M ( q + 1 q , O ∈ X t,M and let O e be the set of t -partial orientations in X t,M whichagree with O away from e . We first observe that O e includes a t -partial orientationwith e bioriented if and only if it includes t -partial orientations with e orientedin each direction. Furthermore, this is the case if and only if O /e ∈ X t,M/e . Wealways have that O \ e ∈ X t,M \ e as deleting e cannot cause a t -partial orientationto stop being circuit connected. Therefore, |O e | = 1 if and only if O /e / ∈ X t,M/e ,and |O e | = t +2 if and only if O /e ∈ X t,M/e . The claim now follows by partitioning X t,M into maximal sets of t -partial orientations which agree on every element in M \ e . (cid:3) Remark 5.1.3.
The realizable part of the Bohne-Dress theorem states that theregular tilings of Z A by paralleletopes are dual to the generic perturbations of ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 31 the central hyperplane arrangement defined by A . Hopkins and Perkinson [25]investigated generic bigraphical arrangements, i.e. generic perturbations of twicethe graphical arrangement, and associated certain partial orientations, which theycalled admissible , to the regions in the complement of such an arrangement. Theaforementioned duality induces a geometric bijection between these regions andthe lattice points in the twice-dilated graphical zonotope. This in turn gives abijection between the admissible partial orientations and the circuit connectedpartial orientations. The enumeration of these two different classes of partialorientations both appear as specializations of the 12-variable expansion of theTutte polynomial from [5], and the aforementioned duality interchanges a pair ofsymmetric variables.5.2. Ehrhart reciprocity.
Ehrhart reciprocity states that if P is an integralpolytope, and E P ( q ) is its Ehrhart polynomial, then the number of interior pointsof the q -th dilate of P is | E P ( − q ) | . Combining Ehrhart reciprocity and Stanley’sresult, one obtains the following corollary: Corollary 5.2.1.
The number of interior lattice points in qZ A is q rk( M ) T M (1 − /q, . Remark 5.2.2.
Our proof of Stanley’s formula also allows for a direct verifica-tion of Corollary 5.2.1 in this setting without appealing to Ehrhart reciprocity.Each facet of qZ is determined by a positive cocircuit in qM . Thus a point liesin the interior of qZ if and only if the corresponding σ q -compatible orientationof qM contains no positive cocircuits, or equivalently, if every element in thecorresponding circuit connected ( q − M is contained in apotential circuit. One can verify that these objects are enumerated by the corre-sponding Tutte polynomial specialization via deletion-contraction as illustratedabove, although the argument is slightly more involved as one needs to take careto show that potential circuits and cocircuits can be treated separately.For the case of graphs, various generalizations of the arguments used in theproof of Theorem 5.1.2 are given in [5].5.3. Other invariants of unimodular zonotopes.
The following theorem col-lects some known connections between evaluations of the Tutte polynomial andgeometric quantities associated to unimodular zonotopes.
Theorem 5.3.1.
Let Z be a unimodular zonotope. Then: • T M (2 ,
1) is the number of lattice points in Z . • T M (0 ,
1) is the number of interior lattice points in Z . • T M (1 ,
1) is the lattice volume of Z . • T M (2 ,
0) is the number of vertices of Z . Proof.
The first two formulas follow from evaluating the Ehrhart polynomial at q = 1 and q = −
1. The third follows from interpreting the lattice volume of Z asVol( Z ) = lim q →∞ | Z n ∩ qZ | q rk( M ) = lim q →∞ T M (1 + 1 /q,
1) = T M (1 , . The fourth enumeration follows from the classical observation that the normalfan of the zonotope is the central hyperplane arrangement defined by A andthen applying Zaslavsky’s theorem which says that the number of such regions is T M (2 , (cid:3) Remark 5.3.2.
Recall that T M (1 ,
1) is equal to the number of bases of M , whichis equal to | Jac( M ) | . One can show that each maximal cell in our polyhedral de-composition of Z A has volume 1, which gives an alternate proof of the thirdevaluation in Theorem 5.3.1. Taking the limit of qZ A as q goes to infinity whilescaling the lattice by q , the set X q − ,M approaches the set of σ -compatible con-tinuous orientations of M and we recover the subdivision from Proposition 3.4.1(see Figure 5.3). Figure 5.
The set X ,K associated to the lattice points of 2 Z K using the acyclic signature σ from Figure 3. The bioriented edgesare colored red. Taking the (suitably rescaled) limit of qZ K as q goes to infinity, the set X q − ,K induces the subdivision depicted inFigure 1. Acknowledgements
The first author was partially supported by DFG-Collaborative Research Cen-ter, TRR 109 Discretization in Geometry and Dynamics, and a Zuckerman STEM
ONOTOPAL BIJECTIONS FOR REGULAR MATROIDS 33
Postdoctoral Scholarship; he thanks Sam Hopkins for introducing him to The-orem 5.3.1, and Raman Sanyal for explaining that regular tilings of a zonotopecan alternately be viewed as dual to generic perturbations of the associated hy-perplane arrangement. The second author’s work was partially supported by theNSF research grant DMS-1529573. The third author was partially supported byNWO Vici grant 639.033.514; he thanks Yin Tat Lee for the discussion on linearprogramming. All three authors thank the anonymous referee for the helpfulfeedback.
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Einstein Institute of MathematicsThe Hebrew University of JerusalemGivat Ram. Jerusalem, 9190401, IsraelEmail address: [email protected]
School of Mathematics, Georgia Institute of TechnologyAtlanta, Georgia 30332-0160, USAEmail address: [email protected]