View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector The Tutte Group of a Weakly Orientable Matroid Marc Wagowski Universit6 Pierre et Marie Curie -Paris 6 U.E.R. 48 - Mathbmatiques 4 place Jzmieu, 75005 Paris, France Submittedby Richard A. Brualdi ABSTRACT We show that a matroid M is weakly orientable if and only if the element cM (an abstraction of - 1) of its Tutte group is not a square. 1. INTRODUCTION A circuit signature of a matroid is an assignment of a sign + or - to all elements of each circuit. Four classes of signatures have been studied so far; they are, in their order of appearance, digraphoids [7], orientations [2], weak orientations [l, 61, and ternary signatures [8]. Digraphoids are an abstraction of directed graphs defined by Minty, which are exactly signatures arising from representations of regular matroids by unimodular subspaces over Iw. Orientations, a generalization of vector spaces over an ordered field, have received considerable attention during the past ten years and have been given many applications to convex polytopes, arrangements of hyperplanes, and linear programming. Weak orientations are obtained by a weakening, in a natural way, of the definition of orientations, and this class contains the three others. Ternary signatures are signatures arising from representations of ternary matroids over GF(3). Recently Dress has introduced the concept of matroids with coefficients [3], a generalization of the Tutte representation, providing a common frame- work for ordinary matroids, matroid representation over a field, and oriented matroids. In [lo] it has been shown that signatures, coordinatizable over a semiring, in the sense of Dress, are exactly the four aforementioned classes. Associated with a matroid M is an Abelian group TM, christened the “Tutte group” because it allows one to reinterpret Tutte’s homotopy theory LINEAR ALGEBRA AND ITS APPLICATIONS 117: 21-24 (1989) 21 0 Elsevier Science Publishing Co., Inc., 1989 655 Avenue of the Americas, New York, NY 10010 0024-3795/89/$3.50 22 MARC WAGOWSKI in algebraic terms. Furthermore the universal representation ring of a matroid M can be obtained as a factor ring of the group algebra Z[U,] [5]. A study of this group can be found in [5], [12], and [13]. In Corollary 3.3 we derive from our characterization of weakly oriented matroids a reformulation, due to A. Dress, of a deep result by Tutte [9, 7.51 in terms of the Tutte group. A characterization of ternary matroids whose statement is very similar to Theorem 3.2 is given in [5] and [13]. 2. NOTATION AND DEFINITIONS Let M be a matroid, and %? (V*) be its set of circuits (cocircuits). The Tutte group of M, T,, is the factor group of the free Abelian group, IL,,,, generated by the symbols .sM and Xc;.,, for all C E %’ G g* and all e, f E C (e # f ),modulo the relations EL = 1, X,; e, f. Xc. f e = 1, Xc.e eOX,; e2 e3. for all CE%? and’all C*E%?*‘wi% Cn’C* XC;e3,e,=1’XC,e,f*X&,f=EM = {e,f}. A signed set is a set X together with a partition into two subsets Xt and X- [the pair (X’, X- ) is also called a signature of the underlying set]. We define the opposite - X of the signed set X to be the signed set of signature (X-, X+ ). A circuit signature of M is a set Y of signed sets such that each element of Sp is a signature of a circuit, and for every circuit C there exists exactly two signed sets in Y with underlying set C, and these two signed sets are opposite. For X a signed set and e one of its elements, X(e) denotes thesignof e on X,i.e. + if eEX+ and - if eEX_. A circuit signature 9 of a matroid M and a cocircuit signature z?‘* of M form a dual pair of weakly oriented mutroids if for any X E 9 and for any Y E y* with IX n Y] = 2 we have [l] (X+nY+)U(X- fly-)+0 ifandonlyif (X+nY-)u(X- nY+)#0, equivalently, ](X’ n Yt ) U (X- n Y- )I = 1 [and therefore also ili+nY-)u(x- nY+)I=l]. 3. PROPERTIES OF THE TUTTE GROUP OF A WEAKLY ORIENTABLE MATROID The following theorem is the particular case of [4, Theorem 101 dealing with weak orientations. But in [4] the results are stated without proof, therefore we give for completeness a proof of Theorem 3.1. WEAKLY ORIENTABLE MATROIDS 23 THEOREM 3.1 (Dress [4, 6.11). A mutroid M is weakly orientable if and only if there exists a morphism Cpj%m 8, into the group { + , - } such that ia = - . Proof. Assume M to be weakly orientable. By definition there exists a dual pair of weak orientations (9’,9’*) of M. Let \k denote the morphism from ILM into the group { + , - } defined by \Ir(l) = + , \k(~~) = - , and for all C~gbIj* and all e,f EC with ef f, \k(X,G.,f)=Y(e).Y(f), where Y denotes one of the two opposite signed sets associated with C. For simplicity, in the remainder of this proof we will denote a signed circuit and its associated circuit by the same letter. We have ‘k( EL) = + , ‘k( X,; e, f. X C;fie) = [C(e)*C(f )I2= +, \k(Xc;.,,e2.XC;ep.e3.XC;e3.e,)= [CGOC(e2)~ C(e,)12=+, and for all CE~’ and all CLEF* with CnC*= {e, f}, since by definition of a weak orientation C(e) * C(f) +C*(e) * C*(f) = - , we have \~(E~.X~;~,~ . Xc*, e,f) = + . From these equalities we deduce that \k induces a morphism Q from the factor group UM into the group { + , - }, satisfying <a(.sy) = - . Conversely, assume the existence of a morphism Q from T, into the group { +, - } satisfying @(sM) = - . Take C E % 6 V*, choose e E C, and denote by Y the signature of C defined by Y(e) = + , and for f E C\e by Y(f) = + when @(Xc;.,,) = + and Y(f) = - otherwise. In such a manner we define a circuit signature 9’ and a cocircuit signature 9* of M. We leave it to the reader to check that the relations inducing the factor group U,, together with the hypothesis a’(~~~) = - , imply that (9’,9 *) is a dual pair of weak orientations. n As mentioned in [l], matroids coordinatizable over a finite field in which - 1 is not a square are weakly orientable. Theorem 3.2 generalizes this result to the Tutte group. THEOREM 3.2. A matroid M is weakly orientable if and only if Ed 4 (W2. Proof. If M is weakly orientable and if @ is a morphism from U, into { +, - } with @(.sM) = -, then sM @ (UM)2, since Ed = a2 for some (YE UM would imply @( eM) = @(02) = Cp(o)2 = + . Vice versa, if eM @ (UM)2,then we may view U,/(UM)2 as a vector space over GF(2) to find a morphism into { +, - } [ G GF(2)] which maps the nontrivial vector eM .(UM)2into - . n COROLLARY 3.3 (Tutte; see also [5] or [13]). A binary matroid M is regular if and only if Ed 4 (UM)2. 24 MARC WAGOWSKI Proof. Since regular matroids are exactly binary weakly orientable ma- troids (see [ll, II 2.3.5]), the conclusion follows from Theorem 3.2. n I am grateful to the referee fora simplification in the proof of Theorem 3.2 and forseveral helpful suggestions. REFERENCES R. Bland and D. L. Jensen, Weakly Oriented Matroids, Cornell University School of OR/IE Technical Report No. 732, Apr. 1987. R. Bland and M. Las Vergnas, Orientabihty of matroids, J. Cornbin. Theory Ser. B 24:94-123 (1978). A. Dress, Duality theory for finite and infinite matroids with coefficients, Ado. in Math. 59 (2):97-123 (1986). A. Dress, unpublished notes. A. Dress and W. Wenzel, Geometric algebra for combinatorial geometries, Preprint, 1988. D. L. Jensen, Coloring and Duality: Combinatorial Augmentation Methods, Thesis (under the supervision of R. Bland), Cornell Univ., 1985, 371 pp. G. Minty, On the axiomatic foundations of the theories of directed linear graphs, electrical networks, and network-programming, J. Math. Mech. 15:485-520 (1966). 8 J. P. Roundneff and M. Wagowski, Characterizations of ternary matroids in terms of circuit signatures, J. Combin. Theory Ser. B, to appear. 9 W. T. Tutte, Lectures on matroids, J. Res. Nat. Bur. Standards B69:1-47 (1965). 10 M. Wagowski, Matroid signatures coordinatizable over a semiring, European J. Combin., to appear. 11 M. Wagowski, Representations Algebriques des Matroides, These, Univ. Paris VI, 1987. 12 W. Wenzel, ijber die Stmktur der Tutte-Gruppe von Matroiden, Thesis, Univ. of Bielefeld, 1986. 13 W. Wenzel, A Grouptheoretic interpretation of Tutte’s homotopy theory, preprint, 1988. Received 6 April 1987; final manuscript accepted 25 July 1988 .