P.R. Scott posed the problem of determining the minimum number of directions determined by n poin... more P.R. Scott posed the problem of determining the minimum number of directions determined by n points which are not all collinear in the plane. We consider a generalization of this problem for oriented marroids. We prove the following theorem: Let M denote an oriented matroid of rank 3. Suppose M has a modular line L, such that the n points of M not in L arc not all collinear. Then L has at least $( n + 3) points.
Let A be a simplicial hyperplane arrangement in R d. We prove that the center of the fundamental ... more Let A be a simplicial hyperplane arrangement in R d. We prove that the center of the fundamental group of the manifold C d \ {H ⊗ C : H ∈ A} is a direct product of infinite cyclic subgroups.
Let E be a finite set of points in IWd. Then {A, E-A} is a non-Radon partition of E iff there is ... more Let E be a finite set of points in IWd. Then {A, E-A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E-A. Or equivalently iff ,-0 is an acyclic reorientation of (M,,(E), 0). the oriented matroid canonically determined by E. If (M(E), (0) is an oriented matroid without loops then the set NR(E, fl) = {(A, E-A): ,-B is acyclic} determines (M(E), 0). In particular the matroidal properties of a finite set of points in IWd are precisely the properties which can be formulated in non-Radon partitions terms. The Mobius function of the poset .01= {A: A G E, ,-I? is acyclic) and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of aB" dissected by hyperplanes, Trans.
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a ma... more In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.
A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a... more A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a transformation of gain graphs that generalizes conjugation in a group. A weak chromatic function of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws are analogous to those of the chromatic polynomial of an ordinary graph, though they are different from those usually assumed of gain graphs or matroids. The three laws lead to the weak chromatic group of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for all switching-invariant fu...
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by... more The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal (M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called χ-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, András Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to the χ-algebras. We give a survey of the results obtained by the authors concerning the construction of Gröbner bases of χ(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an "iterative residue formula" introduced by Szenes.
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the not... more A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.
Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orie... more Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orientation O of G(P) is good if, for every non-empty face F of P , the induced graph G(F) has exactly one sink. Gil Kalai gave a simple way to tell a simple polytope from the good orientations of its graph. This article is a broader study of "good orientations" (of the graphs) on matroid polytopes.
Oriented matroids were introduced in the 1970s. From an algebraic point of view, they constitute ... more Oriented matroids were introduced in the 1970s. From an algebraic point of view, they constitute a combinatorial axiomization of sign properties of the linear calculus in real vector spaces. They also have a nice topological interpretation: by the topological representation theorem, oriented matroids are equivalent to arrangements of pseudohyperplanes in real projective spaces, a higher-dimensional generalization of pseudolines arrangements, a topic studied since the beginning of the 20th century. The firs applications of oriented matroids came naturally from discrete geometry: convex polytopes, linear programming, arrangements of hyperplanes. Rapidly, significan contributions appeared in more distant domains: topology of complements of complexificatio of real arrangements, equivalence of oriented matroid realization with real semi-algebraic varieties (Mnëv universality theorem), application to Pontrjagin classes in differential geometry (I. M. Gelfand and R. MacPherson), stratificatio of Grassmann varieties (Gelfand and his school). A short but comprehensive survey of research in current oriented matroid theory can be found in the appendix 'Some current frontiers of research' of the 2nd edition (1999) of 'Oriented Matroids', by A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler (Cambridge University Press, First edition, 1993). The present special issue originated during the meeting 'Combinatorial Geometries: Oriented Matroids, Matroids, and Applications', held at the C.I.R.M. † (Centre International de Rencontres Mathématiques) in the campus of the University of Marseille-Luminy, November 8-12, 1999 ‡. The firs and previous meeting 'Combinatorics and Geometry: Oriented Matroids', organised by M. Las Vergnas and G. Ziegler was also held in C.I.R.M. but eight years earlier (April 15-19, 1991). This meeting was mainly devoted to oriented matroids and coincided with the completion of the firs textbook on the subject: 'Oriented Matroids'. At the time, the emphasis was on the foundations. In the 1999 meeting the emphasis moved to recent applications of oriented matroids, ranging from algebra and the geometry of polytopes to differential geometry. The subjects covered a large part of the current frontiers of research: such as realization spaces (Babson, Roy), spaces of oriented matroids (Anderson, Santos, Azaola), triangulations of polytopes (Below, Mnev), zonotopal tilings (Athanasiadis), realizations algorithms (Streinu), polyhedral 2-manifolds (Bokowski). A few talks have been devoted to other aspects of oriented matroids: construction of oriented matroids (Guedes de Oliveira, Fukuda), applications of oriented matroids to combinatorics (Ziegler) or discrete geometry (Roudneff). Finally, 4 talks dealt with structures related to oriented matroids: two on Coxeter matroids (Borovik, White), two on, or partially on, Orlik-Solomon algebras (Falk, Las Vergnas). We feel that the present volume offers a significat ve picture of the state-of-the-art in the domain, and that it should be a motivating reference for future research.
A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C... more A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We introduce and study an Orlik-Solomon type algebra determined by (M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon algebra of M.
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by... more The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called X-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, Andras Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to X-algebras. We give a survey of the results obtained by the authors concerning the construction of Groebner bases of I(M) and diagonal bases of Orlik-Solomo...
Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cy... more Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable i# G is chordal (rigid): this is an other way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M , a maximal chain of modular flats of M determines canonically a chordal graph.
This special volume is dedicated to the memory of late Michel Las Vergnas who passed away on Janu... more This special volume is dedicated to the memory of late Michel Las Vergnas who passed away on January 19, 2013 at age 72. Michel Las Vergnaswas a combinatorialmathematicianwith artistic sense and creativity. He is also well known as a cofounder of the theory of oriented matroids which acquired the firm acceptance as fundamental concept in mathematics. Michel was one of the founding editors of European Journal of Combinatorics. On personal side,Michel had broad and deep interest inmusic, arts and culture.Michel was a warm-hearted individual caring for his family and friends with kind and genuine smiles. Michel was the first doctoral student of another great combinatorialist Claude Berge. They were two of the leaders of the Paris school of combinatorics (L’Equipe Combinatoire, Université Paris VI). They influenced greatly the advances of combinatorial mathematics in the world and guided many excellent graduate students. Both Claude and Michel had unusual talents to formulate beautiful conjectures. Throughout his whole career as Directeur de Recherche au CNRS and professor of Université Paris VI (Pierre et Marie Curie), Michel’s devotion to combinatorial mathematics and to the guidance of doctoral students was extraordinary. The late Yahya Ould Hamidoune and the first editor Raul were the first doctoral students ofMichel, graduated in 1978 and 1979, respectively. The third editor Emeric was the last student who completed his study in 2002. Michel had supervised 15 doctoral students. He won the silver medal of the CNRS in 1985. On his pioneering research front, Michel gave many equivalent axiomatizations of oriented matroids in an extensive manuscript,1 which unfortunately has never been fully published. Michel’s theorem on single-element extensions2 of oriented matroids turned out to be crucial in constructing fascinating examples and in resolving degeneracy in the abstract combinatorial setting of optimization. Michel had strong interest in a wide range of combinatorics beyond the oriented
P.R. Scott posed the problem of determining the minimum number of directions determined by n poin... more P.R. Scott posed the problem of determining the minimum number of directions determined by n points which are not all collinear in the plane. We consider a generalization of this problem for oriented marroids. We prove the following theorem: Let M denote an oriented matroid of rank 3. Suppose M has a modular line L, such that the n points of M not in L arc not all collinear. Then L has at least $( n + 3) points.
Let A be a simplicial hyperplane arrangement in R d. We prove that the center of the fundamental ... more Let A be a simplicial hyperplane arrangement in R d. We prove that the center of the fundamental group of the manifold C d \ {H ⊗ C : H ∈ A} is a direct product of infinite cyclic subgroups.
Let E be a finite set of points in IWd. Then {A, E-A} is a non-Radon partition of E iff there is ... more Let E be a finite set of points in IWd. Then {A, E-A} is a non-Radon partition of E iff there is a hyperplane H separating A strictly from E-A. Or equivalently iff ,-0 is an acyclic reorientation of (M,,(E), 0). the oriented matroid canonically determined by E. If (M(E), (0) is an oriented matroid without loops then the set NR(E, fl) = {(A, E-A): ,-B is acyclic} determines (M(E), 0). In particular the matroidal properties of a finite set of points in IWd are precisely the properties which can be formulated in non-Radon partitions terms. The Mobius function of the poset .01= {A: A G E, ,-I? is acyclic) and in a special case its homotopy type are computed. This paper generalizes recent results of P. Edelman (A partial order on the regions of aB" dissected by hyperplanes, Trans.
In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a ma... more In this note we introduce a sufficient condition for the Orlik-Solomon algebra associated to a matroid M to be l-adic and we prove that this condition is necessary when M is binary (in particular graphic). Moreover, this result cannot be extended to the class of all matroids.
A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a... more A gain graph is a graph whose edges are labelled invertibly by gains from a group. Switching is a transformation of gain graphs that generalizes conjugation in a group. A weak chromatic function of gain graphs with gains in a fixed group satisfies three laws: deletion-contraction for links with neutral gain, invariance under switching, and nullity on graphs with a neutral loop. The laws are analogous to those of the chromatic polynomial of an ordinary graph, though they are different from those usually assumed of gain graphs or matroids. The three laws lead to the weak chromatic group of gain graphs, which is the universal domain for weak chromatic functions. We find expressions, valid in that group, for a gain graph in terms of minors without neutral-gain edges, or with added complete neutral-gain subgraphs, that generalize the expression of an ordinary chromatic polynomial in terms of monomials or falling factorials. These expressions imply relations for all switching-invariant fu...
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by... more The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal (M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called χ-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, András Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to the χ-algebras. We give a survey of the results obtained by the authors concerning the construction of Gröbner bases of χ(M) and diagonal bases of Orlik-Solomon type algebras and we present the combinatorial analogue of an "iterative residue formula" introduced by Szenes.
A mixed graph is a graph with some directed edges and some undirected edges. We introduce the not... more A mixed graph is a graph with some directed edges and some undirected edges. We introduce the notion of mixed matroids as a generalization of mixed graphs. A mixed matroid can be viewed as an oriented matroid in which the signs over a fixed subset of the ground set have been forgotten. We extend to mixed matroids standard definitions from oriented matroids, establish basic properties, and study questions regarding the reorientations of the unsigned elements. In particular we address in the context of mixed matroids the P-connectivity and P-orientability issues which have been recently introduced for mixed graphs.
Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orie... more Let P a polytope and let G(P) be the graph of P. Following Gil Kalai, we say that an acyclic orientation O of G(P) is good if, for every non-empty face F of P , the induced graph G(F) has exactly one sink. Gil Kalai gave a simple way to tell a simple polytope from the good orientations of its graph. This article is a broader study of "good orientations" (of the graphs) on matroid polytopes.
Oriented matroids were introduced in the 1970s. From an algebraic point of view, they constitute ... more Oriented matroids were introduced in the 1970s. From an algebraic point of view, they constitute a combinatorial axiomization of sign properties of the linear calculus in real vector spaces. They also have a nice topological interpretation: by the topological representation theorem, oriented matroids are equivalent to arrangements of pseudohyperplanes in real projective spaces, a higher-dimensional generalization of pseudolines arrangements, a topic studied since the beginning of the 20th century. The firs applications of oriented matroids came naturally from discrete geometry: convex polytopes, linear programming, arrangements of hyperplanes. Rapidly, significan contributions appeared in more distant domains: topology of complements of complexificatio of real arrangements, equivalence of oriented matroid realization with real semi-algebraic varieties (Mnëv universality theorem), application to Pontrjagin classes in differential geometry (I. M. Gelfand and R. MacPherson), stratificatio of Grassmann varieties (Gelfand and his school). A short but comprehensive survey of research in current oriented matroid theory can be found in the appendix 'Some current frontiers of research' of the 2nd edition (1999) of 'Oriented Matroids', by A. Björner, M. Las Vergnas, B. Sturmfels, N. White and G. Ziegler (Cambridge University Press, First edition, 1993). The present special issue originated during the meeting 'Combinatorial Geometries: Oriented Matroids, Matroids, and Applications', held at the C.I.R.M. † (Centre International de Rencontres Mathématiques) in the campus of the University of Marseille-Luminy, November 8-12, 1999 ‡. The firs and previous meeting 'Combinatorics and Geometry: Oriented Matroids', organised by M. Las Vergnas and G. Ziegler was also held in C.I.R.M. but eight years earlier (April 15-19, 1991). This meeting was mainly devoted to oriented matroids and coincided with the completion of the firs textbook on the subject: 'Oriented Matroids'. At the time, the emphasis was on the foundations. In the 1999 meeting the emphasis moved to recent applications of oriented matroids, ranging from algebra and the geometry of polytopes to differential geometry. The subjects covered a large part of the current frontiers of research: such as realization spaces (Babson, Roy), spaces of oriented matroids (Anderson, Santos, Azaola), triangulations of polytopes (Below, Mnev), zonotopal tilings (Athanasiadis), realizations algorithms (Streinu), polyhedral 2-manifolds (Bokowski). A few talks have been devoted to other aspects of oriented matroids: construction of oriented matroids (Guedes de Oliveira, Fukuda), applications of oriented matroids to combinatorics (Ziegler) or discrete geometry (Roudneff). Finally, 4 talks dealt with structures related to oriented matroids: two on Coxeter matroids (Borovik, White), two on, or partially on, Orlik-Solomon algebras (Falk, Las Vergnas). We feel that the present volume offers a significat ve picture of the state-of-the-art in the domain, and that it should be a motivating reference for future research.
A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C... more A family C of circuits of a matroid M is a linear class if, given a modular pair of circuits in C}, any circuit contained in the union of the pair is also in C. The pair (M,C) can be seen as a matroidal generalization of a biased graph. We introduce and study an Orlik-Solomon type algebra determined by (M,C). If C is the set of all circuits of M this algebra is the Orlik-Solomon algebra of M.
The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by... more The Orlik-Solomon algebra of a matroid M is the quotient of the exterior algebra on the points by the ideal I(M) generated by the boundaries of the circuits of the matroid. There is an isomorphism between the Orlik-Solomon algebra of a complex matroid and the cohomology of the complement of a complex arrangement of hyperplanes. In this article a generalization of the Orlik-Solomon algebras, called X-algebras, are considered. These new algebras include, apart from the Orlik-Solomon algebras, the Orlik-Solomon-Terao algebra of a set of vectors and the Cordovil algebra of an oriented matroid. To encode an important property of the "no broken circuit bases" of the Orlik-Solomon-Terao algebras, Andras Szenes has introduced a particular type of bases, the so called "diagonal bases". This notion extends naturally to X-algebras. We give a survey of the results obtained by the authors concerning the construction of Groebner bases of I(M) and diagonal bases of Orlik-Solomo...
Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cy... more Let G be a finite simple graph. From the pioneering work of R. P. Stanley it is known that the cycle matroid of G is supersolvable i# G is chordal (rigid): this is an other way to read Dirac's theorem on chordal graphs. Chordal binary matroids are not in general supersolvable. Nevertheless we prove that, for every supersolvable binary matroid M , a maximal chain of modular flats of M determines canonically a chordal graph.
This special volume is dedicated to the memory of late Michel Las Vergnas who passed away on Janu... more This special volume is dedicated to the memory of late Michel Las Vergnas who passed away on January 19, 2013 at age 72. Michel Las Vergnaswas a combinatorialmathematicianwith artistic sense and creativity. He is also well known as a cofounder of the theory of oriented matroids which acquired the firm acceptance as fundamental concept in mathematics. Michel was one of the founding editors of European Journal of Combinatorics. On personal side,Michel had broad and deep interest inmusic, arts and culture.Michel was a warm-hearted individual caring for his family and friends with kind and genuine smiles. Michel was the first doctoral student of another great combinatorialist Claude Berge. They were two of the leaders of the Paris school of combinatorics (L’Equipe Combinatoire, Université Paris VI). They influenced greatly the advances of combinatorial mathematics in the world and guided many excellent graduate students. Both Claude and Michel had unusual talents to formulate beautiful conjectures. Throughout his whole career as Directeur de Recherche au CNRS and professor of Université Paris VI (Pierre et Marie Curie), Michel’s devotion to combinatorial mathematics and to the guidance of doctoral students was extraordinary. The late Yahya Ould Hamidoune and the first editor Raul were the first doctoral students ofMichel, graduated in 1978 and 1979, respectively. The third editor Emeric was the last student who completed his study in 2002. Michel had supervised 15 doctoral students. He won the silver medal of the CNRS in 1985. On his pioneering research front, Michel gave many equivalent axiomatizations of oriented matroids in an extensive manuscript,1 which unfortunately has never been fully published. Michel’s theorem on single-element extensions2 of oriented matroids turned out to be crucial in constructing fascinating examples and in resolving degeneracy in the abstract combinatorial setting of optimization. Michel had strong interest in a wide range of combinatorics beyond the oriented
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