Homological properties of Orlik–Solomon algebras

The Orlik–Solomon algebra of a matroid can be considered as a quotient ring over the exterior algebra E. At first, we study homological properties of E-modules as e.g., complexity, depth and regularity. In particular, we consider modules with linear injective resolutions. We apply our results to Orlik–Solomon algebras of matroids and give formulas for the complexity, depth and regularity of such rings in terms of invariants of the matroid. Moreover, we characterize those matroids whose Orlik–Solomon ideal has a linear projective resolution and compute in these cases the Betti numbers of the ideal.     

A Note on Totally Free Matroids

It is known matroids obtained from a totally free uniform matroid U _2,n by a sequence of segment–cosegment and cosegment–segment exchanges are totally free (Geelen et al., in J Combin Theory Ser B 92:55–67, 2004). In this paper, we prove matroids obtained from any totally free matroid by a sequence of segment–cosegment and cosegment–segment exchanges are also totally free.     

Classifying spaces of Kač-Moody groups

We study the structure of classifying spaces of Kač-Moody groups from a homotopy theoretic point of view. They behave in many respects as in the compact Lie group case. The mod p cohomology algebra is noetherian and Lannes'T functor computes the mod p cohomology of classifying spaces of centralizers of elementary abelian p-subgroups. Also, spaces of maps from classifying spaces of finite p-groups to classifying spaces of Kač-Moody groups are described in terms of classifying spaces of centralizers while the classifying space of a Kač-Moody group itself can be described as a homotopy colimit of classifying spaces of centralizers of elementary abelian p-subgroups, up to p-completion. We show that these properties are common to a larger class of groups, also including parabolic subgroups of Kač-Moody groups, and centralizers of finite p-subgroups. Received: 15 June 2000 / in final form: 20 September 2001 / Published online: 29 April 2002     

Quasi‐graphic matroids*

Frame matroids and lifted‐graphic matroids are two interesting generalizations of graphic matroids. Here, we introduce a new generalization, quasi‐graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted‐graphic matroids, it is easy to certify that a 3‐connected matroid is quasi‐graphic. The main result is that every 3‐connected representable quasi‐graphic matroid is either a lifted‐graphic matroid or a frame matroid.     

On reid’s 3-simplicial matroid theorem

In this paper we prove the following result of Ralph Reid (which was never published nor completely proved). Theorem. Let M be a matroid coordinatizable (representable) over a prime field F. Then there is a 3-simplicial matroid M′ over F which is a series extension of M. The proof we give is different from the original proof of Reid which uses techniques of algebraic topology. Our proof is constructive and uses elementary matrix operations.     

Circuit Admissible Triangulations of Oriented Matroids

All triangulations of euclidean oriented matroids are of the same PL-homeo-morphism type by a result of Anderson. That means all triangulations of euclidean acyclic oriented matroids are PL-homeomorphic to PL-balls and that all triangulations of totally cyclic oriented matroids are PL-homeomorphic to PL-spheres. For non-euclidean oriented matroids this question is wide open. One key point in the proof of Anderson is the following fact: for every triangulation of a euclidean oriented matroid the adjacency graph of the set of all simplices ``intersecting' a segment [p _- p ₊ ] is a path. We call this graph the [p _- p ₊ ] -adjacency graph of the triangulation. While we cannot solve the problem of the topological type of triangulations of general oriented matroids we show in this note that for every circuit admissible triangulation of an arbitrary oriented matroid the [p _- p ₊ ] -adjacency graph is a path. Received December 8, 2000, and in revised form May 23, 2001. Online publication November 7, 2001.     

Matroid representation over GF(3)

We show that a matroid is representable over GF(3) if and only if no minor is the five-point line or the Fano matroid, or their duals. Tutte's famous characterization of the regular matroids is a corollary. A key lemma states that two representations of the same matroid in the same vector space over GF(3) may be transformed one into the other by inverting some points through the origin and taking a linear transformation; no result of this kind holds in larger fields.     

Matroid Enumeration for Incidence Geometry

Matroids are combinatorial abstractions for point configurations and hyperplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinearity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of incidence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algorithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in ℝ³ and that of Motzkin on ≤12 lines in ℝ², extending previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of oriented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n≤12 elements and rank 4 on n≤9 elements. We further list several new minimal non-orientable matroids.     

Oriented Interval Greedoids

We propose a definition of an oriented interval greedoid that simultaneously generalizes the notion of an oriented matroid and the construction on antimatroids introduced by L.J. Billera, S.K. Hsiao, and J.S. Provan in Enumeration in convex geometries and associated polytopal subdivisions of spheres (Discrete Comput. Geom. 39(1–3):123–137, 2008). As for oriented matroids, associated to each oriented interval greedoid is a spherical simplicial complex whose face enumeration depends only on the underlying interval greedoid.     

Representable Functions on the Unit Ball of a Banach Space

In this paper we treat the problem of integral representation of analytic functions over the unit ball of a complex Banach space X using the theory of abstract Wiener spaces. We define the class of representable functions on the unit ball of X and prove that this set of functions is related with the classes of integral k–homogeneous polynomials, integral holomorphic functions and also with the set of L ^p –representable functions on a Banach space.     

Cyclotomic and simplicial matroids

We show that two naturally occurring matroids representable over ℚ are equal: thecyclotomic matroid μ_n represented by then ^th roots of unity 1, ζ, ζ², …, ζ^n-1 inside the cyclotomic extension ℚ(ζ), and a direct sum of copies of a certain simplicial matroid, considered originally by Bolker in the context of transportation polytopes. A result of Adin leads to an upper bound for the number of ℚ-bases for ℚ(ζ) among then ^th roots of unity, which is tight if and only ifn has at most two odd prime factors. In addition, we study the Tutte polynomial of μ_n in the case thatn has two prime factors. First author supported by NSF Postdoctoral Fellowship. Second author supported by NSF grant DMS-0245379.     

Mod 2 cohomology of combinatorial Grassmannians

Matroid bundles, introduced by MacPherson, are combinatorial analogues of real vector bundles. This paper sets up the foundations of matroid bundles. It defines a natural transformation from isomorphism classes of real vector bundles to isomorphism classes of matroid bundles. It then gives a transformation from matroid bundles to spherical quasifibrations, by showing that the geometric realization of a matroid bundle is a spherical quasifibration. The poset of oriented matroids of a fixed rank classifies matroid bundles, and the above transformations give a splitting from topology to combinatorics back to topology. A consequence is that the mod 2 cohomology of the poset of rank k oriented matroids (this poset classifies matroid bundles) contains the free polynomial ring on the first k Stiefel-Whitney classes.     

The incidence structure of subspaces with well-scaled frames

In this paper, the incidence structure of classes of subspaces that generalize the regular (unimodular) subspaces of rational coordinate spaces is studied. Let F the a field and S - F β {0}. A subspace, V, of a coordinate space over F is S-regular if every elementary vector of V can be scaled by an element of F β {0} so that all of its non-zero entries are elements of S. A subspace that is {−1, +1 }-regular over the rational field is regular.Associated with a subspace, V, over an arbitrary (respectively, ordered) field is a matroid (oriented matroid) having as circuits (signed circuits) the set of supports (signed supports) of elementary vectors of V. Fundamental representation properties are established for the matroids that arise from certain classes of subspaces. Matroids that are (minor) minimally non-representable by various classes of subspaces are identified. A unique representability results is established for the oriented matroids of subspaces that are dyadic (i.e., {±2⁰, ±2¹, ±2², …}-regular) over the rationals. A self-dual characterization is established for the matroids of S-regular subspaces which generalizes Minty's characterization of regular spaces as digraphoids.     

On critical exponents of Dowling matroids

The critical problem in matroid theory is the problem to determine the critical exponent of a given representable matroid over a finite field. In this paper, we study the critical exponents of a class of representable matroids over finite fields, called Dowling matroids. Then the critical problem for a Dowling matroid is corresponding to the classical problem in coding theory to determine the maximum dimension k such that there exists an \([n,k,d]_q\) code for given n, d and q. We give a necessary and sufficient condition on the critical exponents of Dowling matroids by using a coding theoretical approach.     

On Maximum-sized k-Regular Matroids

 Let k be an integer exceeding one. The class of k-regular matroids is a generalization of the classes of regular and near-regular matroids. A simple rank-r regular matroid has the maximum number of points if and only if it is isomorphic to M(K _r+1), the cycle matroid of the complete graph on r+1 vertices. A simple rank-r near-regular matroid has the maximum number of points if and only if it is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding a point to a 3-point line of M(K _r+2) and then contracting this point. This paper determines the maximum number of points that a simple rank-r k-regular matroid can have and determines all such matroids having this number. With one exception, there is exactly one such matroid. This matroid is isomorphic to the simplification of , that is, the simplification of the matroid obtained, geometrically, by freely adding k independent points to a flat of M(K _r+k+1) isomorphic to M(K _k+2) and then contracting each of these points. Revised: July 27, 1998     

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.     

Normalizers of maximal tori

Normalizers and p-normalizers of maximal tori in p-compact groups can be characterized by the Euler characteristic of the associated homogeneous spaces. Applied to centralizers of elementary abelian p-groups these criteria show that the normalizer of a maximal torus of the centralizer is given by the centralizer of a preferred homomorphism to the normalizer of the maximal torus; i.e. that “normalizer” commutes with “centralizer”. Received April 1, 1995; in final form August 11, 1997     

Almost simple groups with socle 3D4(q) act on finite linear spaces

After the classification of flag-transitive linear spaces, attention has now turned to line-transitive linear spaces. Such spaces are first divided into the point-imprimitive and the point-primitive, the first class is usually easy by the theorem of Delandtsheer and Doyen. The primitive ones are now subdivided, according to the O’Nan-Scotte theorem and some further work by Camina, into the socles which are an elementary abelian or non-abelian simple. In this paper, we consider the latter. Namely, T ≤ G ≤ Aut(T) and G acts line-transitively on finite linear spaces, where T is a non-abelian simple. We obtain some useful lemmas. In particular, we prove that when T is isomorphic to ³ D ₄(q), then T is line-transitive, where q is a power of the prime p.     

A Note on the Sticky Matroid Conjecture

A matroid is sticky if any two of its extensions by disjoint sets can be glued together along the common restriction (that is, they have an amalgam). The sticky matroid conjecture asserts that a matroid is sticky if and only if it is modular. Poljak and Turzík proved that no rank-3 matroid having two disjoint lines is sticky. We show that, for r ≥ 3, no rank−r matroid having two disjoint hyperplanes is sticky. These and earlier results show that the sticky matroid conjecture for finite matroids would follow from a positive resolution of the rank-4 case of a conjecture of Kantor.     

Linked tree-decompositions of represented infinite matroids

We prove that a represented infinite matroid having finite tree-width w has a linked tree-decomposition of width at most 2w. This result should be a key lemma in showing that any class of infinite matroids representable over a fixed finite field and having bounded tree-width is well-quasi-ordered under taking minors. We also show that for every finite w, a represented infinite matroid has tree-width at most w if and only if all its finite submatroids have tree-width at most w. Both proofs rely on the use of a notion of chordality for represented matroids.