2,586 research outputs found
Topological representations of matroid maps
The Topological Representation Theorem for (oriented) matroids states that
every (oriented) matroid can be realized as the intersection lattice of an
arrangement of codimension one homotopy spheres on a homotopy sphere. In this
paper, we use a construction of Engstr\"om to show that structure-preserving
maps between matroids induce topological mappings between their
representations; a result previously known only in the oriented case.
Specifically, we show that weak maps induce continuous maps and that the
process is a functor from the category of matroids with weak maps to the
homotopy category of topological spaces. We also give a new and conceptual
proof of a result regarding the Whitney numbers of the first kind of a matroid.Comment: Final version, 21 pages, 8 figures; Journal of Algebraic
Combinatorics, 201
A module-theoretic approach to matroids
Speyer recognized that matroids encode the same data as a special class of
tropical linear spaces and Shaw interpreted tropically certain basic matroid
constructions; additionally, Frenk developed the perspective of tropical linear
spaces as modules over an idempotent semifield. All together, this provides
bridges between the combinatorics of matroids, the algebra of idempotent
modules, and the geometry of tropical linear spaces. The goal of this paper is
to strengthen and expand these bridges by systematically developing the
idempotent module theory of matroids. Applications include a geometric
interpretation of strong matroid maps and the factorization theorem; a
generalized notion of strong matroid maps, via an embedding of the category of
matroids into a category of module homomorphisms; a monotonicity property for
the stable sum and stable intersection of tropical linear spaces; a novel
perspective of fundamental transversal matroids; and a tropical analogue of
reduced row echelon form.Comment: 22 pages; v3 minor corrections/clarifications; to appear in JPA
Proto-exact categories of matroids, Hall algebras, and K-theory
This paper examines the category of pointed matroids
and strong maps from the point of view of Hall algebras. We show that
has the structure of a finitary proto-exact category -
a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We
define the algebraic K-theory of
via the Waldhausen construction, and show that it is
non-trivial, by exhibiting injections from the stable homotopy groups of spheres for
all . Finally, we show that the Hall algebra of is
a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.Comment: 29 page
Topological representation of matroids from diagrams of spaces
Swartz proved that any matroid can be realized as the intersection lattice of
an arrangement of codimension one homotopy spheres on a sphere. This was an
unexpected extension from the oriented matroid case, but unfortunately the
construction is not explicit. Anderson later provided an explicit construction,
but had to use cell complexes of high dimensions that are homotopy equivalent
to lower dimensional spheres.
Using diagrams of spaces we give an explicit construction of arrangements in
the right dimensions. Swartz asked if it is possible to arrange spheres of
codimension two, and we provide a construction for any codimension. We also
show that all matroids, and not only tropical oriented matroids, have a
pseudo-tropical representation.
We determine the homotopy type of all the constructed arrangements.Comment: 18 pages, 6 figures. Some more typos fixe
Testing Linear-Invariant Non-Linear Properties
We consider the task of testing properties of Boolean functions that are
invariant under linear transformations of the Boolean cube. Previous work in
property testing, including the linearity test and the test for Reed-Muller
codes, has mostly focused on such tasks for linear properties. The one
exception is a test due to Green for "triangle freeness": a function
f:\cube^{n}\to\cube satisfies this property if do not all
equal 1, for any pair x,y\in\cube^{n}.
Here we extend this test to a more systematic study of testing for
linear-invariant non-linear properties. We consider properties that are
described by a single forbidden pattern (and its linear transformations), i.e.,
a property is given by points v_{1},...,v_{k}\in\cube^{k} and
f:\cube^{n}\to\cube satisfies the property that if for all linear maps
L:\cube^{k}\to\cube^{n} it is the case that do
not all equal 1. We show that this property is testable if the underlying
matroid specified by is a graphic matroid. This extends
Green's result to an infinite class of new properties.
Our techniques extend those of Green and in particular we establish a link
between the notion of "1-complexity linear systems" of Green and Tao, and
graphic matroids, to derive the results.Comment: This is the full version; conference version appeared in the
proceedings of STACS 200
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