262 research outputs found
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic
matroids and (b) finite and infinite matroids whose ground set have some
canonical symmetry, for example row and column symmetry and transposition
symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible
to techniques from commutative algebra. This allows us to introduce for each
circuit in an algebraic matroid an invariant called circuit polynomial,
generalizing the minimal poly- nomial in classical Galois theory, and studying
the matroid structure with multivariate methods.
For (b) matroids with symmetries we introduce combinatorial invariants
capturing structural properties of the rank function and its limit behavior,
and obtain proofs which are purely combinatorial and do not assume algebraicity
of the matroid; these imply and generalize known results in some specific cases
where the matroid is also algebraic. These results are motivated by, and
readily applicable to framework rigidity, low-rank matrix completion and
determinantal varieties, which lie in the intersection of (a) and (b) where
additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the
matroidal structure and the circuit polynomials completely
Algebras related to matroids represented in characteristic zero
Let k be a field of characteristic zero. We consider graded subalgebras A of
k[x_1,...,x_m]/(x_1^2,...,x_m^2) generated by d linearly independant linear
forms. Representations of matroids over k provide a natural description of the
structure of these algebras. In return, the numerical properties of the Hilbert
function of A yield some information about the Tutte polynomial of the
corresponding matroid. Isomorphism classes of these algebras correspond to
equivalence classes of hyperplane arrangements under the action of the general
linear group.Comment: 11 pages AMS-LaTe
Combinatorics of Toric Arrangements
In this paper we build an Orlik-Solomon model for the canonical gradation of
the cohomology algebra with integer coefficients of the complement of a toric
arrangement. We give some results on the uniqueness of the representation of
arithmetic matroids, in order to discuss how the Orlik-Solomon model depends on
the poset of layers. The analysis of discriminantal toric arrangements permits
us to isolate certain conditions under which two toric arrangements have
diffeomorphic complements. We also give combinatorial conditions determining
whether the cohomology algebra is generated in degree one.Comment: 29 pages, 1 figur
Algebraic matroids with graph symmetry
This paper studies the properties of two kinds of matroids: (a) algebraic matroids
and (b) finite and infinite matroids whose ground set have some canonical symmetry,
for example row and column symmetry and transposition symmetry.
For (a) algebraic matroids, we expose cryptomorphisms making them accessible to
techniques from commutative algebra. This allows us to introduce for each circuit in an
algebraic matroid an invariant called circuit polynomial, generalizing the minimal polynomial
in classical Galois theory, and studying the matroid structure with multivariate
methods.
For (b) matroids with symmetries we introduce combinatorial invariants capturing
structural properties of the rank function and its limit behavior, and obtain proofs which
are purely combinatorial and do not assume algebraicity of the matroid; these imply
and generalize known results in some specific cases where the matroid is also algebraic.
These results are motivated by, and readily applicable to framework rigidity, low-rank
matrix completion and determinantal varieties, which lie in the intersection of (a) and (b)
where additional results can be derived. We study the corresponding matroids and their
associated invariants, and for selected cases, we characterize the matroidal structure
and the circuit polynomials completely
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
Matroids in OSCAR
OSCAR is an innovative new computer algebra system which combines and extends
the power of its four cornerstone systems - GAP (group theory), Singular
(algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic
(number theory). Here, we present parts of the module handeling matroids in
OSCAR, which will appear as a chapter of the upcoming OSCAR book. A matroid is
a fundamental and actively studied object in combinatorics. Matroids generalize
linear dependency in vector spaces as well as many aspects of graph theory.
Moreover, matroids form a cornerstone of tropical geometry and a deep link
between algebraic geometry and combinatorics. Our focus lies in particular on
computing the realization space and the Chow ring of a matroid.Comment: 13 pages, 1 figur
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