109 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
Binary Matroids and Quantum Probability Distributions
We characterise the probability distributions that arise from quantum
circuits all of whose gates commute, and show when these distributions can be
classically simulated efficiently. We consider also marginal distributions and
the computation of correlation coefficients, and draw connections between the
simulation of stabiliser circuits and the combinatorics of representable
matroids, as developed in the 1990s.Comment: 24 pages (inc appendix & refs
Specht Polytopes and Specht Matroids
The generators of the classical Specht module satisfy intricate relations. We
introduce the Specht matroid, which keeps track of these relations, and the
Specht polytope, which also keeps track of convexity relations. We establish
basic facts about the Specht polytope, for example, that the symmetric group
acts transitively on its vertices and irreducibly on its ambient real vector
space. A similar construction builds a matroid and polytope for a tensor
product of Specht modules, giving "Kronecker matroids" and "Kronecker
polytopes" instead of the usual Kronecker coefficients. We dub this process of
upgrading numbers to matroids and polytopes "matroidification," giving two more
examples. In the course of describing these objects, we also give an elementary
account of the construction of Specht modules different from the standard one.
Finally, we provide code to compute with Specht matroids and their Chow rings.Comment: 32 pages, 5 figure
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
Displaying blocking pairs in signed graphs
A signed graph is a pair (G, S) where G is a graph and S is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of S. A blocking pair of (G, S) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor
On Excluded Minors for Even Cut Matroids
In this thesis we will present two main theorems that can be used to study
minor minimal non even cut matroids.
Given any signed graph we can associate an even cut matroid. However, given
an even cut matroid, there are in general, several signed graphs which
represent that matroid. This is in contrast to, for instance graphic (or
cographic) matroids, where all graphs corresponding to a particular
graphic matroid are essentially equivalent. To tackle the multiple
non equivalent representations of even cut matroids we use the concept of
Stabilizer first introduced by Wittle. Namely, we show the following:
given a "substantial" signed graph, which represents a matroid N that is a
minor of a matroid M, then if the signed graph extends to a signed graph
which represents M then it does so uniquely. Thus the representations of the
small matroid determine the representations of the larger matroid containing
it. This allows us to consider each representation of an even cut matroid
essentially independently.
Consider a small even cut matroid N that is a minor of a matroid M that is
not an even cut matroid. We would like to prove that there exists a
matroid N' which contains N and is contained in M such that the size of N'
is small and such that N' is not an even cut matroid (this would imply in
particular that there are only finitely many minimally non even cut
matroids containing N). Clearly, none of the representations of N extends to
M. We will show that (under certain technical conditions) starting from a
fixed representation of N, there exists a matroid N' which contains N
and is contained in M such that the size of N' is small and such that the
representation of N does not extend to N'
Totally free expansions of matroids
The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension M′ of M by an element x′ such that {x, x′} is independent and M′ is unaltered by swapping the labels on x and x′. When x is fixed, a representation of M.\x extends in at most one way to a representation of M. A 3-connected matroid N is totally free if neither N nor its dual has a fixed element whose deletion is a series extension of a 3-connected matroid. The significance of such matroids derives from the theorem, established here, that the number of inequivalent representations of a 3-connected matroid M over a finite field F is bounded above by the maximum, over all totally free minors N of M, of the number of inequivalent F-representations of N. It is proved that, within a class of matroids that is closed under minors and duality, the totally free matroids can be found by an inductive search. Such a search is employed to show that, for all r ≥ 4, there are unique and easily described rank-r quaternary and quinternary matroids, the first being the free spike. Finally, Seymour\u27s Splitter Theorem is extended by showing that the sequence of 3-connected matroids from a matroid M to a minor N, whose existence is guaranteed by the theorem, may be chosen so that all deletions and contractions of fixed and cofixed elements occur in the initial segment of the sequence. © 2001 Elsevier Science
Torus orbits on homogeneous varieties and Kac polynomials of quivers
In this paper we prove that the counting polynomials of certain torus orbits
in products of partial flag varieties coincides with the Kac polynomials of
supernova quivers, which arise in the study of the moduli spaces of certain
irregular meromorphic connections on trivial bundles over the projective line.
We also prove that these polynomials can be expressed as a specialization of
Tutte polynomials of certain graphs providing a combinatorial proof of the
non-negativity of their coefficients
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