All Questions
Tagged with algebraic-combinatorics matroids
29
questions
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23
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Affiness, $U_{2,4}$ and $M(K_4).$
I do not know why $M(K_4)$ is not affine over $GF(2)$ or $GF(3)$ but it is affine over all fields with more than 3 elements. I proved that $U_{2,4}$ is $\mathbb F$-representable iff $|\mathbb F| \geq ...
- 3,139
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0
answers
20
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Lift and frame matroids.
I want to read more about lift matroid and frame matroid and their flats and relations to signed graphs, do you know any basic resources for this?
- 2,087
1
vote
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answers
23
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Characteristic polynomial and bounded regions.
I know that the number of bounded regions of a homogeneous hyperplane arrangement $\mathcal{A}$(a collection of n hyperplanes) in $\mathbb R^d$is ${ n - 1 \choose d}$ but how can this be expressed in ...
- 3,139
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1
answer
53
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Understanding contraction in hyperplane arrangements.
Here are two figures that shows a hyperplane arrangement's contraction (this is from McNulty book, "Matroids, a geometric introduction"):
I am not sure why a became a line in the right ...
- 3,139
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90
views
Hyperplane Areangements and contraction.
I am trying to understand an idea presented in McNulty book, matriods a geometric introduction about the new hyperplane arrangement $\mathcal{A}^{''} = \{ H \cap H_x | H \in \mathcal{A}\}$ where $\...
- 3,139
1
vote
1
answer
144
views
No minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I am trying to find all the matroids having no minors isomorphic to $U_{0,1}$ and $U_{1,2}.$
I know that the matroid $U_{0,n}$ is a matroid of rank zero with n edges and so it is just a vertex with n ...
- 95
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votes
1
answer
60
views
Contraction, loops and flats.
This idea is being used a lot, but I cannot justify why it is correct:
If M is a matroid and $T$ a subset of $E(M).$ Then $M/T$ has no loops iff $T$ is a flat of $M.$
I know how to proof that in a ...
- 3,139
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votes
1
answer
46
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The basis of a regular matroid.
I know that a regular matroid is one that can be represented by a totally unimodular matrix.
I also know that a rank r totally unimodular matrix is a matrix over $\mathbb R$ for which every submatrix ...
- 3,139
1
vote
1
answer
56
views
affine geometries that are self-dual matroids.
I want to know which of the affine geometries $AG(n,q)$ are self-dual matroids?
I have proved before that uniform matroids $U_{n,m}$ that are self duals are those who satisfies that $m = 2n.$ I am ...
- 3,139
3
votes
1
answer
101
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Understanding how to find the dual of a matroid.
I am trying to understand the following picture of a matroid and its dual but I found myself not understanding exactly what we are doing to find the dual:
]1
Roughly speaking, according to some ...
- 95
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votes
1
answer
68
views
if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.
here is the question I am trying to solve:
In a matroid $M,$ if $X$ is independent and $E(M) - X$ is coindependent, show that $X$ is a basis and $E(M) - X$ is a cobasis.
I know how to prove that a set ...
- 3,139
0
votes
1
answer
68
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What is the definition of affine independence then?
The following is the definition of affine dependence (This definition is from James Oxley book, second edition "matroid theory")
Definition of affine dependence:
A multiset $\{ \underline{...
- 2,087
0
votes
1
answer
36
views
Let $A\subset B$ be flats of a matroid $M$ such that $r(A)=r(B)<\infty$. Then $A = B$.
I want to prove the following lemma:
Let $r$ denotes the rank.
Lemma. Let $A\subset B$ be any flats of a matroid $M$ such that $r(A)=r(B)<\infty$. Then $A = B$.
My thoughts are:
I know that $cl(A) =...
- 3,139
0
votes
1
answer
82
views
Proving that $ \beta(M) = \beta (M - e) + \beta (M /e).$
Here is the statement I am trying to prove:
If $e \in E$ is neither a loop nor an isthmus, then $$ \beta(M) = \beta (M - e) + \beta (M /e).$$
Here are all the properties I know about the Crapo's beta ...
- 3,139
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1
answer
71
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Why always the Crapo beta invariant value greater than or equal zero?
Here are the definitions of the Crapo beta invariant I know:
My definition of the Crapo's beta invariant of a matroid from the book "Combinatorial Geometries" from page 123 and 124 is as ...
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