specificMatroid -- creates built-in matroid

Synopsis

• Usage:

  specificMatroid(S)

• Inputs:
  • S, a string, the name of the matroid
• Outputs:
  • a matroid,

Description

• U24
• C5
• P6
• Q6
• fano
• nonfano
• V8+
• vamos
• pappus
• nonpappus
• AG32
• R9A
• R9B
• R10
• betsyRoss

• U24 is the uniform matroid of rank 2 on 4 elements, i.e. the 4 point line, and is the unique forbidden minor for representability over the field of 2 elements
• The Fano matroid F7 is the matroid of the projective plane over F_2, and is representable only in characteristic 2. The non-Fano matroid is a relaxation of F7, and is representable only in characteristic not equal to 2.
• The Pappus matroid is an illustration of Pappus' theorem. By the same token, the non-Pappus matroid is a relaxation which is not representable over any field.
• The Vamos matroid V, which is a relaxation of V8+, is the smallest (size) matroid which is not representable over any field - indeed, it is not even algebraic. V8+ is identically self-dual, while V is isomorphic to its dual.
• AG32 is the affine geometry corresponding to a 3-dimensional vector space over F_2, and is identically self-dual, with circuits equal to its hyperplanes. A relaxation of AG32 is the smallest matroid not representable over any field, with fewer basis elements than V.
• R9A and R9B (along with their duals) are the only matroids on <= 9 elements that are not representable over any field, although their foundations do not have $1$ as a fundamental element.
• R10 is a rank 5 matroid on 10 elements, which is the unique splitter for the class of regular matroids.
• The Betsy Ross matroid is a matroid which is representable over the Golden Mean partial field

i1 : F7 = specificMatroid "fano"

o1 = a matroid of rank 3 on 7 elements

o1 : Matroid

i2 : all(F7_*, x -> areIsomorphic(matroid completeGraph 4, F7 \ {x}))

o2 = true

i3 : AG32 = specificMatroid "AG32"

o3 = a matroid of rank 4 on 8 elements

o3 : Matroid

i4 : representationOf AG32

o4 = | 1 1 1 1 1 1 1 1 |
     | 0 0 0 0 1 1 1 1 |
     | 0 0 1 1 0 0 1 1 |
     | 0 1 0 1 0 1 0 1 |

             ZZ 4       ZZ 8
o4 : Matrix (--)  <--- (--)
              2          2

i5 : AG32 == dual AG32

o5 = true

i6 : R10 = specificMatroid "R10"

o6 = a matroid of rank 5 on 10 elements

o6 : Matroid

i7 : representationOf R10

o7 = | 1 0 0 0 0 1 1 0 0 1 |
     | 0 1 0 0 0 1 1 1 0 0 |
     | 0 0 1 0 0 0 1 1 1 0 |
     | 0 0 0 1 0 0 0 1 1 1 |
     | 0 0 0 0 1 1 0 0 1 1 |

             ZZ 5       ZZ 10
o7 : Matrix (--)  <--- (--)
              2          2

i8 : areIsomorphic(R10 \ set{0}, matroid completeMultipartiteGraph {3,3})

o8 = true

Caveat

See also

• matroid -- constructs a matroid
• affineGeometry -- affine geometry of rank n+1 over F_p
• uniformMatroid -- uniform matroid
• allMatroids -- returns all n-element matroids