# specificMatroid -- creates built-in matroid

## Synopsis

• Usage:
specificMatroid(S)
• Inputs:
• S, , or symbol, the name of the matroid
• Outputs:
• ,

## Description

Returns one of the named matroids below.

• U24
• C5
• P6
• Q6
• fano
• nonfano
• V8+
• vamos
• pappus
• nonpappus
• nondesargues
• betsyRoss
• AG32
• AG32'
• F8
• J
• L8
• O7
• P6
• P7
• P8
• P8=
• Q3(GF(3)*)
• Q6
• Q8
• R6
• R8
• R9
• R9A
• R9B
• R10
• R12
• S8
• S5612
• T8
• T12

The matroids provided in this function (together with the infinite families given by the functions affineGeometry, projectiveGeometry, binarySpike, spike, swirl, wheel, whirl, thetaMatroid, uniformMatroid) includes all "interesting matroids" listed in Oxley, p. 639 - 664 (except for the general Dowling geometry).

Many of these matroids are interesting for their (non-)representability or duality properties:

• 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 : getRepresentation 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 : getRepresentation 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

Notice that the ground set is a subset of $\{0, ..., n-1\}$ rather than $\{1, ..., n\}$.