The projective geometry of dimension n over F_p is the matroid whose ground set consists of points in an n-dimensional projective space over F_p. The matroid structure is precisely the simple matroid associated to the realizable matroid of (F_p)^(n+1) (i.e. all vectors in an (n+1)-dimensional vector space over F_p) - the origin (being a loop) has been removed, and a representative is chosen for all parallel classes (= lines).
Note that projective space has a stratification into affine spaces (one of each smaller dimension). In particular, deleting any hyperplane from PG(n, p) gives AG(n, p).
i1 : PG22 = projectiveGeometry(2, 2) o1 = a matroid of rank 3 on 7 elements o1 : Matroid |
i2 : PG22 == specificMatroid "fano" o2 = true |
i3 : A = transpose sub(matrix toList(((3:0)..(3:2-1))/toList), ZZ/2) -- all vectors in (ZZ/2)^3
o3 = | 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 3 ZZ 8
o3 : Matrix (--) <--- (--)
2 2
|
i4 : areIsomorphic(PG22, simpleMatroid matroid A) o4 = true |
i5 : PG32 = projectiveGeometry(3, 2) o5 = a matroid of rank 4 on 15 elements o5 : Matroid |
i6 : representationOf PG32
o6 = | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 |
| 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 |
| 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 |
| 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 |
ZZ 4 ZZ 15
o6 : Matrix (--) <--- (--)
2 2
|
i7 : H = first hyperplanes PG32
o7 = set {8, 9, 10, 11, 12, 13, 14}
o7 : Set
|
i8 : areIsomorphic(affineGeometry(3, 2), PG32 \ H) o8 = true |
The object projectiveGeometry is a method function.