The matroid (basis) polytope of a matroid on n elements lives in R^n, and is the convex hull of the indicator vectors of the bases.
For uniform matroids, the basis polytope is precisely the hypersimplex:
i1 : U24 = uniformMatroid(2, 4) o1 = a matroid of rank 2 on 4 elements o1 : Matroid |
i2 : A = basisIndicatorMatrix U24 o2 = | 1 1 0 1 0 0 | | 1 0 1 0 1 0 | | 0 1 1 0 0 1 | | 0 0 0 1 1 1 | 4 6 o2 : Matrix ZZ <--- ZZ |
In order to obtain an actual polytope, one must take the convex hull of the columns of the indicator matrix, which is provided by the Polyhedra package:
i3 : needsPackage "Polyhedra" o3 = Polyhedra o3 : Package |
i4 : P = convexHull A o4 = P o4 : Polyhedron |
i5 : vertices P o5 = | 1 1 0 1 0 0 | | 1 0 1 0 1 0 | | 0 1 1 0 0 1 | | 0 0 0 1 1 1 | 4 6 o5 : Matrix QQ <--- QQ |
The Gelfand-Goresky-MacPherson-Serganova characterizes which polytopes are basis polytopes for a matroid: namely, each edge is of the form $e_i - e_j$ for some $i, j$, where $e_i$ are the standard basis vectors.
The object basisIndicatorMatrix is a method function.