BergmanFan -- the Bergman fan of a matroid

Synopsis

Usage:

BergmanFan(M)
Inputs:
- M, a matroid,
Outputs:
- T, a tropical cycle,

Description

Computes the Bergman fan of a matroid, with the fine fan structure. This uses the Matroids package; the input should be a matroid in the sense of that package. The output is a tropical cycle T whose underlying fan is the fine fan structure in the sense of Ardila-Klivans on the Bergman fan of the matroid. This has underlying simplicial complex the order complex of the lattice of flats of the matroid M.

If the ground set of the matroid has size n, then the fan given is in R^n, so there is always a lineality space of dimension at least one.

i1 : M=uniformMatroid(2,3);

i2 : T=BergmanFan(M);

i3 : rays T

o3 = | 0 0  0 |
     | 1 -1 0 |
     | 0 -1 1 |

              3       3
o3 : Matrix ZZ  <-- ZZ

i4 : maxCones T

o4 = {{2}, {0}, {1}}

o4 : List

i5 : linealitySpace T

o5 = | 1 |
     | 1 |
     | 1 |

              3       1
o5 : Matrix ZZ  <-- ZZ

For the programmer

The object BergmanFan is a function closure.