gfan.sm1 is a package to compute global and local Grobner fans.
See  R.Bahloul and N.Takayama, arxiv, math.AG/0412044 and references as to algorithms.
At the beginning of the source code gfan.sm1, there are sample inputs cone.sample and cone.sample2.
gfan.sm1 works only with polymake 2.0. We provide a web service of computing 
with polymake 2.0.  /@@@polymake.web 1 def is set by default in gfan.sm1.
See changelog-ja.tex as to details on the difference between 2.0 and later versions.
*cone.sample ; is an example. See the source code. The state polytope is the hexagon.
*cone.Wt cone.Lpt {vertices in the output} are weights on the rays of the Grobner cone.
*cone.L gives a basis of the linearity space.
*cone.Lp gives a basis of the pointed cone. cone.Lpt is the transpose of cone.Lp.
*When v is a row vector in an ouput cone, (v cone.Lp cone.W) gives 
  the corresponding weight vector in the full variable space in D
*cone.incidence is a list of [[cone num1,facet num1], [cone num2,facet num2]]
  which means that cone num1 and cone num2 are adjacent and shares 
  the facet num1 and the facet num2
*/cone.withGblist 1 def saves the Grobner basis standing for each cone.
*Cone descriptions:
**A facet is given by its normal vector n of a cone. It gives facet num of the cone.
**A cone is defined by facet normal vectors n1, n2, ... as n1.x>=0 and n2.x >=0 and ...
**facetsv is a list of facets expressed by generators.
**nextcid is a list of the adjacent cone numbers.
**nextfid is a list of the shared facet numbers.
**vertices is the generators of a cone.
**inequalities are not necessarily unique.

Nobuki Takayama 2020-11-24