Here is the GAP session for Example 3.6.2 :
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gap> Read("weights");


The fundamental weights:[ [ -1/2, -1/2, 1/2 ], [ 1/4, -3/4, 1/4 ], 
  [ -1/4, -1/4, -1/4 ] ]


The monoid generators:[ [ -1, -1, -1 ], [ 0, -1, 0 ], [ 1, -3, 1 ],
  [ -1, -1, 0 ], [ 0, -2, 1 ], [ -1, -1, 1 ] ]


The coefficients (in terms of the fundamental weights):
[ [ 0, 0, 4 ], [ 0, 1, 1 ], [ 0, 4, 0 ], [ 1, 0, 2 ], [ 1, 2, 0 ], 
  [ 2, 0, 0 ] ]


The orbits of the fundamental weights:
[ [ [ -1/2, -1/2, 1/2 ], [ -1/2, 1/2, -1/2 ], [ -1/2, 1/2, 1/2 ], 
      [ 1/2, -1/2, -1/2 ], [ 1/2, -1/2, 1/2 ], [ 1/2, 1/2, -1/2 ] ], 
  [ [ -3/4, 1/4, 1/4 ], [ 1/4, -3/4, 1/4 ], [ 1/4, 1/4, -3/4 ], 
      [ 1/4, 1/4, 1/4 ] ], 
  [ [ -1/4, -1/4, -1/4 ], [ -1/4, -1/4, 3/4 ], [ -1/4, 3/4, -1/4 ], 
      [ 3/4, -1/4, -1/4 ] ] ]
gap> Runtime();
2030
gap> quit;
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Explanation:

After reading in the file "weights", the following are printed out.
First, the coordinates of the fundamental weights lambda_1,...,lambda_3,
then the explicit monoid generators m_1,...,m_6.
Next, come the coefficients used in expressing the  m_i's  as linear
combinations of the fundamental weights. For example, the 2nd sublist
[ 0, 1, 1 ] tells us that  m_2 ( = [ 0, -1, 0 ]) is equal to lambda_2+lambda_3.
(In the paper, the numbering of the m's is different; e.g., m_2 here is m_4
in the paper.)
Finally, the orbits of  lambda_1,...,lambda_3  under the acting group  G
are given; these lead immediately to the orbit sums  sigma(lambda_i)
given in the paper.

The runtime was approx. 2 seconds (on a 166MHz Pentium PC).