GAP (toric) - Chapter 2: Cones and semigroups

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2 Cones and semigroups
2.1 Cones
2.1-1 InsideCone
2.1-2 InDualCone
2.1-3 PolytopeLatticePoints
2.1-4 Faces
2.1-5 ConesOfFan
2.1-6 NumberOfConesOfFan
2.1-7 ToricStar
2.2 Semigroups
2.2-1 DualSemigroupGenerators

### 2 Cones and semigroups

#### 2.1 Cones

This section introduces the toric commands which deal with cones and related combinatorial-geometric objects. Recall, a cone is a strongly convex polyhedral cone ([Ful93], page 4).

##### 2.1-1 InsideCone

`‣ InsideCone`( v, L ) | ( function )

This command returns `true` if the vector v belongs to the interior of the (strongly convex polyhedral) cone generated by the vectors in L.

This procedure does not check if L generates a strongly convex polyhedral cone.

    gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
    gap> InsideCone(v,L);
    false
    gap> L:=[[1,0],[3,4]];;
    gap> v:=[1,-7]; InsideCone(v,L);
    [ 1, -7 ]
    false
    gap> v:=[4,-3]; InsideCone(v,L);
    [ 4, -3 ]
    false
    gap> v:=[4,-4]; InsideCone(v,L);
    [ 4, -4 ]
    false
    gap> v:=[4,1]; InsideCone(v,L);
    [ 4, 1 ]
    true

##### 2.1-2 InDualCone

`‣ InDualCone`( v, L ) | ( function )

This command returns `true` if v belongs to the dual of the cone generated by the vectors in L.

    gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
    gap> InDualCone(v,L);
    true
    gap> L:=[[1,0],[3,4]];
    [ [ 1, 0 ], [ 3, 4 ] ]
    gap> v:=[1,-7]; InDualCone(v,L);
    [ 1, -7 ]
    false
    gap> v:=[4,-3]; InDualCone(v,L);
    [ 4, -3 ]
    true
    gap> v:=[4,-4]; InDualCone(v,L);
    [ 4, -4 ]
    false
    gap> v:=[4,1]; InDualCone(v,L);
    [ 4, 1 ]
    true

##### 2.1-3 PolytopeLatticePoints

`‣ PolytopeLatticePoints`( A, Perps ) | ( function )

Input: Perps=[v_1,...,v_k] is the list of ``inward normal" vectors perpendicular to the walls of a polytope P in the vector space L_0^*⊗ Q,
A=[a_1,...,a_k] is a k-tuple of integers, where a_i denotes the amount the i-th ``wall" (defined by the normal v_i) is shifted from the origin (each a_i is assumed non-negative).
For example, the polytope P with faces `[x=0, x=a, y=0, y=b]` has Perps=[[1,0],[-1,0],[0,1],[0,-1]] and A=[0,a,0,b].
Output: the list of points in P ∩ L_0^*.

    gap> Perps:=[[1,0],[-1,0],[0,1],[0,-1]];
    [ [ 1, 0 ], [ -1, 0 ], [ 0, 1 ], [ 0, -1 ] ]
    gap> A:=[0,4,0,3];
    [ 0, 4, 0, 3 ]
    gap> PolytopeLatticePoints(A,Perps);
    [ [ 0, 0 ], [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 0 ], [ 1, 1 ], [ 1, 2 ],
      [ 1, 3 ], [ 2, 0 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 0 ], [ 3, 1 ],
      [ 3, 2 ], [ 3, 3 ], [ 4, 0 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ] ]
    gap> Length(last);
    20

##### 2.1-4 Faces

`‣ Faces`( Rays ) | ( function )

Input: Rays is a list of rays for the fan ∆
Output: All the normals to the faces (hyperplanes of the cone).

    gap> Cones1:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
    gap> Faces(Cones1[1]);
    [ [ 1/2, 1 ], [ 2, 1 ] ]
    gap> Faces(Cones1[2]);
    [ [ -2, -1 ], [ -1, 1 ] ]
    gap> Cones2:=[[[ 2,0,0],[0,2,0],[0,0,2]], [[2,0,0], [0,2,0], [2,-2,1],[1,2,-2]]];;
    gap> Faces(Cones2[1]);
    [ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ]
    gap> Faces(Cones2[2]);
    [ [ 1/3, 5/6, 1 ], [ 1/2, 0, -1 ], [ 2, 0, 1 ] ]

##### 2.1-5 ConesOfFan

`‣ ConesOfFan`( Delta, k ) | ( function )

Input: Delta is the fan of cones,
k is the dimension of the cones desired.
Output: The k-dimensional cones in the fan.

##### 2.1-6 NumberOfConesOfFan

`‣ NumberOfConesOfFan`( Delta, k ) | ( function )

Input: Delta is the fan of cones in V=Q^n,
k is the dimension of the cones counted.
Output: The number of k-dimensional cones in the fan.

Idea: The fan Delta is represented as a set of maximal cones. For each maximal cone, look at the k-dimensional faces obtained by taking n choose k subsets of the rays describing the cone. Certain of these k-subsets yield the desired cones.

    gap> Delta0:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
    gap> NumberOfConesOfFan(Delta0,2);
    6
    gap> ConesOfFan(Delta0,2);
    [ [ [ 0, 0, 2 ], [ 0, 2, 0 ] ], [ [ 0, 0, 2 ], [ 2, 0, 0 ] ],
      [ [ 0, 2, 0 ], [ 1, 2, -2 ] ], [ [ 0, 2, 0 ], [ 2, -2, 1 ] ],
      [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 1, 2, -2 ], [ 2, -2, 1 ] ] ]
    gap> ConesOfFan(Delta0,1);
    [ [ [ 0, 0, 2 ] ], [ [ 0, 2, 0 ] ], [ [ 1, 2, -2 ] ],
      [ [ 2, -2, 1 ] ], [ [ 2, 0, 0 ] ] ]
    gap> NumberOfConesOfFan(Delta0,1);
    5

##### 2.1-7 ToricStar

`‣ ToricStar`( sigma, Delta ) | ( function )

Input: sigma is a cone in the fan, represented by its set of maximal (i.e., highest dimensional) cones.
Delta is the fan of cones in V=Q^n.
Output: The star of the cone sigma in Delta, i.e., the cones τ which have sigma as a face.

    gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ],
    >                 [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
    gap> #this is the set of maximal cones in the fan Delta
    gap> ToricStar([[1,0]],MaxCones);
    [  ]
    gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
    [ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
      [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 2, -2, 1 ], [ 1, 2, -2 ] ] ]
    gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[1,1,-2] ] ];;
    gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
    [ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
      [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 1, 1, -2 ] ] ]
    gap> ToricStar([[1,0]],MaxCones);
    [  ]

#### 2.2 Semigroups

##### 2.2-1 DualSemigroupGenerators

`‣ DualSemigroupGenerators`( L ) | ( function )

Input: L is a list of integral n-vectors generating a cone σ.
Output: the generators of S_σ,

Idea: let M be the maximum of the absolute values of the coordinates of the L[i]'s, for each vector v in [1..M]^n, test if v is in the dual cone σ^*. If so, add v to list of possible generators. Once this for loop is finished, one can check this list for redundant generators. The trick is to simply omit those elements which are of the form d_1+d_2, where d_1 and d_2 are ``small" elements in the integral dual cone.

This program is not very efficient and should not be used in ``large examples'' involving semigroups with ``many'' generators. For example, if you take L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]]; then `DualSemigroupGenerators(L);` can exhaust GAP's memory allocation.

    gap> L:=[[1,0],[3,4]];; DualSemigroupGenerators([[1,0],[3,4]]);
    [ [ 0, 0 ], [ 0, 1 ], [ 1, 0 ], [ 2, -1 ], [ 3, -2 ], [ 4, -3 ] ]
    gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
    gap> DualSemigroupGenerators(L);
    [ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, -1, 0 ], [ 1, 0, -1 ] ]

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