Product

*(G1::GKMGraph{R}, G2::GKMGraph{R}) where {R} -> GKMGraph{R}

Construct the Cartesian product of two smooth GKM graphs, or of two orbifold GKM graphs. The character lattices are combined by direct sum.

Examples

julia> G = generalized_gkm_flag(root_system(:A, 1))
GKM graph with 2 nodes, valency 1 and axial function:
s1 -> id => (1, -1)
Birkhoff-Grothendieck connection for GKM graph with 2 nodes and valency 1

julia> G*G
GKM graph with 4 nodes, valency 2 and axial function:
s1,id -> id,id => (1, -1, 0, 0)
id,s1 -> id,id => (0, 0, 1, -1)
s1,s1 -> s1,id => (0, 0, 1, -1)
s1,s1 -> id,s1 => (1, -1, 0, 0)
Product(Birkhoff-Grothendieck, Birkhoff-Grothendieck) connection for GKM graph with 4 nodes and valency 2

julia> W = stacky_weighted_projective_space_fan([1, 2, 4]);

julia> Wp = gkm_graph_of_orbifold_toric(W);

julia> Wp*Wp
Orbifold GKM graph with 9 nodes, valency 4 and axial function:
2,1 -> 1,1 => (0, -1, 2, 0, 0, 0)
3,1 -> 1,1 => (-1, 0, 1, 0, 0, 0)
3,1 -> 2,1 => (-2, 1, 0, 0, 0, 0)
1,2 -> 1,1 => (0, 0, 0, 0, -1, 2)
2,2 -> 2,1 => (0, 0, 0, 0, -1, 2)
2,2 -> 1,2 => (0, -1, 2, 0, 0, 0)
3,2 -> 3,1 => (0, 0, 0, 0, -1, 2)
3,2 -> 1,2 => (-1, 0, 1, 0, 0, 0)
3,2 -> 2,2 => (-2, 1, 0, 0, 0, 0)
1,3 -> 1,1 => (0, 0, 0, -1, 0, 1)
1,3 -> 1,2 => (0, 0, 0, -2, 1, 0)
2,3 -> 2,1 => (0, 0, 0, -1, 0, 1)
2,3 -> 2,2 => (0, 0, 0, -2, 1, 0)
2,3 -> 1,3 => (0, -1, 2, 0, 0, 0)
3,3 -> 3,1 => (0, 0, 0, -1, 0, 1)
3,3 -> 3,2 => (0, 0, 0, -2, 1, 0)
3,3 -> 1,3 => (-1, 0, 1, 0, 0, 0)
3,3 -> 2,3 => (-2, 1, 0, 0, 0, 0)
Vertex Isotropy:
2,1 => Orbifold Vertex Isotropy, Group Structure (cyclic factors): [2]
  Tangent Representation: 
[1   1   0   0]
1,2 => Orbifold Vertex Isotropy, Group Structure (cyclic factors): [2]
  Tangent Representation: 
[0   0   1   1]
2,2 => Orbifold Vertex Isotropy, Group Structure (cyclic factors): [2, 2]
  Tangent Representation: 
[1   1   0   0]
[0   0   1   1]
3,2 => Orbifold Vertex Isotropy, Group Structure (cyclic factors): [2]
  Tangent Representation: 
[0   0   1   1]
2,3 => Orbifold Vertex Isotropy, Group Structure (cyclic factors): [2]
  Tangent Representation: 
[1   1   0   0]
Flag Isotropy:
2,1.3 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,1.4 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
1,2.1 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
1,2.2 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,2.1 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,2.2 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,2.3 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,2.4 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
3,2.1 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
3,2.2 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,3.3 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
2,3.4 => Orbifold Flag Isotropy, Group Structure (cyclic factors): [2]
  Embedding Matrix: 
[1]
Product(Algorithmic, Algorithmic) connection for GKM graph with 9 nodes and valency 4

*(pair::Tuple{GKMGraph{R}, GKMVectorBundle},
  pairs::Tuple{GKMGraph{R}, GKMVectorBundle}...) where {R}

Return (G1 * G2 * ..., (v1, v2, ...)) for one or more graph–bundle pairs, where vi is the pullback of the i-th bundle to the product graph. Equivalently, each vi is the external tensor product with the trivial line bundles on all other factors. The pullbacks share the direct sum of all bundle character lattices. For one pair, return (G1, (V1,)).