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Subgraph

The following figure shows the GKM subgraph of a Schubert variety $X_w$ in the full flag variety of $\mathbb{C}^3$. Edge labels illustrate the axial function, while the vertex labels describe the equivariant Poincaré dual of $X_w$ via localization. Illustration of GKM subspaces

GKMtools.subgraph_from_vertices — Function
subgraph_from_vertices(G::AbstractGKMGraph, vertices_to_keep::AbstractVector{<:Integer})

Return the induced GKM subgraph of G, with the vertices of the subgraph given by vertices_to_keep. The GKM subgraph structure keeps both the ambient graph and the subgraph. The connection on the subgraph is the algorithmic connection. See also ambient_graph and subgraph.

Examples

julia> G = projective_space(GKMGraph, 3)
GKM graph with 4 nodes, valency 3 and axial function:
2 -> 1 => (1, -1, 0, 0)
3 -> 1 => (1, 0, -1, 0)
3 -> 2 => (0, 1, -1, 0)
4 -> 1 => (1, 0, 0, -1)
4 -> 2 => (0, 1, 0, -1)
4 -> 3 => (0, 0, 1, -1)
Birkhoff-Grothendieck connection for GKM graph with 4 nodes and valency 3

julia> S = subgraph_from_vertices(G, [2, 3]) # GKM subgraph structure
GKM subgraph of:
GKM graph with 4 nodes, valency 3 and axial function:
2 -> 1 => (1, -1, 0, 0)
3 -> 1 => (1, 0, -1, 0)
3 -> 2 => (0, 1, -1, 0)
4 -> 1 => (1, 0, 0, -1)
4 -> 2 => (0, 1, 0, -1)
4 -> 3 => (0, 0, 1, -1)
Birkhoff-Grothendieck connection for GKM graph with 4 nodes and valency 3
Subgraph:
GKM graph with 2 nodes, valency 1 and axial function:
3 -> 2 => (0, 1, -1, 0)
Algorithmic connection for GKM graph with 2 nodes and valency 1

julia> subgraph(S) # forget the ambient graph
GKM graph with 2 nodes, valency 1 and axial function:
3 -> 2 => (0, 1, -1, 0)
Algorithmic connection for GKM graph with 2 nodes and valency 1

julia> ambient_graph(S) # forget the subgraph
GKM graph with 4 nodes, valency 3 and axial function:
2 -> 1 => (1, -1, 0, 0)
3 -> 1 => (1, 0, -1, 0)
3 -> 2 => (0, 1, -1, 0)
4 -> 1 => (1, 0, 0, -1)
4 -> 2 => (0, 1, 0, -1)
4 -> 3 => (0, 0, 1, -1)
Birkhoff-Grothendieck connection for GKM graph with 4 nodes and valency 3
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subgraph_from_vertices(G::AbstractGKMGraph, labels_to_keep::AbstractVector{<:AbstractString})

As before, but the vertices are given by their labels.

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GKMtools.subgraph — Function
subgraph(G::AbstractGKMSubgraph)

Return the subgraph of G as a smooth or orbifold GKM graph, thus forgetting the ambient space.

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In order to construct the Poincaré dual class of a subgraph, see poincare_dual.