Creating subquiver representations

A subquiver is a subgraph of a quiver. Specifically, it is formed by selecting a subset $I$ of the arrows from the original quiver, ensuring that the tails and heads of the chosen arrows align with the selected vertices. In this context, there are two ways to approach a subquiver. One approach is to recall the original quiver and represent the subquiver as a subset of its arrows and vertices, denoted as $Q^I$. The flow of the resultant quiver will be zero along the arrows not in $I$. Alternatively, we can disregard the original quiver and focus solely on the arrows and vertices of the new subquiver, represented as $Q_I$. The weight of the new quiver $Q_I$ is derived from the flows of the original quiver $Q$.

The two methods corresponding to these ideas are referenced in the examples below.

i1 : Q = bipartiteQuiver(2, 3)

o1 = ToricQuiver{flow => {1, 1, 1, 1, 1, 1}                            }
                 IncidenceMatrix => | -1 -1 -1 0  0  0  |
                                    | 0  0  0  -1 -1 -1 |
                                    | 1  0  0  1  0  0  |
                                    | 0  1  0  0  1  0  |
                                    | 0  0  1  0  0  1  |
                 Q0 => {0, 1, 2, 3, 4}
                 Q1 => {{0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}}
                 synonym => toric quiver
                 weights => {-3, -3, 2, 2, 2}

o1 : ToricQuiver

i2 : Q_{0,1,3}

o2 = ToricQuiver{flow => {1, 1, 1}              }
                 IncidenceMatrix => | -1 -1 0  |
                                    | 0  0  -1 |
                                    | 1  0  1  |
                                    | 0  1  0  |
                 Q0 => {0, 1, 2, 3}
                 Q1 => {{0, 2}, {0, 3}, {1, 2}}
                 synonym => toric quiver
                 weights => {-2, -1, 2, 1}

o2 : ToricQuiver

i3 : Q^{0,1,3}

o3 = ToricQuiver{flow => {1, 1, 0, 1, 0, 0}                            }
                 IncidenceMatrix => | -1 -1 -1 0  0  0  |
                                    | 0  0  0  -1 -1 -1 |
                                    | 1  0  0  1  0  0  |
                                    | 0  1  0  0  1  0  |
                                    | 0  0  1  0  0  1  |
                 Q0 => {0, 1, 2, 3, 4}
                 Q1 => {{0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}}
                 synonym => toric quiver
                 weights => {-2, -1, 2, 1, 0}

o3 : ToricQuiver

Creating subquiver representations

See also