Creating toric quivers

toric quivers are represented as a type of HashTable with the following keys:

IncidenceMatrix: weighted incidence matrix giving the vertex-edge connectivity structure of $Q$
Q0: list of vertices
Q1: list of edges
flow: list of integers giving the flow on each edge
weights: induced weights on vertices given by the image of the flow

One can generate the quiver Q associated to the bipartite graph K_{2,3} with a random flow w as follows:

i1 : Q0 = {{0,2},{0,3},{0,4},{1,2},{1,3},{1,4}}

o1 = {{0, 2}, {0, 3}, {0, 4}, {1, 2}, {1, 3}, {1, 4}}

o1 : List

i2 : Q = toricQuiver(Q0, Flow => "Random")

o2 = ToricQuiver{flow => {8, 1, 3, 7, 8, 3}                            }
                 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 => {-12, -18, 15, 9, 6}

o2 : ToricQuiver

Alternatively, one can construct a toric quiver using any of the following constructions:

create a toric quiver from matrix

i3 : Q = toricQuiver matrix({{-1,-1,-1,-1},{1,1,0,0},{0,0,1,1}})

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

o3 : ToricQuiver

create a toric quiver from matrix with specified flow

i4 : Q = toricQuiver(matrix({{-1,-1,-1,-1},{1,1,0,0},{0,0,1,1}}), {3, 1, 0, 5})

o4 = ToricQuiver{flow => {3, 1, 0, 5}                  }
                 IncidenceMatrix => | -1 -1 -1 -1 |
                                    | 1  1  0  0  |
                                    | 0  0  1  1  |
                 Q0 => {0, 1, 2}
                 Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                 synonym => toric quiver
                 weights => {-9, 4, 5}

o4 : ToricQuiver

create a toric quiver from a list of edges

i5 : Q = toricQuiver {{0,1},{0,1},{0,2},{0,2}}

o5 = ToricQuiver{flow => {1, 1, 1, 1}                  }
                 IncidenceMatrix => | -1 -1 -1 -1 |
                                    | 1  1  0  0  |
                                    | 0  0  1  1  |
                 Q0 => {0, 1, 2}
                 Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                 synonym => toric quiver
                 weights => {-4, 2, 2}

o5 : ToricQuiver

create a toric quiver from a list of edges and a flow

i6 : Q = toricQuiver ({{0,1},{0,1},{0,2},{0,2}}, {1,2,3,4})

o6 = ToricQuiver{flow => {1, 2, 3, 4}                  }
                 IncidenceMatrix => | -1 -1 -1 -1 |
                                    | 1  1  0  0  |
                                    | 0  0  1  1  |
                 Q0 => {0, 1, 2}
                 Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                 synonym => toric quiver
                 weights => {-10, 3, 7}

o6 : ToricQuiver

create a toric quiver from a matrix with keyword flow

i7 : Q = toricQuiver(matrix({{-1,-1,-1,-1},{1,1,0,0},{0,0,1,1}}), Flow => "Canonical")

o7 = ToricQuiver{flow => {1, 1, 1, 1}                  }
                 IncidenceMatrix => | -1 -1 -1 -1 |
                                    | 1  1  0  0  |
                                    | 0  0  1  1  |
                 Q0 => {0, 1, 2}
                 Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                 synonym => toric quiver
                 weights => {-4, 2, 2}

o7 : ToricQuiver

create a toric quiver from a matrix with random flow

i8 : Q = toricQuiver(matrix({{-1,-1,-1,-1},{0,0,1,1},{1,1,0,0}}), Flow => "Random")

o8 = ToricQuiver{flow => {8, 8, 3, 7}                  }
                 IncidenceMatrix => | -1 -1 -1 -1 |
                                    | 1  1  0  0  |
                                    | 0  0  1  1  |
                 Q0 => {0, 1, 2}
                 Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                 synonym => toric quiver
                 weights => {-26, 16, 10}

o8 : ToricQuiver

create a toric quiver copied from another one

i9 : R = toricQuiver(Q)

o9 = ToricQuiver{flow => {8, 8, 3, 7}                  }
                 IncidenceMatrix => | -1 -1 -1 -1 |
                                    | 1  1  0  0  |
                                    | 0  0  1  1  |
                 Q0 => {0, 1, 2}
                 Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                 synonym => toric quiver
                 weights => {-26, 16, 10}

o9 : ToricQuiver

create a toric quiver copied from another, but with alternative flow

i10 : R = toricQuiver(Q, {1,2,3,4})

o10 = ToricQuiver{flow => {1, 2, 3, 4}                  }
                  IncidenceMatrix => | -1 -1 -1 -1 |
                                     | 1  1  0  0  |
                                     | 0  0  1  1  |
                  Q0 => {0, 1, 2}
                  Q1 => {{0, 1}, {0, 1}, {0, 2}, {0, 2}}
                  synonym => toric quiver
                  weights => {-10, 3, 7}

o10 : ToricQuiver