isStable -- whether a subquiver is semistable with respect to a given weight

Synopsis

Usage:

isStable (L, Q)

isStable (SQ, Q)
Inputs:
- L, a list, of the indices of arrows in Q that make up the subquiver in question
- Q, an instance of the type ToricQuiver,
- SQ, an instance of the type ToricQuiver, A subquiver of the quiver Q
Outputs:
- a Boolean value, true if the subquiver is stable, and false otherwise.

Description

This function determines if a given subquiver is stable with respect to the weight saved on Q. A subquiver SQ of the quiver Q is stable if for every subset V of the vertices of Q that is also SQ-successor closed, the sum of the weights associated to V is positive.

i1 : Q = bipartiteQuiver(2, 3);

i2 : P = Q^{0,1,4,5};

i3 : isStable(P, Q)

o3 = true

i4 : isStable ({0, 1}, bipartiteQuiver(2, 3))

o4 = false

i5 : Q = bipartiteQuiver(2, 3)

o5 = 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}

o5 : ToricQuiver

i6 : S = first(subquivers(Q, Format => "quiver", AsSubquiver => true))

o6 = ToricQuiver{flow => {1, 0, 0, 0, 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 => {-1, 0, 1, 0, 0}

o6 : ToricQuiver

i7 : isStable (S, Q)

o7 = false

Ways to use isStable :

isStable(List,ToricQuiver)
isStable(ToricQuiver,ToricQuiver)

For the programmer

The object isStable is a method function.