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ThinSincereQuivers :: isStable

isStable -- determines if a subquiver is semistable with respect to a given weight

Synopsis

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}}
                 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}}
                 weights => {-1, 0, 1, 0, 0}

o6 : ToricQuiver
 
i7 : isStable (S, Q)

o7 = false

Ways to use isStable :

For the programmer

The object isStable is a method function.