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

Synopsis

Usage:

isSemistable (L, Q)

isSemistable (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 semistable, and false otherwise.

Description

This function determines if a given subquiver is semistable with respect to the weight saved on Q. A subquiver SQ of the quiver Q is semistable 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 nonnegative.

i1 : isSemistable ({0, 1}, bipartiteQuiver(2, 3))

o1 = false

i2 : isSemistable ({0, 1, 3, 4, 5}, bipartiteQuiver(2, 3))

o2 = true

i3 : Q = bipartiteQuiver(2, 3);

i4 : S = first(subquivers(Q, Format => "quiver", AsSubquiver => true))

o4 = 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}

o4 : ToricQuiver

i5 : isSemistable (S, Q)

o5 = false

Ways to use isSemistable :

isSemistable(List,ToricQuiver)
isSemistable(ToricQuiver,ToricQuiver)

For the programmer

The object isSemistable is a method function.