makeTight -- return a tight quiver with the same flow polytope

Synopsis

Usage:

makeTight(W, Q)
Inputs:
- W, a list, of values corresponding to a weight on each arrow of Q
- Q, an instance of the type ToricQuiver,
Outputs:
- Q, an instance of the type ToricQuiver, that is tight with respect to the flow on the input, and which has the same flow polytope as the input.

Description

Let $\theta$ be an integral weight assigned to the vertices of a quiver $Q$. The quiver $Q$ is called $\theta$-tight if for every arrow $\alpha$, the subquiver $Q\setminus \alpha$ is $\theta$-stable. Every quiver can be tightened by contraction of certain arrows in Q and changing the weight accordingly, see Section 4 at Altmann, Klaus, and Duco van Straten. "Smoothing of quiver varieties." manuscripta mathematica 129 (2009): 211-230.

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 : w = {-5,-1,2,2,2}

o2 = {-5, -1, 2, 2, 2}

o2 : List

i3 : makeTight(w, Q)

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

o3 : ToricQuiver

For the programmer

The object makeTight is a function closure.