incInverse -- compute a flow in the preimage for a given weight

Synopsis

Usage:

incInverse(W, Q)
Inputs:
- W, a list, of integers, specifying the weight on each vertex
- Q, an instance of the type ToricQuiver,
Outputs:
- F, a list, giving a flow in the preimage of the input weight.

Description

Every integral flow induces a weight on the vertices $Q_0$ as follows: Let $a \in Q_1$ be an arrow. The symbol $a^{+} \in Q_0$ denotes its head and the symbol $a^{-} \in Q_0$ denotes its tail. The so called incidence map generates a weight from the flow $\mathbf{w}$ by $$ \text{inc}(\mathbf{w})(i) := \sum_{a\in Q_1 \atop a^{+} = i} \mathbf{w}(a) - \sum_{a\in Q_1 \atop a^{-} = i} \mathbf{w}(a) \quad \text{ for all } i \in Q_0. $$ The function incInverse calculates a vector in the preimage of such a map.

i1 : Q = toricQuiver(bipartiteQuiver(2,3));

i2 : th = {-5,-1,2,2,2};

i3 : incInverse(th, Q)

o3 = {1, 2, 2, 1, 0, 0}

o3 : List

Ways to use incInverse :

incInverse(List,ToricQuiver)

For the programmer

The object incInverse is a method function.

incInverse -- compute a flow in the preimage for a given weight

Synopsis

Description

See also

Ways to use incInverse :

For the programmer