# basisForFlowPolytope -- compute the necessary basis vectors for the hyperplane of a flow polytope

## Synopsis

• Usage:
basisForFlowPolytope Q
basisForFlowPolytope (T, Q)
• Inputs:
• Q, an instance of the type ToricQuiver,
• T, a list, of edges comprising the spanning tree to use to generate the basis.
• Outputs:
• B, ,

## Description

The polytope associated to a toric quiver is defined in terms of the stable spanning trees for that given quiver, and hence its vertices are in a lower dimensional subspace of the space with dimension $|Q_1|$. Thus a lower dimensional basis is useful for viewing polytopes in the appropriate dimension.

 i1 : basisForFlowPolytope bipartiteQuiver(2,3) o1 = | -1 0 | | 0 -1 | | 1 1 | | 1 0 | | 0 1 | | -1 -1 | 6 2 o1 : Matrix ZZ <--- ZZ
 i2 : basisForFlowPolytope ({0,1,4,5}, bipartiteQuiver(2,3)) o2 = | 0 1 | | 1 -1 | | -1 0 | | 0 -1 | | -1 1 | | 1 0 | 6 2 o2 : Matrix ZZ <--- ZZ

## Ways to use basisForFlowPolytope :

• "basisForFlowPolytope(List,ToricQuiver)"
• "basisForFlowPolytope(ToricQuiver)"

## For the programmer

The object basisForFlowPolytope is .