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, a matrix, whose rows are the basis vectors.

Description

For a generic weight, theta, in $C(Q)$, the flow polytope has the same dimension as the kernel of the inc map, which is $|Q_0| - |Q_1| + 1$. Moreover, given a spanning tree of the quiver, there exists a natural basis for the kernel of the inc map constructed from the combinatorics of the quiver, see Dániel Joó, Toric Quiver Varieties, Ph.D thesis, 2015. Therefore, we can translate the flow polytope to this kernel and express the polytope on such basis. With basisForFlowPolytope Q, we calculate the basis for inc map from a spanning tree of it. If none is provided, then one is randomly chosen.

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 a method function.

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

Synopsis

Description

See also

Ways to use basisForFlowPolytope :

For the programmer