potentialWalls -- lists the potential walls in the weight chamber decomposition for a given quiver

Synopsis

Usage:

potentialWalls Q

potentialWalls M
Inputs:
- Q, an instance of the type ToricQuiver,
Outputs:
- a list, of potential walls for he cone system associated to the toric quiver

Description

Every wall can be represented uniquely by a partition of the vertices Q0 of Q into two sets Qplus and Qminus. As a partition can be expressed in terms of only one of the subsets, only one of the two sets Qplus and Qminus is used in every case. Thus we denote the wall W by the subset of vertices Qplus used for defining it.

This method is written to compute the potential walls based on the incidence matrix, in the case of a flow of all unit values, or based on the entire quiver if the flow is nontrivial.

i1 : potentialWalls toricQuiver {{0,1},{0,2},{0,3},{1,2},{1,3},{2,3}}

o1 = {Wall{Qplus => {1, 2, 3}}, Wall{Qplus => {0, 2, 3}}, Wall{Qplus =>
           WallType => (0, 3)        WallType => (1, 2)        WallType
     ------------------------------------------------------------------------
     {0, 1, 3}}, Wall{Qplus => {0, 1, 2}}, Wall{Qplus => {2, 3}   },
     => (2, 1)        WallType => (3, 0)        WallType => (0, 4)  
     ------------------------------------------------------------------------
     Wall{Qplus => {1, 3}   }, Wall{Qplus => {0, 3}   }}
          WallType => (1, 3)        WallType => (2, 2)

o1 : List

i2 : potentialWalls chainQuiver {1}

o2 = {Wall{Qplus => {1}      }}
           WallType => (0, 1)

o2 : List

Ways to use potentialWalls :

potentialWalls(Matrix)
potentialWalls(ToricQuiver)

For the programmer

The object potentialWalls is a method function.