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### Obtaining admissible relations for acyclic tree quivers with Sage for QPA

Let $Q$ be a finite acyclic quiver which is a tree as an undirected graph.

Question: Is there a way to use Sage to obtain all admissible ideals $I$ in the quiver algebra $KQ$ (those are simply the ideals generated by paths of lenghts at least two)?

The output should be so that it can be read by the GAP-package QPA.

Here an example:

Let $Q$ be the quiver

Quiver( ["v1","v2","v3","v4"], [["v1","v2","a1"],["v2","v3","a2"],["v3","v4","a3"]] )


Then there are 5 admissible ideals given as follows:

[ [],[ (1)*a1*a2, (1)*a2*a3 ], [ (1)*a2*a3 ], [ (1)*a1*a2 ], [ (1)*a1*a2*a3 ] ]


(more generally the number of admissible ideals for a linear oriented line quiver is given by the Catalan numbers)

### Obtaining admissible relations for acyclic tree quivers with Sage for QPA

Let $Q$ be a finite acyclic quiver which is a tree as an undirected graph.

Question: Is there a way to use Sage to obtain all admissible ideals $I$ in the quiver algebra $KQ$ (those are simply the ideals generated by paths of lenghts at least two)?

The output should be so that it can be read by the GAP-package QPA.

Here an example:

Let $Q$ be The input is the quiver $Q$ given by

Quiver( ["v1","v2","v3","v4"], [["v1","v2","a1"],["v2","v3","a2"],["v3","v4","a3"]] )


Then there are The output is the 5 admissible ideals given as follows:

[ [],[ (1)*a1*a2, (1)*a2*a3 ], [ (1)*a2*a3 ], [ (1)*a1*a2 ], [ (1)*a1*a2*a3 ] ]


(more generally the number of admissible ideals for a linear oriented line quiver is given by the Catalan numbers)

### Obtaining admissible relations for acyclic tree quivers with Sage for QPA

Let $Q$ be a finite acyclic quiver which is a tree as an undirected graph.

Question: Is there a way to use Sage to obtain all admissible ideals $I$ in the quiver algebra $KQ$ (those are simply the ideals generated by paths of lenghts at least two)?

The output should be so that it can be read by the GAP-package QPA.

Here an example:

The input is the quiver $Q$ given by

Quiver( ["v1","v2","v3","v4"], [["v1","v2","a1"],["v2","v3","a2"],["v3","v4","a3"]] )


The output is the 5 admissible ideals given as follows:

[ [],[ (1)*a1*a2, (1)*a2*a3 a1*a2, a2*a3 ], [ (1)*a2*a3 a2*a3 ], [ (1)*a1*a2 a1*a2 ], [ (1)*a1*a2*a3 a1*a2*a3 ] ]


(more generally the number of admissible ideals for a linear oriented line quiver is given by the Catalan numbers)