Obtaining admissible relations for acyclic tree quivers with Sage for QPA

asked 2021-07-11 18:20:13 +0200

klaaa gravatar image

updated 2021-07-11 18:21:34 +0200

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:

[ [],[ a1*a2, a2*a3 ], [ a2*a3 ], [ a1*a2 ], [ a1*a2*a3 ] ]

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

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