Quiver and Relations

Quiver(A) : AlgBas -> SeqEnum
For a basic algebra A, this intrinsic returns the quiver of A. The quiver is returned as a sequence of integer pairs where each pair represents an arrow. The first integer of a pair is the index of the source and the second is the index of the target of the arrow.
QuiverAndRelations(A) : AlgBas -> SeqEnum, SeqEnum, SeqEnum
For a basic algebra A, this intrinsic returns the quiver of A and associated relations. The first return value is the quiver where each arrow is given as a pair consisting of the index of the source and the index of the target of the arrow. The second return value is a sequence containing relations on the quiver. The third return value is a sequence containing a a condensed sequence of relations that does not contain the obvious relations of degree two where the head of the first arrow does not match the tail of the second.

Example AlgBas_Quiver (H92E12)

In this example we construct the quiver and relations for the basic algebra of the principal block of the alternating group A7 over GF(2). 
> G := AlternatingGroup(7);
> A := BasicAlgebraOfPrincipalBlock(G, GF(2));
> A;
Basic algebra of dimension 19 over GF(2)
Number of projective modules: 3
Number of generators: 8
> quiv, rels, crels := QuiverAndRelations(A);
> quiv;
[ <1, 2>, <1, 3>, <2, 1>, <2, 2>, <3, 1> ]
> rels;
[
    $.1*$.3*$.2*$.5 + $.2*$.5*$.1*$.3,
    $.3*$.2*$.5*$.1 + $.4^2,
    $.1*$.4,
    $.3*$.1 + $.4^2,
    $.4*$.3,
    $.5*$.2,
    $.1^2,
    $.2*$.1,
    $.4*$.1,
    $.1*$.2,
    $.2^2,
    $.4*$.2,
    $.2*$.3,
    $.3^2,
    $.5*$.3,
    $.2*$.4,
    $.3*$.4,
    $.5*$.4,
    $.1*$.5,
    $.3*$.5,
    $.4*$.5,
    $.5^2
]
> crels;
[
    $.1*$.3*$.2*$.5 + $.2*$.5*$.1*$.3,
    $.3*$.2*$.5*$.1 + $.4^2,
    $.1*$.4,
    $.3*$.1 + $.4^2,
    $.4*$.3,
    $.5*$.2
]
V2.28, 13 July 2023