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 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: Quiver (ex-3af24d)

In this example we construct the quiver and relations for the basic algebra of the principal block of the alternating group \(A_7\) 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
]

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