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Obtaining the Solomon-Orlik algebra in QPA with the help of Sage

asked 2021-06-07 18:35:55 +0200

klaaa gravatar image

updated 2021-06-07 20:35:07 +0200

slelievre gravatar image

The Solomon-Orlik algebra (see Sage documentation: Orlik-Solomon algebra) associates to a finite matroid a finite dimensional algebra. It is implemented in Sage to obtain generators and relations of those algebras but there seems to be no easy way to translate them into the GAP-package QPA where many more functions are available to study those algebras.

Let M be a finite matroid which we can assume for simplicity given as a finite set X of vectors with set of circuits C, which are simply the minimal linearly dependent subsets of X.

Recall that the exterior algebra on a ground set of variables x1,...,xn (assume them to be ordered x1 < ... < xn) is given as the quotient of the free algebra on x1,...,xn modulo the relations xi^2 and xi xj +xj xi for i < j.

In QPA the exterior algebra for n=4 is for example obtained as follows:

Q := Quiver(1,[[1,1,"x1"],[1,1,"x2"],[1,1,"x3"],[1,1,"x4"]]);
KQ := PathAlgebra(Rationals,Q);
AssignGeneratorVariables(KQ);
rel := [x1^2,x2^2,x3^2,x4^2,x1*x2+x2*x1,x1*x3+x3*x1,x1*x4+x4*x1,x2*x3+x3*x2,x2*x4+x4*x2,x3*x4+x4*x3];
A := KQ/rel;

Now the Solomon-Orlik algebra (see Sage documentation: Orlik-Solomon algebra) of a matroid M is given by adding the relations $$\sum_{r=1}^{t}{(-1)^{r-1}{x_{j_1} ... \hat{x_{j_r}} .... x_{j_t}}$$ for every element (x1< ... < xt) in the circuits C.

For example when M is the matroid given by the vectors x1=t=(1,-1,0), x2=u=(0,1,-1), x3=v=(-1,0,1), x4=w=(1,1,1). then we have one circut given by x1 x2 x3 and thus one additional relation given by x2 x3 - x1 x3 +x1x2 and the algebra in QPA can be obtained as follows:

Q := Quiver(1,[[1,1,"x1"],[1,1,"x2"],[1,1,"x3"],[1,1,"x4"]]);
KQ := PathAlgebra(Rationals,Q);
AssignGeneratorVariables(KQ);
rel := [x1^2,x2^2,x3^2,x4^2,x1*x2+x2*x1,x1*x3+x3*x1,x1*x4+x4*x1,x2*x3+x3*x2,x2*x4+x4*x2,x3*x4+x4*x3,x2*x3-x1*x3+x1*x2];
A := KQ/rel;

Question: Is there a direct way to directly obtain the quiver and relations for QPA in the format as in the examples using Sage for a given matroid?

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answered 2021-06-10 06:16:50 +0200

tkarn gravatar image

Here is one way to get the generators of the Orlik-Solomon ideal:

sage: M = matroids.CompleteGraphic(3)
sage: OS = M.orlik_solomon_algebra(QQ)
sage: E.<x,y,z> = algebras.Exterior(QQ)
sage: d = [[((-1)^i,tuple(list(c)[:i]+list(c)[i+1:])) for i in range(len(c))] for c in M.circuits()]
sage: rels = [E.sum([sgn*E.basis()[idx] for sgn,idx in c]) for c in d]; rels
[x*y - x*z + y*z]
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Comments

Thank you very much! This works. I am not so experienced in Sage: Is there a quick way to make Sage automatically write "E.<x1,x2,...,xn> = algebras.Exterior(QQ)" (so that the variables names are xi for i=1,...,n where n is len(G) with G = OS.algebra_generators()))

klaaa gravatar imageklaaa ( 2021-06-11 09:18:30 +0200 )edit

One way to do it is algebras.Exterior(QQ,[f'x{i}' for i in range(1, len(G)+1)])

tkarn gravatar imagetkarn ( 2021-06-11 15:57:07 +0200 )edit

Thanks again!

klaaa gravatar imageklaaa ( 2021-06-11 16:51:49 +0200 )edit

A better way to do it might be: E = algebras.Exterior(QQ, 'x', len(G)). Sorry for the clunky previous answer.

tkarn gravatar imagetkarn ( 2021-06-16 20:04:10 +0200 )edit

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Asked: 2021-06-07 18:35:55 +0200

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Last updated: Jun 10