Group algebra/matrix space homomorphism

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Hi,

it seems to me that R.module_morphism for R a parent inheriting from CombinatorialFreeModule does not support a definition on_gens (probably some code by Robert Bradshaw do that but I do not remember where it is in Sage...). So the only current way to define your morphism imply to do some math to get a definition on_basis instead on_gens.

For example, this work but is ugly :

sage: S3 = SymmetricGroup(3)
sage: QG = GroupAlgebra(S3, QQ)
sage: QG.gens()
Finite family {(1,2,3): (1,2,3), (1,2): (1,2)}
sage: M_3 = MatrixSpace(QQ, 2)
sage: ma = M_3([[0,-1],[1,-1]])
sage: mb = M_3([[1,-1],[0,-1]])
sage: def img_of_perm(p, gens_mat):
    w = p.reduced_word()
    R = gens_mat[0].parent()
    return R.prod([gens_mat[i-1] for i in w])
....: 
sage: f = QG.module_morphism(on_basis=lambda x: img_of_perm(x, [ma,mb]), 
codomain=ma.parent(), category=Algebras(QQ))
sage: f(QG.an_element())    
[ 2 -5]
[ 2 -3]

Another bug : if you replace category=Algebras(QQ) by category=AlgebrasWithBasis(QQ), you get the error : TypeError: Full MatrixSpace of 2 by 2 dense matrices over Rational Field is not in Category of algebras with basis over Rational Field

In a perfect world, the category should be at least FiniteDimensionnalAlgebraWithBasis(QQ) (because the parents should lies in it if I don't say something false)

The other way is trying to find stuff from Robert to define maps between parents on generators... I am not sure but I think it won't work either because it assume that elements of the two parents are not based on RingElement (old category stuff) and your group algebra is a CombinatorialFreeModule in data structure.

anyway, generally, define an algebra homomorphism between a group algebra and a matrix space implies there exist an algorithm to decompose each element of the domain in a polynomial over the algebra generators (I am not very sure it is a prerequisite but I think so...). For the symmetric group, we know how to do that with reduced word but a lot of combinatorial free module in Sage have a method algebra_generators() but do not know how to decompose an element along these algebra generators...

GAP do probably that very well but I also never tried (GAP knows how to decompose a group element along group generators...).

I hope that help a little...

Cheers,

Nicolas B.

Hi,

I reproduce here the answer I got from Travis Scrimshaw on the mailing list sage-combinat devel.

First method

sage: S3 = SymmetricGroup(3)
sage: SGA = GroupAlgebra(S3, QQ)
sage: MS = MatrixSpace(QQ,2)
sage: mtx = [MS([[0,-1],[1,-1]]), MS([[1,-1],[0,-1]])]
sage: f = lambda x: MS.prod(mtx[i-1] for i in x.reduced_word())
sage: phi = SGA.module_morphism(f, codomain=MS)

Where the -1 is needed because of indexing. Now you can check:

sage: elt = SGA.gen(0) + SGA.gen(1); elt
(1,2) + (1,2,3)
sage: phi(elt)
[-1 -1]
[ 0  0]

Second method An alternative method is to build a group morphism and then build the associate algebra morphism. There is a subtle issue as there is currently no method to express an element of a finite group in terms of generators. In particular, a morphism can not be built by providing only the image of the generators.

sage: MG = MatrixGroup(mtx)    # take some time because it launches GAP
sage: H = Hom(S3, MG)
sage: psi = H(lambda x: MG.prod(MG.gen(i-1) for i in x.reduced_word()))
sage: kappa = SGA.module_morphism(lambda x: MS(psi(x)), codomain=MS)

You can again check

sage: elt = SGA.gen(0) + SGA.gen(1); elt
(1,2) + (1,2,3)
sage: kappa(elt)
[-1 -1]
[ 0  0]

Alternative to the second method The following shows how in general, given a finite group, one may express a given element in terms of generators using the Cayley graph:

sage: G = S3.cayley_graph(generators=S3.gens())
sage: path = G.shortest_path(S3.one(), prod(S3.gens())); path
[(), (1,2,3), (2,3)]
sage: labels = [G.edge_label(path[i], path[i+1]) for i in range(len(path)-1)]; labels
[(1,2,3), (1,2)]
sage: map(lambda x: list(S3.gens()).index(x), labels)
[0, 1]