[GAP Forum] PreImageRepresentative and CanonicalBasis

[Alexander Hulpke]

Dear Forum,

On Jun 10, 2011, at 6:27 AM, Øyvind Solberg wrote:

> We are interested in the following question.  Let D=(d_1,d_2,...,d_r)
> be a partition of n where d_i> 0 and let K be a field.  We say a
> vector v=(v_1,v_2,...,v_n) in K^n is D-partitioned if there is j, 1<=
> j <= r such that the only nonzero v_i's occur in positions
> d_1+d_2+---+d_(j-1)+1 to d_1+d_2+---+d_j.  E.g. if D=(2,1,3) then
> (7,6,0,0,0,0) and (0,0,4,0,0,0) and (0,0,0,4,5,0) are all
> D-partitioned whereas (7,0,3,0,0,0) and (0,4,0,0,5,0) are not
> D-partitioned.
> 
> Let W be a subspace of K^n:=FullRowSpace(K,n) generated by
> D-partitioned vectors and let p:=NaturalHomomorphismBySubspace(V,W) be
> the canonical surjection.
> 
> Now let B:=CanonicalBasis(Range(p)) and then call
> PreImagesRepresentative to the elements of B. Our question is, must
> these preimages in V be D-partitioned? (In a few examples, this seems
> to be the case.


I think what is happening is that the homomorphism extends a basis of W to one of V by adding basis vectors from V. Usually this is the standard basis. Then a basis for V/W is formed correspondingly to the newly added basis vectors. Pre-Images of these basis vectors then are the elements added from the basis of V, i.e. typically elements from the standard basis which of course are D-partitioned.

However I would not want to guarantee this behaviour, in particular if V was created somehow with a different basis. Also I could imagine NaturalHomomorphism BySubspace at some point changing its method for efficiency.

Instead, I think the following would be cleaner (and then is guaranteed to work -- essentially this is building the homomorphism with the properties as above):
Extend the basis for W to a basis for V using D-partitioned vectors, e.. vectors from the standard basis. Let C be this extended basis
Let Q=Field^(dim V-dim W) and B the canonical basis of Q.
Use p:=LeftModuleHomomorphismByMatrix(C,mat,B)   with the first dim W rows of mat zero and the last dim Q rows an identity 

Best,

  Alexander Hulpke

-- Colorado State University, Department of Mathematics,
Weber Building, 1874 Campus Delivery, Fort Collins, CO 80523-1874, USA
email: hulpke at math.colostate.edu, Phone: ++1-970-4914288
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