[GAP Forum] PreImageRepresentative and CanonicalBasis

[Øyvind Solberg]

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.

Best regards, Oeyvind Solberg.