Non-full tensors

[joshmaglione]

Here's an interesting bug that Eamonn O'Brien found. With

t := 
Tensor(GF(5), \[ 3, 3, 18 ], [ GF(5) | 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], TensorCategory(\[ 1, 1, 1 ], { 
PowerSet(IntegerRing()) |
{ IntegerRing() | 0 },
{ IntegerRing() | 1, 2 }
}));

Then the fully nondegenerate tensor is

> s := FullyNondegenerateTensor(t);
> s;
Tensor of valence 3, U2 x U1 >-> U0
U2 : Full Vector space of degree 3 over GF(5)
U1 : Full Vector space of degree 3 over GF(5)
U0 : Vector space of degree 18, dimension 3 over GF(5)
Generators:
(0 0 1 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0)
(0 0 0 0 1 0 4 0 0 0 0 0 0 0 0 0 0 0)
(0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 0)
Echelonized basis:
(0 0 1 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0)
(0 0 0 0 1 0 4 0 0 0 0 0 0 0 0 0 0 0)
(0 0 0 0 0 0 0 0 0 0 0 0 1 4 0 0 0 0)

And when one asks for the densor subspace, the user gets an error.

> UniversalDensorSubspace(s);

UniversalDensorSubspace(
    t: Tensor of valence 3, U2 x U1 >-> U0 U2 : Full Vector space o...
)
In file "/home/josh/Densor/src/Densor.m", line 173, column 32:
>>     densor := DerivationClosure(Parent(t), Basis(DerivationAlgebra(t)));
                                  ^
Runtime error in 'DerivationClosure': Incompatible operators.

I am not entirely sure how to fix this---not in terms of coding. The problem is coming from the fact that, when considering only dimensions the tensor is 3 x 3 >-> 3, but the ambient vector space in the codomain is 18-dimensional.

[joshmaglione]

This might only be an issue with co-radicals. I cannot test now, but does the same phenomenon occur with radicals? Do we have proper subspaces in the coordinates with nontrivial radicals? Will come back soon.

[algeboy]

Formally the derivation closure should be in the parent tensor space of the fully nondegenerate tensor, not its embedding inside a larger tensor. The bug seems to me to be that when passing to the fully nondegenerate tensor it is storing the full 18-dimensional space as the "Parent" -- see for example the print out:

> t;
...
U0 : Vector space of degree 18, dimension 3 over GF(5)
Generators:

What needs to happen is a final reduction, rather than leaving it as a sub we need to replace U0 by

U00 :=VectorSpaceWithBasis(Basis(U0));

Then Parent(U00) is itself rather than the 18 dimensional thing.

[joshmaglione]

That is a great solution if it will work that way.