Inconsistent handling of empty tensors when using multifusion sectors

[leburgel]

There is some inconsistency in how tensor maps without any valid fusion channels are handled when using multifusion sectors. Depending on how the spaces of such a tensor are partitioned between the domain and codomain, construction and manipulation eithers returns an empty tensor with norm zero, or throws an ArgumentError.

For example:

using TensorKit, TensorKitSectors

C0, C1 = IsingBimodule(1, 1, 0), IsingBimodule(1, 1, 1)
D0, D1 = IsingBimodule(2, 2, 0), IsingBimodule(2, 2, 1)
M, Mop = IsingBimodule(1, 2, 0), IsingBimodule(2, 1, 0)

V = (
    Vect[IsingBimodule](C0 => 2, C1 => 1), Vect[IsingBimodule](Mop => 1)',
    Vect[IsingBimodule](D0 => 2, D1 => 2), Vect[IsingBimodule](M => 2)',
    Vect[IsingBimodule](C0 => 3, C1 => 2),
)
t = randn(Float64, V[1] ⊗ V[2] ← V[3] ⊗ V[4] ⊗ V[5])
# t is fine: norm(t) == 0.0, blocksectors(t) == IsingBimodule[]
@show norm(t)
@show blocksectors(t)

repartition(t, 0)   # ok, norm = 0.0
repartition(t, 1)   # ok, norm = 0.0
repartition(t, 2)   # ok, norm = 0.0
repartition(t, 3)   # ArgumentError: invalid fusion channel
repartition(t, 4)   # ArgumentError: invalid fusion channel
repartition(t, 5)   # ok, norm = 0.0

The issue seems to be that blocksectors calls corresponding to an empty set sometimes return and empty set and sometimes throw, depening on the specific arguments:

codom = Vect[IsingBimodule](C0 => 1) ⊗ Vect[IsingBimodule](C0 => 1)
blocksectors(codom ← Vect[IsingBimodule](D0 => 1))   # ArgumentError: invalid fusion channel
blocksectors(codom ← Vect[IsingBimodule](C1 => 1))   # IsingBimodule[]

I'm not sure whether it's better to always throw or never throw. @lkdvos, @borisdevos.

[borisdevos]

Just tagging my response in the PR where this issue originated, so we can discuss here: #513 (comment)

[lkdvos]

Looking at the last example of the blocksectors, that to me looks like the behavior we would expect, in the sense that the bottom line is an empty set of sectors because fusion is allowed but there is no overlap between domain and codomain, which is equivalent to U1Irrep(0) otimes U1Irrep(0) -> U1Irrep(1) behavior.
On the other hand, the first is not because this truly is an invalid fusion diagram, connecting regions with different colors together, which should error.

Given this, it might be the case of investigating a bit more in detail what is going on and being slightly more consistent with that, since I think the original tensor you construct is of the allowed but no overlap kind, so presumably all repartitions should indeed work and behave equivalently to something like t = rand(U1Space(0 => 1) * U1Space(0 => 1), U1Space(1 => 1)). (Which we might want to test works as well)