define DualSector

[ogauthe]

This PR defines DualSector <: AbstractSector and changes the logic of dual to put it at the level of the SymmetrySector.

Linked PR with ITensor/GradedUnitRanges.jl#15, ITensor/BlockSparseArrays.jl#45, ITensor/TensorAlgebra.jl#25

[ogauthe]

I am working on replacing DualSector{SectorProduct{Sectors}} with SectorProduct{DualArguments}. I see an issue with the tricky empty case: how to define the dual of SectorProduct{Tuple{}} or SectorProduct{NamedTuple{()}}?

Of course me may have dedicated code to handle them, making an exception and defining DualSector{SectorProduct{Tuple{}}}, but this is not a satisfying solution.

[mtfishman]

I am working on replacing DualSector{SectorProduct{Sectors}} with SectorProduct{DualArguments}. I see an issue with the tricky empty case: how to define the dual of SectorProduct{Tuple{}} or SectorProduct{NamedTuple{()}}?

Can't those be defined to be self-dual?

[mtfishman]

Another way to phrase the question is, what goes wrong if you define dual(::SectorProduct{NamedTuple{()}}) = SectorProduct((;))? I could imagine a few places where that might cause some trouble but I'm curious if we can work through those with good generic code patterns.

[mtfishman]

One way to view this particular issue is in a broader context that we have a few self-dual objects, such as TrivialSector, non-graded unit ranges, and empty SectorProduct. So, from that perspective we could lump all of those cases together and decide how we want to define isdual for those cases.

In ITensors.jl, we use "arrow directions" as the analog of duality information, and we have three options (captured by an enum), In, Out, and Neither. Neither is used when the space doesn't have any symmetry sectors. We could consider a similar strategy, which would mean in cases where objects are self-dual we have a third choice for isdual besides true or false, for example missing or nothing.

[ogauthe]

Replace by ITensor/GradedArrays.jl#22