feat: group algebras with sparse representation

[thofma]

Due to popular demand, group algebras with sparse representation (aka "don't crash with large groups"). Needs some work/clean up:

julia> G = SymmetricGroup(10)
Full symmetric group over 10 elements

julia> QG = group_algebra(QQ, G)
Group algebra
  of full symmetric group over 10 elements
  over rational field

julia> a = QG(rand(G)); b = QG(rand(G));

julia> c = (a + b) * (a - b)
(1,2,4,8,7)(3,9,10,5,6) - (1,4,9)(2,10,3,6,7,5,8) + (1,7,3,8,6,10,4)(2,9,5) - (1,2,6)(3,9)(4,5)

@royess this might interest you. It is similar to your approach.

Small todo:

• Add a FG(::Dict) constructor.
• Add tests.
• Add documentation.

[codecov]

Codecov Report

Attention: Patch coverage is 90.79755% with 15 lines in your changes missing coverage. Please review.

Project coverage is 75.90%. Comparing base (2af99ca) to head (4c08764).
Report is 6 commits behind head on master.

Files with missing lines	Patch %	Lines
src/AlgAss/Elem.jl	84.14%	13 Missing ⚠️
src/AlgAss/AlgGrp.jl	85.71%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master    #1655      +/-   ##
==========================================
+ Coverage   75.85%   75.90%   +0.04%     
==========================================
  Files         361      361              
  Lines      113719   113783      +64     
==========================================
+ Hits        86262    86366     +104     
+ Misses      27457    27417      -40     

Files with missing lines	Coverage Δ
src/AlgAss/AbsAlgAss.jl	88.26% <100.00%> (+0.06%)	⬆️
src/AlgAss/Types.jl	85.39% <100.00%> (+2.49%)	⬆️
src/Grp/GenGrp.jl	70.11% <100.00%> (+0.11%)	⬆️
src/GrpAb/GrpAbFinGen.jl	76.01% <100.00%> (+0.02%)	⬆️
src/AlgAss/AlgGrp.jl	74.41% <85.71%> (+1.35%)	⬆️
src/AlgAss/Elem.jl	88.34% <84.14%> (-0.26%)	⬇️

... and 30 files with indirect coverage changes

[lgoettgens]

Does this change allow me to work with group algebras of infinite groups as well?

[thofma]

Yes:

julia> F = free_group(2)
Free group of rank 2

julia> x, y = gens(F);

julia> QF = group_algebra(QQ, F; sparse = true)
Group algebra
  of free group of rank 2
  over rational field

julia> (QF(x) + QF(y))^2
f1^2 + f1*f2 + f2*f1 + f2^2

[Fe-r-oz]

This is wonderful! I will shift to sparse = true as soon as this is merged! Thank you.

[fingolfin]

Great! Some quick comments and questions (all nitpicks, not meant to block anything)

[thofma]

Thanks for the comments. I think I addressed them all. I also removed the sparse option from the user-facing version. It is now chosen automatically (as in Magma).