Include pointers in docstring

[blegat]

It would be helpful to give pointers to references with more details on the algorithms used in this package.
For instance, in the docstrings of

• isotypical_basis
• matrix_projection

[kalmarek]

I don't think there's much to add here. There aren't really any advanced algorithms involved, it's all "simple" representation theory of groups ;)

If π: G → GL(n,K) is a matrix representation, then this homomorphism has canonical extension to a homomorphism of group algebra into the algebra M(n,n,K) of matrices over K. The matrix_projection (probably not a great name) is just realization of this extension. If χ:G → K is a function on the group (e.g. a character) (often written as Σ χ(g)·g), then matrix_projection is just Σ χ(g)·π(g), a weighted sum of matrices. Since every character is an idempotent in the group algebra, matrix_projection is a projection (e.g. in the sense of linear algebra) in the fixed basis.
Character (as an idempotent) and matrix_projection (as matrix) play the same game as quadratic form and its matrix realization in classical linear algebra.

isotypical_basis just finds a maximal set of independent vectors in the image of a projection. One could use rref for this, or SVD as I did here:
https://github.com/kalmarek/PropertyT.jl/blob/db044301059cecc3dfe7aa22325aa39e4109b981/src/blockdecomposition.jl#L28
Isotypical refers to the fact that a representation may decompose into constitutents (~irreducible summands) with multiplicities (so called isotypical decomposition). The returned basis is a basis for the whole block of such type.
Since characters (in the group algebra) are orthogonal, in the sense of this dot:

SymbolicWedderburn.jl/src/characters.jl

Line 5 in 5ec5eff

function LinearAlgebra.dot(

(which is just a specialization of a general dot on the group algebra), the derived matrix projections are orthogonal (i.e. their images are orthogonal subspaces), so combining/concatenating their images gives you blocks of orthogonality.

[kalmarek]

Probably at a later stage we may opt to provide more thorough explanation in e.g. a case-study documentation as you do here:
https://jump.dev/SumOfSquares.jl/latest/generated/Symmetry%20reduction/

??

[blegat]

Thanks for the details!
So the primitive central idempotents of CG are e_i = χ_i(1)/|G| * sum_g χ_i(inv(g)) * g where χ_i are the characters of the irreps.
We drop the χ_i(1)/|G| as it's just constant and we compute instead sum_g χ_i(g) * inv(g). The regular representation of inv(g) is the matrix with i^g in each row i which gives

SymbolicWedderburn.jl/src/projections.jl

Lines 45 to 51 in 5ec5eff

	for (val, cc) in zip(vals, ccls)
	for g in cc
	for i = 1:dim
	result[i, i^g] += val
	end
	end
	end

since val is χ_i(g), correct ?

Probably at a later stage we may opt to provide more thorough explanation in e.g. a case-study documentation

Yes, it's a good idea. I do something similar in https://jump.dev/SumOfSquares.jl/latest/generated/Polynomial%20Optimization/#A-deeper-look-into-atom-extraction.

[kalmarek]

that's correct, with a small amendment, that it's not the regular representation (group acting by left/right multiplication on itself), but the representation given by the action of the group on the basis.

In our case we have π:G → GL(basis) and to obtain projections for central idempotents (~characters) one need to evaluate the action matrix for each element g∈G. (yes, I forgot about g⁻¹ above).

For matrix groups precisely this function needs to be extended.

EDIT:
We actually don't drop this χ_i(1)/|G| factor. it is computed here

SymbolicWedderburn.jl/src/projections.jl

Line 15 in 5ec5eff

return U, T(deg) / ordG

and applied to the projection here:

SymbolicWedderburn.jl/src/projections.jl

Line 110 in 5ec5eff

u .*= weight

[blegat]

Thanks, that clarifies it :)