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update README with working examples
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kalmarek committed May 2, 2024
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----

The package implements `*`-algebras with basis. The prime example use is group/monoid algebras (or rings) (or their finite dimensional subspaces). An example usage can be as follows.
The package implements `*`-algebras with basis. The prime example use are group
or monoid algebras (or their finite dimensional subspaces).
An example usage can be as follows.

```julia
julia> using StarAlgebras
julia> using StarAlgebras; import StarAlgebras as SA

julia> using PermutationGroups

Expand All @@ -19,21 +21,15 @@ Permutation group on 2 generators generated by
(1,2)
(1,2,3)

julia> b = StarAlgebras.Basis{UInt8}(collect(G))
6-element StarAlgebras.Basis{Permutation{Int64, …}, UInt8, Vector{Permutation{Int64, …}}}:
()
(2,3)
(1,2)
(1,3,2)
(1,3)
(1,2,3)
julia> b = SA.DiracBasis{UInt32}(G);

julia> RG = StarAlgebra(G, b)
*-algebra of Permutation group on 2 generators of order 6
*-algebra of PermGroup( (1,2), (1,2,3) )

```

This creates the group algebra of the symmetric group. How do we compute inside the group algebra? There are a few ways to comstruct elements:
This creates the group algebra of the symmetric group. There are a few ways to
comstruct its elements:
```julia
julia> zero(RG)
0·()
Expand All @@ -48,98 +44,112 @@ julia> RG(-5.0) # coerce a scalar to the ring
-5.0·()

julia> RG(rand(G)) # the indicator function on a random element of G
1·(1,3,2)
1·()

julia> f = AlgebraElement(rand(-3:3, length(b)), RG) # an element given by vectors of coefficients in the basis
1·() -1·(2,3) +3·(1,2) -1·(1,3,2) -3·(1,3) +3·(1,2,3)
julia> f = sum(rand(-3:3)*RG(rand(G)) for _ in 1:12) # an element given by vectors of coefficients in the basis
-1·() +4·(1,2) +5·(1,2,3) -2·(1,3,2) -2·(1,3)

julia> SA.star(g::PermutationGroups.AbstractPermutation) = inv(g); star(f) # the star involution
-1·() +4·(1,2) +5·(1,3,2) -2·(1,2,3) -2·(1,3)

julia> f' # the same
-1·() +4·(1,2) +5·(1,3,2) -2·(1,2,3) -2·(1,3)

julia> f' == f
false

julia> f - f' # the alternating part
7·(1,2,3) -7·(1,3,2)

julia> StarAlgebras.aug(p::PermutationGroups.Permutation) = 1

julia> StarAlgebras.aug(f) # sum of coefficients
4

julia> StarAlgebras.aug(f - f') # sum of coefficients
0

julia> using LinearAlgebra; norm(f, 2) # 2-norm
25.0

```
One may work with such element using the following functions:
By default `f` here is stored as (sparse) coefficients against `basis(RG)`:
```julia
julia> StarAlgebras.coeffs(f)
6-element Vector{Int64}:
1
-1
3
-1
-3
3

julia> StarAlgebras.star(p::PermutationGroups.AbstractPerm) = inv(p); star(f) # the star involution
1·() -1·(2,3) +3·(1,2) +3·(1,3,2) -3·(1,3) -1·(1,2,3)
julia> coeffs(f) # same as SA.coeffs(f, basis(RG))
StarAlgebras.SparseCoefficients{...}(Permutation{...}[(), (1,2), (1,2,3), (1,3,2), (1,3)], [-1, 4, 5, -2, -2])
```

julia> f' # the same
1·() -1·(2,3) +3·(1,2) +3·(1,3,2) -3·(1,3) -1·(1,2,3)
Working directly with elements of `G` one can also write

julia> g = rand(G); g
```julia
julia> g = Permutation(perm"(1,2,3)", G)
(1,2,3)

julia> StarAlgebras.coeffs(RG(g)) # note the type of coefficients
6-element SparseArrays.SparseVector{Int64, UInt8} with 1 stored entry:
[6] = 1

julia> x = one(RG) - 3RG(g); supp(x) # support of the funtion
2-element Vector{Permutation{...}:
2-element Vector{Permutation{…}}:
()
(1,2,3)

julia> x(g) # value of x at g
-3

julia> x[g] += 3; x # modification of x in-place
1·()

julia> aug(f) # sum of coefficients
2

julia> using LinearAlgebra; norm(f, 2) # 2-norm
5.477225575051661
julia> x(inv(g))
0
```

Using this we can define e.g. a few projections in `RG` and check their orthogonality:
## Example: Projections in group algebra

Using these functions let's define a few projections in `RG` and check their
orthogonality:
```julia
julia> using Test

julia> Base.sign(p::PermutationGroups.Permutation) = sign(p.perm);

julia> l = length(b)
julia> l = length(basis(RG))
6

julia> P = sum(RG(g) for g in b) // l # projection to the subspace fixed by G
1//6·() +1//6·(2,3) +1//6·(1,2) +1//6·(1,3,2) +1//6·(1,3) +1//6·(1,2,3)
julia> P = sum(RG(g) for g in b) // l # projection to the subspace fixed by all elements of G
1//6·() +1//6·(2,3) +1//6·(1,2) +1//6·(1,2,3) +1//6·(1,3,2) +1//6·(1,3)

julia> @test P * P == P
Test Passed

julia> P3 = 2 * sum(RG(g) for g in b if sign(g) > 0) // l # projection to the subspace fixed by Alt(3) = C₃
1//3·() +1//3·(1,3,2) +1//3·(1,2,3)
1//3·() +1//3·(1,2,3) +1//3·(1,3,2)

julia> @test P3 * P3 == P3
Test Passed

julia> P2 = (RG(1) + RG(b[2])) // 2 # projection to the C₂-fixed subspace
1//2·() +1//2·(2,3)
julia> h = PermutationGroups.gens(G,1)
(1,2)

julia> P2 = (RG(one(G)) + RG(h)) // 2 # projection to the C₂-fixed subspace
1//2·() +1//2·(1,2)

julia> @test P2 * P2 == P2
Test Passed

julia> @test P2 * P3 == P3 * P2 == P # their intersection is precisely the same as the one for G
Test Passed

julia> P2m = (RG(1) - RG(b[2])) // 2 # orthogonal C₂-fixed subspace
1//2·() -1//2·(2,3)
julia> P2m = (RG(1) - RG(h)) // 2 # projection onto the subspace orthogonal to C₂-fixed subspace
1//2·() -1//2·(1,2)

julia> @test P2m * P2m == P2m
Test Passed

julia> @test iszero(P2m * P2) # indeed P2 and P2m are orthogonal
Test Passed

```

This package originated as a tool to compute sum of hermitian squares in `*`-algebras. These consist not of standard `f*f` summands, but rather `star(f)*f`. You may think of semi-definite matrices: their Cholesky decomposition determines `P = Q'·Q`, where `Q'` denotes transpose. Algebra of matrices with transpose is an (the?) example of a `*`-algebra. To compute such sums of squares one may either sprinkle the code with `star`s, or `'` (aka `Base.adjoint` postfix symbol):
This package originated as a tool to compute sum of hermitian squares in
`*`-algebras. These consist not of standard `f*f` summands, but rather
`star(f)*f`. You may think of semi-definite matrices: their Cholesky
decomposition determines `P = Q'·Q`, where `Q'` denotes (conjugate) transpose.
Algebra of matrices with transpose is an (the?) example of a `*`-algebra.
To compute such sums of squares one may either sprinkle the code with `star`s,
or `'` (aka `Base.adjoint` postfix symbol):
```julia
julia> x = RG(G(perm"(1,2,3)"))
julia> x = RG(Permutation(perm"(1,2,3)", G))
1·(1,2,3)

julia> X = one(RG) - x
Expand All @@ -149,99 +159,77 @@ julia> X'
1·() -1·(1,3,2)

julia> X'*X
2·() -1·(1,3,2) -1·(1,2,3)
2·() -1·(1,2,3) -1·(1,3,2)

julia> @test X'*X == star(X)*X == 2one(X) - x - star(x)
Test Passed

```

### More advanced use
## Fixed basis and translation between bases

`RG = StarAlgebra(G, b)` creates the algebra with `TrivialMStructure`, i.e. a multiplicative structure which computes product of basis elements every time it needs it. This of course may be wastefull, e.g. the computed products could be stored in a cache for future use. There are two options here:
```julia
julia> mt = StarAlgebras.MTable(b, table_size=(length(b), length(b)))
6×6 StarAlgebras.MTable{UInt8, false, Matrix{UInt8}}:
0x01 0x02 0x03 0x04 0x05 0x06
0x02 0x01 0x06 0x05 0x04 0x03
0x03 0x04 0x01 0x02 0x06 0x05
0x04 0x03 0x05 0x06 0x02 0x01
0x05 0x06 0x04 0x03 0x01 0x02
0x06 0x05 0x02 0x01 0x03 0x04
`b = SA.DiracBasis{UInt32}(G)` takes an iterator (in this case a finite
permutation group `G`) and creates a basis with Dirac multiplicative structure.
This means a basis of a linear space with the multiplicative structure where
multiplication of two basis elements results in a third one. This computation
does not depend on the finiteness of `G`, i.e. is fully lazy.

```
creates an eagerly computed multiplication table on elements of `b`. Keyword `table_size` is used to specify the table size (above: it's the whole multiplication table). Since `MTable<:AbstractMatrix`, one can use the indexing syntax `mt[i,j]` to compute the **index** of the product of `i`-th and `j`-th elements of the basis. For example
```julia
julia> g = G(perm"(1,2,3)"); h = G(perm"(2,3)");
When the basis is known a priori one can create an efficient `SA.FixedBasis`
which caches (memoizes) the results of multiplications. For example:

julia> i, j = b[g], b[h] # indices of g and h in basis b
(0x06, 0x02)

julia> k = mt[i,j] # the index of the product
0x05
```julia
julia> fb = let l = length(basis(RG))
StarAlgebras.FixedBasis(b; n=l, mt=l) # mt can be also smaller than l
end;

julia> @test b[k] == g*h
Test Passed
julia> fRG = StarAlgebra(G, fb)
*-algebra of PermGroup( (1,2), (1,2,3) )

```

The second option is
```julia
julia> cmt = StarAlgebras.CachedMTable(b, table_size=(length(b), length(b)));
```
This multiplication table is lazy, i.e. products will be computed and stored only when actually needed. Additionally, one may call
Since the length of the basis is known this time, algebra elements can be stored simply as (sparse) vectors:

```julia
julia> using SparseArrays
julia> coeffs(one(fRG))
6-element SparseArrays.SparseVector{Int64, Int64} with 1 stored entry:
[1] = 1

julia> StarAlgebras.CachedMTable(b, spzeros(UInt8, length(b), length(b)));
julia> AlgebraElement(ones(Int, length(basis(fRG)))//6, fRG)
1//6·() +1//6·(2,3) +1//6·(1,2) +1//6·(1,2,3) +1//6·(1,3,2) +1//6·(1,3)

```
to specify storage type of the matrix (by default it's a simple dense `Matrix`).
This may be advisable when a few products are computed repeatedly on a quite large basis.

```julia
julia> RGc = StarAlgebra(G, b, cmt)
*-algebra of Permutation group on 2 generators of order 6
To translate coefficients between bases one may call

```
should be functinally equivalent to `RG` above, however it will cache computation of products lazily. A word of caution is needed here though. Even though `RGc` and `RG` are functionally equivalent, they are not **comparable** in the sense that e.g.
```julia
julia> @test one(RGc) != one(RG)
Test Passed
julia> coeffs(P2, fb)
6-element SparseArrays.SparseVector{Rational{Int64}, Int64} with 2 stored entries:
[1] = 1//2
[3] = 1//2

```
This is a conscious decision on our part, as comparing algebraic structures is easier said than done ;) To avoid solving this conundrum (are bases equal? are multiplicative structures equal? are these permuted by a compatible permutation? or maybe a linear transformation was applied to the basis, resulting in a different, but equivalent multiplicative structure?), elements could be mixed together **only if their parents are identically** (i.e. `===`) **equal**.
Finally, if the group is infinite (or just too large), but we need specific products, we may reduce the table_size to the required size (it doesn't have to be `length(b) × length(b)`). Note that in such case asking for a product outside of multiplication table will rise `ProductNotDefined` exception.
julia> coeffs(coeffs(P2), b, fb) # same as above, from source basis to target basis
6-element SparseArrays.SparseVector{Rational{Int64}, Int64} with 2 stored entries:
[1] = 1//2
[3] = 1//2

### Even more advanced use (for experts only)
```

For low-level usage `MultiplicativeStructures` follow the sign convention:
```julia
julia> mt = StarAlgebras.CachedMTable(b, table_size=(length(b), length(b)));
Translation in the opposite direction is also possible

```julia
julia> k = mt[-i,j]
0x06
julia> fP2 = AlgebraElement(StarAlgebras.coeffs(P2,fb), fRG)
1//2·() +1//2·(1,2)

julia> @test star(b[i])*b[j] == b[k]
Test Passed
julia> StarAlgebras.coeffs(fP2, b)
StarAlgebras.SparseCoefficients{…}(Permutation{…}[(), (1,2)], Rational{Int64}[1//2, 1//2])

```
Note that this (minus-twisted) "product" is no longer associative! Observe:
```julia
julia> @test mt[mt[3, 5], 4] == mt[3, mt[5, 4]] # (b[3]*b[4])*b[5] == b[3]*(b[4]*b[5])
Test Passed
julia> P2_ = AlgebraElement(StarAlgebras.coeffs(fP2, b), RG)
1//2·() +1//2·(1,2)

julia> @test mt[-signed(mt[-3, 5]), 4] == 0x06 # star(star(b[3])*b[5])*b[4] = star(b[5])*b[3]*b[4]
julia> @test P2 == P2_
Test Passed

julia> @test mt[-3, mt[-5, 4]] == 0x01 # star(b[3])*star(b[5])*b[4]
Test Passed
```



-----
If you happen to use this package please cite either [1712.07167](https://arxiv.org/abs/1712.07167) or [1812.03456](https://arxiv.org/abs/1812.03456). This package superseeds [GroupRings.jl](https://github.com/kalmarek/GroupRings.jl) which was developed and used there. It served its purpose well. Let it rest peacefully.

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