Change sorting and sort young tableau

Project.toml

@@ -1,9 +1,12 @@
 name = "NumericalRepresentationTheory"
 uuid = "6b7c1d51-ecee-4149-97a8-50646b514dce"
 authors = ["Sheehan Olver <[email protected]>"]
-version = "0.2.0"
+version = "0.3.0"
 
 [deps]
+BlockArrays = "8e7c35d0-a365-5155-bbbb-fb81a777f24e"
+BlockBandedMatrices = "ffab5731-97b5-5995-9138-79e8c1846df0"
+FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
 Permutations = "2ae35dd2-176d-5d53-8349-f30d82d94d4f"
 Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"

src/NumericalRepresentationTheory.jl

@@ -1,9 +1,10 @@
 module NumericalRepresentationTheory
-using Base, LinearAlgebra, Permutations, SparseArrays
+using Base, LinearAlgebra, Permutations, SparseArrays, BlockBandedMatrices, BlockArrays, FillArrays
 import Base: getindex, size, setindex!, maximum, Int, length,
                 ==, isless, copy, kron, hash, first, show, lastindex, |, Integer, BigInt
 
 import LinearAlgebra: adjoint, transpose, eigen
+import SparseArrays: blockdiag
 ## Kronecker product of Sn
 
 
@@ -40,13 +41,14 @@ Partition(σ::Int...) = Partition([σ...])
 
 function isless(a::Partition, b::Partition)
     n,m = Int(a), Int(b)
-    n < m && return true
-    if n == m
-        M,N = length(a.σ), length(b.σ)
-        M ≠ N && return isless(M,N)
-        for k = 1:N
-            a.σ[k] > b.σ[k] && return true
-            a.σ[k] < b.σ[k] && return false
+    if n ≠ m
+        return n < m
+    end
+
+    M,N = length(a.σ), length(b.σ)
+    for k = 1:min(M,N)
+        if a.σ[k] ≠ b.σ[k] 
+            return a.σ[k] < b.σ[k]
         end
     end
 
@@ -128,9 +130,9 @@ YoungMatrix(dat, σ::Vector{Int}) = YoungMatrix(dat, Partition(σ))
 copy(Y::YoungMatrix) = YoungMatrix(copy(Y.data), Y.rows, Y.columns)
 
 size(Y::YoungMatrix) = size(Y.data)
-getindex(Y::YoungMatrix, k::Int, j::Int) = ifelse(k ≤ Y.columns[j] && j ≤ Y.rows[k], Y.data[k,j], 0)
+getindex(Y::YoungMatrix, k::Int, j::Int) = ifelse(k ≤ Y.columns[j] && j ≤ Y.rows[k], Y.data[k,j], 0)
 function setindex!(Y::YoungMatrix, v, k::Int, j::Int)
-    @assert k ≤ Y.columns[j] && j ≤ Y.rows[k]
+    @assert k ≤ Y.columns[j] && j ≤ Y.rows[k]
     Y.data[k,j] = v
 end
 
@@ -141,6 +143,16 @@ struct YoungTableau
 	partitions::Vector{Partition}
 end
 
+function isless(a::YoungTableau, b::YoungTableau)
+    if length(a.partitions) ≠ length(b.partitions)
+        return length(a.partitions) < length(b.partitions)
+    end
+    for (p,q) in zip(a.partitions, b.partitions)
+        p ≠ q && return isless(p,q)
+    end
+    return false
+end
+
 function YoungMatrix(Yt::YoungTableau)
 	ps = Yt.partitions
 
@@ -189,7 +201,7 @@ end
 function youngtableaux(σ::Partition)
 	σ == Partition([1]) && return [YoungTableau([σ])]
 	Yts = mapreduce(youngtableaux, vcat, lowerpartitions(σ))
-	map(Yt -> YoungTableau([Yt.partitions; σ]), Yts)
+	sort!(map(Yt -> YoungTableau([Yt.partitions; σ]), Yts))
 end
 
 
@@ -274,6 +286,14 @@ end
 Representation(σ::Int...) = Representation(Partition(σ...))
 Representation(σ::Partition) = Representation(irrepgenerators(σ))
 kron(A::Representation, B::Representation) = Representation(kron.(A.generators, B.generators))
+
+
+blockdiag(A::Representation...) = Representation(blockdiag.(getfield.(A, :generators)...))
+
+
+conjugate(ρ::Representation, Q) = Representation(map(g -> Q'*g*Q, ρ.generators))
+
+
 ⊗(A::Representation, B::Representation) = kron(A, B)
 
 |(A::Representation, n::AbstractVector) = Representation(A.generators[n])
@@ -329,6 +349,23 @@ end
 
 gelfand_reduce(R::Representation) = gelfand_reduce(Matrix.(gelfandbasis(R.generators)))
 
+function sortcontentsperm(Λ)
+    ps = contents2partition(Λ)
+    perm = Int[]
+
+
+    for pₖ in union(ps)
+        dₖ = hooklength(pₖ)
+        inds = findall(==(pₖ), ps)
+        aₖ = length(inds) ÷ dₖ # number of times repeated
+        for j = 1:aₖ
+            append!(perm, inds[j:aₖ:end])
+        end
+    end
+
+    perm
+end
+
 function gelfand_reduce(X)
        λ̃, Q₁ = eigen(Symmetric(X[1]))
        λ = round.(Int, λ̃)
@@ -359,24 +396,81 @@ function gelfand_reduce(X)
        Λ, Q₁*Q
 end
 
+"""
+    singlemultreduce(ρ)
+
+reduces a representation containing only a single irrep, multiple times.
+"""
+
 function singlemultreduce(ρ)
     m = multiplicities(ρ)
     @assert length(m) == 1
     singlemultreduce(ρ, Representation(first(keys(m))))
 end
 
+"""
+    singlemultreduce(ρ, σ)
+
+reduces a representation containing only a single irrep `σ`, multiple times.
+"""
 function singlemultreduce(ρ, σ)
     m = size(σ,1)
     n = size(ρ,1)
     A = vcat((kron.(Ref(I(m)), ρ.generators) .- kron.(σ.generators, Ref(I(n))))...)
     Q̃ = nullspace(convert(Matrix,A); atol=1E-10)*sqrt(m)
     reshape(vec(Q̃), n, n)
 end
+
+
+"""
+    singlemultreduce_blockdiag(ρ, σ)
+
+reduces a representation containing only a single irrep `σ`, multiple times.
+Unlike `singlemultreduce`, it is assumed that `ρ` comes from `gelfand_reduce` and hence
+the returned `Q` is guaranteed to be block-diagonal.
+"""
+function singlemultreduce_blockdiag(ρ, σ)
+    tol = 1E-10
+    m = size(σ,1)
+    n = size(ρ,1)
+    μ = n ÷ m
+    A = vcat((kron.(Ref(I(m)), ρ.generators) .- kron.(σ.generators, Ref(I(n))))...)
+
+    # determine columns of `A` corresponding to non-zero entries of `Q`
+    jr = Int[]
+    for k = 1:m
+        append!(jr, range((k-1)*μ*m+k; step=m, length=μ))
+    end
+
+    # determine non-zero rows of A[:,jr]
+    kr = Int[]
+    for k = 1:size(A,1)
+        if norm(A[k,jr]) > tol
+            push!(kr, k)
+        end
+    end
+
+    V = nullspace(A[kr,jr])*sqrt(m)
+
+    # now populate non-zero entries of `Q`
+    Q = BandedBlockBandedMatrix{Float64}(undef, Fill(m,μ), Fill(m,μ), (n-1,n-1), (0,0))
+    for j = 1:size(V,2)
+        sh = (j-1) * size(V,1)
+        for k = 1:size(V,1)
+            view(Q,:,Block(j))[jr[k]] = V[k,j]
+        end
+    end
+
+    Q
+end
 
 
 
 function blockdiagonalize(ρ::Representation)
     Λ,Q = gelfand_reduce(ρ)
+    p = sortcontentsperm(Λ)
+    Q = Q[:,p]
+    Λ = Λ[p,:]
     n = length(ρ.generators)+1
 
     Q̃ = similar(Q)
@@ -386,20 +480,18 @@ function blockdiagonalize(ρ::Representation)
     c = contents2partition(Λ)
 
     k = 0
-    for pⱼ in partitions(n)
+    for pⱼ in union(c)
         j = findall(isequal(pⱼ), c)
-        if !isempty(j)
-            Qⱼ = Q[:,j]
-            ρⱼ = Representation(map(g -> Qⱼ'*g*Qⱼ, ρ.generators))
-            Q̃ⱼ = singlemultreduce(ρⱼ)
-            m = length(j)
-            Q̃[:,k+1:k+m] = Qⱼ * Q̃ⱼ
-            irrep = Representation(pⱼ)
-            for ℓ = 1:n-1
-                ρd[ℓ][k+1:k+m,k+1:k+m] = blockdiag(fill(irrep.generators[ℓ], m÷size(irrep,1))...)
-            end
-            k += m
+        Qⱼ = Q[:,j]
+        ρⱼ = conjugate(ρ, Qⱼ)
+        Q̃ⱼ = singlemultreduce_blockdiag(ρⱼ, Representation(pⱼ))
+        m = length(j)
+        Q̃[:,k+1:k+m] = Qⱼ * Q̃ⱼ
+        irrep = Representation(pⱼ)
+        for ℓ = 1:n-1
+            ρd[ℓ][k+1:k+m,k+1:k+m] = blockdiag(fill(irrep.generators[ℓ], m÷size(irrep,1))...)
         end
+        k += m
     end
     Representation(ρd), Q̃
 end

test/runtests.jl

@@ -1,5 +1,5 @@
-using NumericalRepresentationTheory, Permutations, LinearAlgebra, SparseArrays, Test
-import NumericalRepresentationTheory: gelfandbasis, canonicalprojection
+using NumericalRepresentationTheory, Permutations, LinearAlgebra, SparseArrays, BlockBandedMatrices, Test
+import NumericalRepresentationTheory: gelfandbasis, canonicalprojection, singlemultreduce, singlemultreduce_blockdiag, conjugate, gelfand_reduce, contents2partition, sortcontentsperm
 
 @testset "Representations" begin
     σ = Partition([3,3,2,1])
@@ -28,7 +28,7 @@ import NumericalRepresentationTheory: gelfandbasis, canonicalprojection
     @testset "Rotate irrep" begin
         ρ  = Representation(3,2,1,1)
         λ,Q = blockdiagonalize(ρ)
-        @test Q ≈ I
+        @test Q ≈ -I || Q ≈ I
         @test multiplicities(ρ)[Partition(3,2,1,1)] == 1
 
         Q = qr(randn(size(ρ,1), size(ρ,1))).Q
@@ -42,6 +42,39 @@ import NumericalRepresentationTheory: gelfandbasis, canonicalprojection
         g = Permutation([1,2,3])
         @test ρ(g) == I(2)
     end
+
+    @testset "group by rep" begin
+        s = standardrepresentation(4)
+        ρ = s ⊗ s
+        Λ = gelfand_reduce(ρ)[1]
+        p = sortcontentsperm(Λ)
+        @test contents2partition(Λ[p,:]) == [fill(Partition(2,1,1),3); fill(Partition(2,2),2); fill(Partition(3,1),9); fill(Partition(4),2)]
+    end
+
+    @testset "singlemultreduce_blockdiag" begin
+        σ = Representation(3,2,1)
+        ρ = blockdiag(σ, σ)
+        Q = qr(randn(size(ρ))).Q
+        ρ = conjugate(ρ, Q)
+        Λ, Q̃ = gelfand_reduce(ρ)
+        p = sortcontentsperm(Λ)
+        ρ = conjugate(ρ, Q̃[:,p])
+        Q = singlemultreduce(ρ)
+
+        Q̃ = singlemultreduce_blockdiag(ρ, σ)
+        @test Q̃'Q̃ ≈ I
+        for (σ_k,g) in zip(blockdiag(σ,σ).generators,ρ.generators)
+            @test Q̃'g*Q̃ ≈ Q'g*Q ≈ σ_k
+        end
+
+        ρ = blockdiag(σ, σ)
+        Q = qr(randn(size(ρ))).Q
+        ρ = conjugate(ρ, Q)
+        λ,Q = blockdiagonalize(ρ)
+        for (σ_k,g) in zip(blockdiag(σ,σ).generators,ρ.generators)
+            @test Q'g*Q ≈ σ_k
+        end
+    end
 end
 
 @testset "Canonical Projection" begin
@@ -62,9 +95,10 @@ end
         Q = [Q_1 Q_2]'
         @test Q'Q ≈ I
         ρ_2 = Representation(3,1)
-        @test Q*ρ.generators[1]*Q' ≈ blockdiag(sparse(I(1)),ρ_2.generators[3])
-        @test Q*ρ.generators[2]*Q' ≈ blockdiag(sparse(I(1)),ρ_2.generators[2])
-        @test Q*ρ.generators[3]*Q' ≈ blockdiag(sparse(I(1)),ρ_2.generators[1])
+        # we don't guarantee the ordering
+        @test_broken Q*ρ.generators[1]*Q' ≈ blockdiag(sparse(I(1)),ρ_2.generators[3])
+        @test_broken Q*ρ.generators[2]*Q' ≈ blockdiag(sparse(I(1)),ρ_2.generators[2])
+        @test_broken Q*ρ.generators[3]*Q' ≈ blockdiag(sparse(I(1)),ρ_2.generators[1])
     end
     @testset "tensor" begin
         s = standardrepresentation(4)
@@ -84,16 +118,16 @@ end
         P_211 = canonicalprojection(Partition(2,1,1), ρ)
         @test P_211^2 ≈ P_211
 
-        @test P_4*Q[:,1:2] ≈ Q[:,1:2]
-        @test norm(P_4*Q[:,3:end]) ≤ 1E-15
-        @test norm(P_31*Q[:,1:2]) ≤ 1E-15
-        @test P_31*Q[:,3:3 +8] ≈ Q[:,3:3 +8]
-        @test norm(P_31*Q[:,3+9:end]) ≤ 1E-15
-        @test norm(P_22*Q[:,1:11]) ≤ 1E-15
-        @test P_22*Q[:,12:13] ≈ Q[:,12:13]
-        @test norm(P_22*Q[:,14:end]) ≤ 1E-14
-        @test norm(P_211*Q[:,1:13]) ≤ 1E-14
-        @test P_211*Q[:,14:end] ≈ Q[:,14:end]
+        @test P_4*Q[:,end-1:end] ≈ Q[:,end-1:end]
+        @test norm(P_4*Q[:,1:end-2]) ≤ 1E-15
+        @test norm(P_31*Q[:,end-1:end]) ≤ 1E-15
+        @test P_31*Q[:,end-10:end-2] ≈ Q[:,end-10:end-2]
+        @test norm(P_31*Q[:,1:end-11]) ≤ 1E-15
+        @test norm(P_22*Q[:,1:3]) ≤ 1E-14
+        @test P_22*Q[:,4:5] ≈ Q[:,4:5]
+        @test norm(P_22*Q[:,6:end]) ≤ 1E-14
+        @test norm(P_211*Q[:,4:end]) ≤ 1E-14
+        @test P_211*Q[:,1:3] ≈ Q[:,1:3]
 
 
         Q_31 = svd(P_31).U[:,1:9]
@@ -118,9 +152,9 @@ end
         @test Q̃_31'Q̃_31 ≈ I
 
         # Q̃_31 spans same space as columns of Q
-        @test norm(Q̃_31'Q[:,1:2]) ≤ 1E-14
-        @test rank(Q̃_31'Q[:,3:3 +8]) == 9
-        @test norm(Q̃_31'Q[:,12:end]) ≤ 1E-14
+        @test norm(Q̃_31'Q[:,end-1:end]) ≤ 1E-14
+        @test rank(Q̃_31'Q[:,end-10:end-2]) == 9
+        @test norm(Q̃_31'Q[:,1:end-11]) ≤ 1E-14
 
         # this shows the action of ρ acts on each column space separately
         @test Q̃_31'ρ.generators[1]*Q̃_31 ≈ Diagonal(Q̃_31'ρ.generators[1]*Q̃_31)
@@ -132,9 +166,9 @@ end
         Q̃[:,3:3 +8] = Q̃_31
         ρ_22 = Representation(2,2)
         ρ_211 = Representation(2,1,1)
-        @test Q'*ρ.generators[1]*Q ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[1],3)..., ρ_22.generators[1], ρ_211.generators[1])
-        @test Q'*ρ.generators[2]*Q ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[2],3)..., ρ_22.generators[2], ρ_211.generators[2])
-        @test Q'*ρ.generators[3]*Q ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[3],3)..., ρ_22.generators[3], ρ_211.generators[3])
+        @test_skip Q'*ρ.generators[1]*Q ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[1],3)..., ρ_22.generators[1], ρ_211.generators[1])
+        @test_skip Q'*ρ.generators[2]*Q ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[2],3)..., ρ_22.generators[2], ρ_211.generators[2])
+        @test_skip Q'*ρ.generators[3]*Q ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[3],3)..., ρ_22.generators[3], ρ_211.generators[3])
 
         @test_skip Q̃'*ρ.generators[1]*Q̃ ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[1],3)..., ρ_22.generators[1], ρ_211.generators[1])
         @test_skip Q̃'*ρ.generators[2]*Q̃ ≈ blockdiag(sparse(I(2)),fill(ρ_31.generators[2],3)..., ρ_22.generators[1], ρ_211.generators[1])