Representations of alternating group of degree 4

src/TensorKitSectors.jl

@@ -21,6 +21,7 @@ export triangle_equation, pentagon_equation, hexagon_equation
 export Trivial
 export Z2Irrep, Z3Irrep, Z4Irrep, ZNIrrep, LargeZNIrrep, U1Irrep
 export D3Irrep, D4Irrep, DNIrrep, CU1Irrep
+export A4Irrep
 export SU2Irrep
 export ZNElement, Z2Element, Z3Element, Z4Element
 export ProductSector, TimeReversed
@@ -35,6 +36,7 @@ export charge, modulus
 # ---------------
 export ⊠, ⊗, ×
 export Cyclic, ℤ, ℤ₂, ℤ₃, ℤ₄, U₁, SU, SU₂, Dihedral, D₃, D₄, CU₁
+export Alternating, A₄
 export fℤ₂, fU₁, fSU₂
 
 # public

src/groups.jl

@@ -38,6 +38,14 @@ group.
 """
 abstract type Dihedral{N} <: Group end
 
+"""
+    abstract type Alternating{N} <: Group
+
+Type to represent the alternating group of order `N!/2`, which is the group
+of even permutations on `N` elements. 
+"""
+abstract type Alternating{N} <: Group end
+
 
 """
     abstract type U₁ <: AbelianGroup
@@ -72,6 +80,7 @@ const ℤ₃ = ℤ{3}
 const ℤ₄ = ℤ{4}
 const D₃ = Dihedral{3}
 const D₄ = Dihedral{4}
+const A₄ = Alternating{4}
 const SU₂ = SU{2}
 
 type_repr(::Type{ℤ₂}) = "ℤ₂"
@@ -80,6 +89,7 @@ type_repr(::Type{ℤ₄}) = "ℤ₄"
 type_repr(::Type{ℤ{N}}) where {N} = "ℤ{$N}"
 type_repr(::Type{D₃}) = "D₃"
 type_repr(::Type{D₄}) = "D₄"
+type_repr(::Type{A₄}) = "A₄"
 type_repr(::Type{SU₂}) = "SU₂"
 type_repr(::Type{U₁}) = "U₁"
 type_repr(::Type{CU₁}) = "CU₁"

src/irreps/a4irrep.jl

@@ -0,0 +1,164 @@
+"""
+    struct A4Irrep <: AbstractIrrep{A₄}
+    A4Irrep(n::Integer)
+    Irrep[A₄](n::Integer)
+
+Represents irreps of the alternating group ``A₄``.
+
+## Fields
+
+- `n::Int8`: the label of the irrep, corresponding to ``1``, ``1′``, ``1″`` and ``3``.
+"""
+struct A4Irrep <: AbstractIrrep{A₄}
+    n::Int8
+    function A4Irrep(n::Integer)
+        0 ≤ n < 4 || throw(ArgumentError("A4Irrep only has irrep labels in `0:3`"))
+        return new(n)
+    end
+end
+
+FusionStyle(::Type{A4Irrep}) = GenericFusion()
+sectorscalartype(::Type{A4Irrep}) = Float64
+
+unit(::Type{A4Irrep}) = A4Irrep(0)
+dual(a::A4Irrep) = a.n == 3 ? a : A4Irrep((3 - a.n) % 3)
+
+Base.hash(a::A4Irrep, h::UInt) = hash(a.n, h)
+Base.convert(::Type{A4Irrep}, n::Integer) = A4Irrep(n)
+
+Base.getindex(::IrrepTable, ::Type{A₄}) = A4Irrep
+
+
+# Sector iterator
+# ---------------
+Base.isless(a::A4Irrep, b::A4Irrep) = isless(a.n, b.n)
+Base.IteratorSize(::Type{SectorValues{A4Irrep}}) = Base.HasLength()
+Base.length(::SectorValues{A4Irrep}) = 4
+
+Base.iterate(v::SectorValues{A4Irrep}, i = 1) = i > length(v) ? nothing : (v[i], i + 1)
+
+@inline function Base.getindex(v::SectorValues{A4Irrep}, i::Int)
+    @boundscheck 1 <= i <= length(v) || throw(BoundsError(v, i))
+    return A4Irrep(i - 1)
+end
+
+findindex(::SectorValues{A4Irrep}, a::A4Irrep) = a.n + 1
+
+# Product iterator
+# ----------------
+
+const A4IrrepProdIterator = SectorProductIterator{A4Irrep}
+⊗(a::A4Irrep, b::A4Irrep) = SectorProductIterator((a <= b ? (a, b) : (b, a))...)
+
+Base.IteratorSize(::Type{A4IrrepProdIterator}) = Base.HasLength()
+Base.length(x::A4IrrepProdIterator) = (x.a == x.b == A4Irrep(3)) ? 4 : 1
+
+function Base.iterate(p::A4IrrepProdIterator, state::Int = 1)
+    a, b = p.a, p.b
+    if state == 1
+        a.n == b.n == 3 && return (A4Irrep(state - 1), state + 1) # 3 ⊗ 3
+        b.n == 3 && return (b, 5) # x ⊗ 3 = 3
+        return (A4Irrep((a.n + b.n) % 3), 5) # 1d irreps ≡ Z3
+    elseif state < 5 # a == b == 3
+        return (A4Irrep(state - 1), state + 1)
+    else
+        return nothing
+    end
+end
+
+# Topological data
+# ----------------
+dim(a::A4Irrep) = a.n == 3 ? 3 : 1
+
+function Nsymbol(a::A4Irrep, b::A4Irrep, c::A4Irrep)
+    # 3d irreps
+    a.n == b.n == 3 && return 1 + (c.n == 3)
+    (a.n == 3 || b.n == 3) && return Int(c.n == 3)
+    # 1d irreps
+    return Int((a.n + b.n) % 3 == c.n)
+end
+
+
+function Fsymbol(a::I, b::I, c::I, d::I, e::I, f::I) where {I <: A4Irrep}
+    T = sectorscalartype(I)
+    Nabe = Nsymbol(a, b, e)
+    Necd = Nsymbol(e, c, d)
+    Nbcf = Nsymbol(b, c, f)
+    Nafd = Nsymbol(a, f, d)
+
+    Nabe > 0 && Necd > 0 && Nbcf > 0 && Nafd > 0 ||
+        return zeros(T, Nabe, Necd, Nbcf, Nafd)
+
+    # fallback through fusiontensor for A4Irrep
+    A = fusiontensor(a, b, e)
+    B = fusiontensor(e, c, d)[:, :, 1, :]
+    C = fusiontensor(b, c, f)
+    D = fusiontensor(a, f, d)[:, :, 1, :]
+    @tensor F[-1, -2, -3, -4] := conj(D[1, 5, -4]) * conj(C[2, 4, 5, -3]) *
+        A[1, 2, 3, -1] * B[3, 4, -2]
+
+    return F
+end
+
+# bosonic
+function Rsymbol(a::I, b::I, c::I) where {I <: A4Irrep}
+    Nabc = Nsymbol(a, b, c)
+    R = zeros(sectorscalartype(I), Nabc, Nabc)
+    Nabc == 0 && return R
+    if a == b == c == A4Irrep(3)
+        R[1, 1] = -1
+        R[2, 2] = 1
+    else
+        R[1, 1] = 1
+    end
+    return R
+end
+
+# choice of basis: https://journals.aps.org/rmp/pdf/10.1103/RevModPhys.82.2701
+# triplet is a real representation -> can make all representation matrices real
+function fusiontensor(a::I, b::I, c::I) where {I <: A4Irrep}
+    T = sectorscalartype(I)
+    Nabc = Nsymbol(a, b, c)
+    C = zeros(T, dim(a), dim(b), dim(c), Nabc)
+    isempty(C) && return C
+
+    ω = cis(2π / 3)
+
+    if a.n == b.n == 3 # 3 ⊗ 3
+        if c.n != 3 # singlets
+            invsqrt3 = 1 / sqrt(3.0)
+            for i in 1:3
+                j = 4 - mod1(i - c.n - 1, 3)
+                C[i, j, 1, 1] = invsqrt3
+            end
+        else # triplet: eq 38 in above reference
+            s2 = 1 / sqrt(2.0)
+            s6 = 1 / sqrt(6.0)
+
+            # antisymmetric channel
+            C[:, :, 1, 1] .= [0 0 0; 0 0 s2; 0 -s2 0]
+            C[:, :, 2, 1] .= [0 s2 0; -s2 0 0; 0 0 0]
+            C[:, :, 3, 1] .= [0 0 -s2; 0 0 0; s2 0 0]
+
+            # symmetric channel
+            C[:, :, 1, 2] .= [-2 * s6 0 0; 0 0 s6; 0 s6 0]
+            C[:, :, 2, 2] .= [0 s6 0; s6 0 0; 0 0 -2 * s6]
+            C[:, :, 3, 2] .= [0 0 s6; 0 -2 * s6 0; s6 0 0]
+        end
+    else
+        if a.n != 3 && b.n != 3 # 1d x 1d
+            C[1, 1, 1] = one(T)
+        elseif a.n == 3 && b.n != 3 # 3 x 1d
+            for i in 1:3
+                j = mod1(i - b.n, 3)
+                C[j, 1, i, 1] = one(T)
+            end
+        else # 1d x 3: reshape of 3 x 1d
+            for i in 1:3
+                j = mod1(i - a.n, 3)
+                C[1, j, i, 1] = one(T)
+            end
+        end
+    end
+    return C
+end

src/irreps/irreps.jl

@@ -68,3 +68,4 @@ include("u1irrep.jl")
 include("dnirrep.jl")
 include("cu1irrep.jl")
 include("su2irrep.jl")
+include("a4irrep.jl")

test/runtests.jl

@@ -9,20 +9,21 @@ const sectorlist = (
     Trivial, PlanarTrivial,
     Z2Irrep, Z3Irrep, Z4Irrep, Irrep[ℤ{200}], U1Irrep,
     DNIrrep{3}, DNIrrep{4}, DNIrrep{5}, CU1Irrep,
-    SU2Irrep, NewSU2Irrep,
+    A4Irrep, SU2Irrep, NewSU2Irrep,
     FibonacciAnyon, IsingAnyon, FermionParity,
     FermionParity ⊠ FermionParity,
     Z3Irrep ⊠ Z4Irrep, FermionParity ⊠ U1Irrep ⊠ SU2Irrep,
     FermionParity ⊠ SU2Irrep ⊠ SU2Irrep, NewSU2Irrep ⊠ NewSU2Irrep,
     NewSU2Irrep ⊠ SU2Irrep, FermionParity ⊠ SU2Irrep ⊠ NewSU2Irrep,
     FibonacciAnyon ⊠ FibonacciAnyon ⊠ Z2Irrep,
+    A4Irrep ⊠ Z2Irrep, A4Irrep ⊠ SU2Irrep,
     Z2Element{0}, Z2Element{1},
     Z3Element{0}, Z3Element{1}, Z3Element{2},
     Z4Element{0}, Z4Element{1}, Z4Element{2},
     Z3Element{1} ⊠ SU2Irrep,
     FibonacciAnyon ⊠ Z4Element{3},
     TimeReversed{Z2Irrep},
-    TimeReversed{Z3Irrep}, TimeReversed{Z4Irrep},
+    TimeReversed{Z3Irrep}, TimeReversed{Z4Irrep}, TimeReversed{A4Irrep},
     TimeReversed{U1Irrep}, TimeReversed{CU1Irrep}, TimeReversed{SU2Irrep},
     TimeReversed{FibonacciAnyon}, TimeReversed{IsingAnyon},
     TimeReversed{FermionParity},
@@ -46,6 +47,28 @@ using .SectorTestSuite
     end
 end
 
+@testset "Intertwiner relation for A4Irrep" begin
+    ω = cis(2π / 3)
+    T3 = [1 0 0; 0 ω 0; 0 0 ω^2]
+    T(a::Int8) = (a == 3) ? T3 : hcat(ω^(a))
+    S3 = 1 / 3 * [-1 2 2; 2 -1 2; 2 2 -1]
+    S(a::Int8) = (a == 3) ? S3 : hcat(1)
+    for a in smallset(A4Irrep), b in smallset(A4Irrep)
+        for c in ⊗(a, b)
+            C = fusiontensor(a, b, c)
+            Ta, Tb, Tc = T(a.n), T(b.n), T(c.n)
+            Sa, Sb, Sc = S(a.n), S(b.n), S(c.n)
+            for μ in 1:Nsymbol(a, b, c)
+                Cmat = reshape(view(C, :, :, :, μ), (dim(a) * dim(b), dim(c)))
+                L_T = Cmat' * kron(Ta, Tb) * Cmat
+                L_S = Cmat' * kron(Sa, Sb) * Cmat
+                @test isapprox(L_T, Tc; atol = 1.0e-12)
+                @test isapprox(L_S, Sc; atol = 1.0e-12)
+            end
+        end
+    end
+end
+
 @testset "Deligne product" begin
     sectorlist′ = (Trivial, sectorlist...)
     for I1 in sectorlist′, I2 in sectorlist′