sync

src/Examples/Haagerup/HaagerupH3.jl

@@ -8,7 +8,7 @@ https://arxiv.org/pdf/1906.01322
 
 where p1,p2 = ±1 are parameters for the different possible sets of associators.
 """
-function haagerup_H3(K::Field = QQ; p1 = 1, p2 = 1)
+function haagerup_H3(K::Ring = QQ; p1 = 1, p2 = 1)
     _,x = QQ["x"]
     if K == QQ
         K,_ = number_field(x^16 - 4*x^14 + 13*x^12 + 4*x^10 + 53*x^8 + 4*x^6 + 13*x^4 - 4*x^2 + 1)

src/Examples/TambaraYamagami/TambaraYamagami.jl

@@ -13,7 +13,7 @@ Construct ``TY(A,τ,χ)`` over ℚ̅ where ``τ = √|A|`` and ``χ`` is a gener
 function tambara_yamagami(A::GAPGroup) 
     m = Int(exponent(A))
     _, x = QQ[:x]
-    K = splitting_field([x^m + 1, x^2 - 2])
+    K = splitting_field([x^m + 1, x^2 - order(A)])
     tambara_yamagami(K, A)
 end
 

src/TensorCategories.jl

@@ -24,7 +24,7 @@ import Oscar: +, @alias, @attributes, AbstractSet, AcbField, StructureConstantAl
     is_rational, is_semisimple, is_simple, is_square, is_subfield, is_subgroup,
     is_independent, is_invertible, iso_oscar_gap,
     jordan_normal_form, kernel, kronecker_product, lcm, leading_coefficient,
-    leading_monomial, left_transversal, lex, load, matrix, matrix_algebra, minpoly,
+    leading_monomial, left_transversal, lex, load, matrix, matrix_algebra, minpoly, quo,
     monomials, multiplication_table, multiplicity, nullspace, nvars, one, orbit,
     orbits, order, parent, permutation_matrix, preimage, primary_decomposition, product, rank, real_solutions, resultant, root_of_unity, roots, rref, save,
     set_attribute!, size, solve, sparse_matrix, splitting_field,
@@ -34,7 +34,9 @@ import Oscar: +, @alias, @attributes, AbstractSet, AcbField, StructureConstantAl
     gens, center, graph_from_adjacency_matrix, connected_components, weakly_connected_components, Directed, Undirected, morphism, algebra,
     radical, is_zero, minimal_submodules, representation_matrix, QQBarField,
     is_irreducible, polynomial, is_univariate, action, is_equivalent, extension_of_scalars, free_module, perm, fraction_field, simplify, CalciumField, CalciumFieldElem, FracFieldElem, PadicField, PadicFieldElem,
-    QadicField, QadicFieldElem, FlintLocalField, FlintLocalFieldElem
+    QadicField, QadicFieldElem, FlintLocalField, FlintLocalFieldElem,
+    MultTableGroup, is_isomorphic_with_map, subgroup_classes, representative,
+    pc_group, permutation_group
 
 using Serialization
 import Oscar: @register_serialization_type,
@@ -71,6 +73,7 @@ export ⊕
 export ⊗ 
 export ⊠ 
 export ⋆
+export ⋊
 export AbstractHomSpace 
 export action
 export action_matrix
@@ -170,6 +173,7 @@ export fpdim
 export Functor 
 export FusionCategory 
 export fusion_coefficient
+export gcrossed_product
 export generic_algebra
 export getindex 
 export graded_vector_spaces
@@ -185,6 +189,8 @@ export GroupRepresentation
 export GroupRepresentationCategory 
 export GroupRepresentationCategory 
 export GroupRepresentationMorphism 
+export gtensor_action
+export GTensorAction
 export GVSHomSpace 
 export GVSMorphism 
 export GVSObject 
@@ -216,6 +222,7 @@ export inv
 export inv_associator 
 export inverse_induction_adjunction
 export inverse_internal_hom_adjunction
+export invertibles
 export involution
 export is_abelian 
 export is_abelian 
@@ -440,12 +447,12 @@ include("CategoryFramework/NaturalTransformations.jl")
 
 
 include("TensorCategoryFramework/AbstractTensorMethods.jl")
-include("TensorCategoryFramework/FusionCategory.jl")
+include("TensorCategoryFramework/SixJCategory/FusionCategory.jl")
 #include("structures/MultiFusionCategories/FusionCategoryExperimental.jl")
 include("TensorCategoryFramework/6j-Solver.jl")
 include("TensorCategoryFramework/Skeletization.jl")
-include("TensorCategoryFramework/PentagonAxiom.jl")
-include("TensorCategoryFramework/HexagonAxion.jl")
+include("TensorCategoryFramework/TensorAxioms/PentagonAxiom.jl")
+include("TensorCategoryFramework/TensorAxioms/HexagonAxion.jl")
 include("TensorCategoryFramework/RingSubcategories.jl")
 include("TensorCategoryFramework/TensorPowerCategory.jl")
 include("TensorCategoryFramework/TensorFunctors.jl")
@@ -455,6 +462,8 @@ include("TensorCategoryFramework/Center/InductionMonad.jl")
 include("TensorCategoryFramework/Center/Centralizer.jl")
 include("TensorCategoryFramework/Center/CentralizerInduction.jl")
 include("TensorCategoryFramework/Center/CenterChecks.jl")
+include("TensorCategoryFramework/GTensorAction.jl")
+include("TensorCategoryFramework/SixJCategory/GCrossedFusion.jl")
 
 
 include("TensorCategoryFramework/InternalModules/InternalAlgebras.jl")

src/TensorCategoryFramework/GTensorAction.jl

@@ -0,0 +1,67 @@
+#=----------------------------------------------------------
+    Define G-actions on Tensor Categories
+----------------------------------------------------------=#
+
+
+struct GTensorAction 
+    C::Category
+    G::GAPGroup
+    images::Vector{<:AbstractFunctor}
+end
+
+group(T::GTensorAction) = T.G
+
+function gtensor_action(C::Category, G::GAPGroup, images::Vector{<:Functor})
+    GTensorAction(C,G,images)
+end
+
+function gtensor_action(C::Category, G::GAPGroup, images::Vector{<:Object})
+    GTensorAction(C,G,[(X ⊗ -) for X ∈ images])
+end
+
+function (T::GTensorAction)(g::GroupElem)
+    i = findfirst(==(g), elements(group(T)))
+    return T.images[i]
+end
+
+
+#=----------------------------------------------------------
+    Cannonical G-action
+----------------------------------------------------------=#
+
+function gtensor_action(C::Category, G::GAPGroup)
+
+    C₁ = invertibles(C)
+    n = length(C₁)
+
+    mult = multiplication_table(C₁)
+    M = [findfirst(!iszero, mult[i,j,:]) for i ∈ 1:n, j ∈ 1:n]
+
+    H = MultTableGroup(M)
+
+    m,r = divrem(order(G), n)
+
+    if r != 0 
+        return gcrossed_product(C, trivial_gtensor_action(C,G))
+    end
+
+    subs = representative.(subgroup_classes(G, order = m))
+
+    i = findfirst(N -> is_isomorphic(quo(G,N)[1], H), subs)
+
+    Q,p = quo(G,subs[i])
+
+    proj = compose(p, is_isomorphic_with_map(Q, H)[2])
+
+    els = elements(H)
+
+    images = [findfirst(==(proj(g)), els) for g ∈ elements(G)]
+    images = [C₁[i] for i ∈ images]
+
+    action = gtensor_action(C, G, images)
+end
+
+#=----------------------------------------------------------
+    Trivial G-action 
+----------------------------------------------------------=#
+

src/TensorCategoryFramework/SixJCategory/GCrossedFusion.jl

@@ -0,0 +1,80 @@
+#=----------------------------------------------------------
+    Build the G-Crossed product 𝒞 ⋊ G of a Fusion category 
+    with a G-action on 𝒞. 
+----------------------------------------------------------=#
+
+function gcrossed_product(C::SixJCategory, T::GTensorAction)
+    S = simples(C)
+    G = group(T)
+    K = base_ring(C) 
+
+    irreducibles = ["($s,$g)" for  g ∈ elements(permutation_group(G)), s ∈ simples_names(C)][:]
+
+    elements_of_G = elements(G)
+    CxG = six_j_category(K, irreducibles)
+
+    m,n = length(S), length(elements_of_G)
+
+    mult = zeros(Int,n*m,n*m,n*m)
+
+    for i1 ∈ 1:m, j1 ∈ 1:n, i2 ∈ 1:m, j2 ∈ 1:n
+        g,h = elements_of_G[[j1,j2]]
+        X = S[i1] ⊗ (T(g)(S[i2]))
+        Y = g * h
+
+        Y_ind = findfirst(==(Y),elements_of_G)
+
+        for k ∈ 1:m
+            mult[(i1-1)*n + j1, (i2-1)*n + j2, (k-1)*n + Y_ind] = X.components[k] 
+        end
+    end
+
+    ass = Array{MatElem,4}(undef, n*m, n*m, n*m, n*m)
+
+    for i1 ∈ 1:m, j1 ∈ 1:n, i2 ∈ 1:m, j2 ∈ 1:n, i3 ∈ 1:m, j3 ∈ 1:n
+        g1,g2,g3 = elements_of_G[[j1,j2,j3]]
+
+        X = S[i1] ⊗ (T(g1)(S[i2])) ⊗ (T(g1*g2)(S[i3]))
+        Y = g1 * g2 * g3
+
+        for k ∈ 1:m, l ∈ 1:n
+            ass[(i1-1)*n + j1, (i2-1)*n + j2, (i3-1)*n + j3, (k-1)*n + l] = C.ass[i1,i2,i3,k]
+        end
+    end
+
+    set_tensor_product!(CxG, mult)
+    set_associator!(CxG, ass)
+
+    one_coeffs = zeros(Int,n*m)
+    one_C = one(C).components
+
+    for i ∈ 1:m
+        one_coeffs[(i-1)*n + 1] = one_C[i]
+    end
+
+    set_one!(CxG, one_coeffs)
+
+    try 
+        spheric = [C.spherical[i] for j ∈ 1:n, i ∈ 1:m][:]
+        set_spherical(CxG, spheric)
+    catch
+    end
+
+    set_name!(CxG, "Crossed product of $(C.name) and $G")
+
+    return CxG
+end
+
+
+function ⋊(C::SixJCategory, G)
+    gcrossed_product(C,G)
+end
+
+function gcrossed_product(C::SixJCategory, G::GAPGroup)
+    # Define a canonical G-action on C. Might be trivial
+
+    action = gtensor_action(C,G)
+
+    return gcrossed_product(C, action)
+end
+

src/TensorCategoryFramework/TensorFunctors.jl

@@ -121,4 +121,18 @@ function induction_mor_map(f::Morphism)
     S = simples(parent(domain(f)))
 
     return direct_sum([id(s)⊗f⊗id(dual(s)) for s ∈ S])
-end
+end
+
+
+#=----------------------------------------------------------
+    Inner Endofunctors
+----------------------------------------------------------=#
+
+abstract type MonoidalFunctor <: AbstractFunctor end 
+
+mutable struct InnerAutoequivalence <: MonoidalFunctor 
+    domain::Category
+    object::Object 
+end 
+
+