hmmm

docs/src/.vitepress/config.mts

@@ -95,6 +95,7 @@ export default defineConfig({
                 { text: 'Introduction', link: 'manual/algebras/intro'},
                 { text: 'Basics', link: 'manual/algebras/basics'},
                 { text: 'Structure constant algebras', link: 'manual/algebras/structureconstant'},
+                { text: 'Group algebras', link: 'manual/algebras/groupalgebras'},
                 { text: 'Ideals', link: 'manual/algebras/ideals'},
               ]
             },

docs/src/manual/algebras/groupalgebras.md

@@ -0,0 +1,70 @@
+# Group algebras
+
+```@meta
+CurrentModule = Hecke
+DocTestSetup = quote
+  using Hecke
+end
+```
+
+As is natural, the basis of a group algebra $K[G]$ correspond to the elements of $G$ with respect
+to some arbitrary ordering.
+
+## Creation
+
+```@docs
+group_algebra(::Field, ::Group)
+```
+
+Note that by default, this construction requires enumerating all elements of
+the group and thus is inefficient for large groups. Using the optional argument `sparse = true`,
+the algebra can be constructed with a different internal model. This allows for much larger groups,
+but not all functionality is available in this case.
+
+```jldoctest
+julia> G = abelian_group([2 for i in 1:10]) # group of order 2^10
+(Z/2)^10
+
+julia> QG = group_algebra(QQ, G; sparse = true);
+```
+
+## Elements
+
+Given a group algebra `A` and an element of a group `g`, the corresponding group algebra element
+can be constructed using the syntax `A(g)`.
+
+```jldoctest
+julia> G = abelian_group([2, 2]); a = G([0, 1]);
+
+julia> QG = group_algebra(QQ, G);
+sparse = false
+
+julia> x = QG(a)
+[0, 0, 1, 0]
+```
+
+Vice versa, one can obtain the coefficient of a group algebra element `x` with respect to a group
+element `a` using the syntax `x[a]`.
+
+```jldoctest
+julia> G = abelian_group([2, 2]); a = G([0, 1]);
+
+julia> QG = group_algebra(QQ, G);
+
+julia> x = QG(a)
+[0, 0, 1, 0]
+
+julia> x[a]
+1
+```
+
+It is also possible to create elements from dictionaries:
+
+```jldoctest
+julia> G = abelian_group([2, 2]); a = G([0, 1]);
+
+julia> QG = group_algebra(QQ, G);
+
+julia> QG(Dict(a => 2, zero(G) => 1)) == 2 * QG(a) + 1 * QG(zero(G))
+true
+```

src/AlgAss/AbsAlgAss.jl

@@ -1,3 +1,18 @@
+################################################################################
+#
+#  Internal thingy
+#
+################################################################################
+
+# check if elements are represented using a sparse row
+
+_is_sparse(A::AbstractAssociativeAlgebra) = false
+
+_is_sparse(A::GroupAlgebra) = A.sparse
+
+# because we are lazy
+_is_dense(A::AbstractAssociativeAlgebra) = !_is_sparse(A)
+
 _base_ring(A::AbstractAssociativeAlgebra) = base_ring(A)
 
 @doc raw"""

src/AlgAss/AlgGrp.jl

@@ -59,17 +59,25 @@ end
 ################################################################################
 
 @doc raw"""
-    group_algebra(K::Ring, G; op = *) -> GroupAlgebra
+    group_algebra(K::Ring, G::Group; sparse::Bool = false,
+                                     cached::Bool = true) -> GroupAlgebra
 
-Returns the group ring $K[G]$.
-$G$ may be any set and `op` a group operation on $G$.
-"""
-group_algebra(K::Ring, G; op = *, cached::Bool = true, sparse::Bool = false) = GroupAlgebra(K, G; op, sparse, cached)
+Return the group algebra of the group $G$ over the ring $R$.
 
-group_algebra(K::Ring, G::FinGenAbGroup; cached::Bool = true, sparse::Bool = false) = GroupAlgebra(K, G; sparse, cached)
+# Examples
 
-function group_algebra(K::Field, G; op = *, sparse::Bool = false, cached::Bool = true)
-  A = GroupAlgebra(K, G; op, sparse, cached)
+```jldoctest
+julia> QG = group_algebra(QQ, small_group(8, 5))
+Group algebra
+  of generic group of order 8 with multiplication table
+  over rational field
+```
+"""
+function group_algebra(K::Ring, G; op = *, sparse::Bool = false, cached::Bool = true)
+  A = GroupAlgebra(K, G; op = op , sparse = sparse, cached = cached)
+  if !(K isa Field)
+    return A
+  end
   if iszero(characteristic(K))
     A.issemisimple = 1
   else
@@ -78,18 +86,8 @@ function group_algebra(K::Field, G; op = *, sparse::Bool = false, cached::Bool =
   return A
 end
 
-function group_algebra(K::Field, G::FinGenAbGroup)
-  A = group_algebra(K, G, op = +)
-  A.is_commutative = true
-  return A
-end
-
-@doc raw"""
-    (K::Ring)[G::Group] -> GroupAlgebra
-    (K::Ring)[G::FinGenAbGroup] -> GroupAlgebra
+group_algebra(K::Ring, G::FinGenAbGroup; cached::Bool = true, sparse::Bool = false) = GroupAlgebra(K, G, cached, sparse)
 
-Returns the group ring $K[G]$.
-"""
 getindex(K::Ring, G::Group) = group_algebra(K, G)
 getindex(K::Ring, G::FinGenAbGroup) = group_algebra(K, G)
 
@@ -629,7 +627,7 @@ const _reps = [(i=24,j=12,n=5,dims=(1,1,2,3,3),
 #
 ################################################################################
 
-mutable struct AbsAlgAssMorGen{S, T, U, V} <: Map{S, T, HeckeMap, AbsAlgAssMorGen}
+mutable struct AbsAlgAssMorGen{S, T, U, V} <: Map{S, T, HeckeMap, Any}#AbsAlgAssMorGen}
   domain::S
   codomain::T
   tempdomain::U

src/AlgAss/Elem.jl

@@ -4,9 +4,9 @@
 #
 ################################################################################
 
-_is_sparse(a::AbstractAssociativeAlgebraElem) = false
+_is_sparse(a::AbstractAssociativeAlgebraElem) = _is_sparse(parent(a))
 
-_is_sparse(a::GroupAlgebraElem) = parent(a).sparse
+_is_dense(a::AbstractAssociativeAlgebraElem) = _is_dense(parent(a))
 
 function AbstractAlgebra.promote_rule(U::Type{<:AbstractAssociativeAlgebraElem{T}}, ::Type{S}) where {T, S}
   if AbstractAlgebra.promote_rule(T, S) === T
@@ -94,7 +94,7 @@ function one(A::AbstractAssociativeAlgebra)
   if !has_one(A)
     error("Algebra does not have a one")
   end
-  if !A.sparse
+  if _is_dense(A)
     return A(deepcopy(A.one)) # deepcopy needed by mul!
   else
     return A(deepcopy(A.sparse_one))
@@ -681,7 +681,17 @@ end
 #end
 
 function (A::GroupAlgebra{T, S, R})(c::R) where {T, S, R}
-  return GroupAlgebraElem{T, typeof(A)}(A, deepcopy(c))
+  return GroupAlgebraElem{T, typeof(A)}(A, c)
+end
+
+function (A::GroupAlgebra{T, S, R})(d::Dict{R, <: Any}) where {T, S, R}
+  K = base_ring(A)
+  dd = sparse_row(base_ring(A), [(__elem_index(A, g), K(i)) for (g, i) in d])
+  if _is_dense(A)
+    return GroupAlgebraElem{T, typeof(A)}(A, Vector(dd, dim(A)))
+  else
+    return GroupAlgebraElem{T, typeof(A)}(A, dd)
+  end
 end
 
 # Generic.Mat needs it
@@ -727,18 +737,22 @@ function show(io::IO, a::AbstractAssociativeAlgebraElem)
   if get(io, :compact, false)
     print(io, coefficients(a, copy = false))
   else
-    if a isa GroupAlgebraElem && parent(a).sparse
+    if _is_sparse(a)
       sum = Expr(:call, :+)
       if !iszero(a)
         for (i, ci) in a.coeffs_sparse
           push!(sum.args,
                 Expr(:call, :*, AbstractAlgebra.expressify(ci, context = io),
-                                AbstractAlgebra.expressify(parent(a).base_to_group[i], context = io)))
+                                AbstractAlgebra.expressify(parent(a).base_to_group[i], context = IOContext(io, :compact => true))))
         end
       end
       print(io, AbstractAlgebra.expr_to_string(AbstractAlgebra.canonicalize(sum)))
     else
-      print(io, coefficients(a, copy = false))
+      ve = Expr(:vect)
+      for ci in coefficients(a, copy = false)
+        push!(ve.args, AbstractAlgebra.expressify(ci, context = io))
+      end
+      print(io, AbstractAlgebra.expr_to_string(AbstractAlgebra.canonicalize(ve)))
     end
   end
 end
@@ -781,7 +795,11 @@ end
 
 function ==(a::AbstractAssociativeAlgebraElem{T}, b::AbstractAssociativeAlgebraElem{T}) where {T}
   parent(a) != parent(b) && return false
-  return coefficients(a, copy = false) == coefficients(b, copy = false)
+  if !_is_sparse(a)
+    return coefficients(a, copy = false) == coefficients(b, copy = false)
+  else
+    return a.coeffs_sparse == b.coeffs_sparse
+  end
 end
 
 ################################################################################

src/AlgAss/Types.jl

@@ -130,6 +130,13 @@ end
 @attributes mutable struct GroupAlgebra{T, S, R} <: AbstractAssociativeAlgebra{T}
   base_ring::Ring
   group::S
+  # We represent elements using a coefficient vector,
+  # so all we have to keep track of is which group element corresponds to
+  # which basis element of the algebra
+  # This is what group_to_base, base_to_group are for. They realize the map
+  # G -> {1,...,n}
+  # {1,...n} -> G
+
   group_to_base::Dict{R, Int}
   base_to_group::Vector{R}
   one::Vector{T}
@@ -145,18 +152,21 @@ end
   center
   maps_to_numberfields
   maximal_order
+
+  # For the sparse presentation
   sparse::Bool
-  ind::Int
-  sparse_one
+  ind::Int      # This is the number of group elements currently stored in
+                # group_to_base and base_to_group.
+  sparse_one    # Store the sparse row for the one element
 
-  function GroupAlgebra(K::Ring, G::FinGenAbGroup, cached::Bool = true)
-    A = GroupAlgebra(K, G, op = +, cached = cached)
+  function GroupAlgebra(K::Ring, G::FinGenAbGroup, cached::Bool = true, sparse::Bool = false)
+    A = GroupAlgebra(K, G; op = +, cached = cached, sparse = sparse)
     A.is_commutative = true
     return A
   end
 
-  function GroupAlgebra(K::Ring, G; op = *, cached = true, sparse::Bool = false)
-    return get_cached!(GroupAlgebraID, (K, G, op), cached) do
+  function GroupAlgebra(K::Ring, G; op = *, cached::Bool = true, sparse::Bool = false)
+    return get_cached!(GroupAlgebraID, (K, G, op, sparse), cached) do
       A = new{elem_type(K), typeof(G), elem_type(G)}()
       A.sparse = sparse
       A.ind = -1
@@ -165,9 +175,14 @@ end
       A.issemisimple = 0
       A.base_ring = K
       A.group = G
-      d = Int(order(G))
       A.group_to_base = Dict{elem_type(G), Int}()
-      A.base_to_group = Vector{elem_type(G)}(undef, d)
+      if !sparse
+        @assert is_finite(G)
+        d = order(Int, G)
+        A.base_to_group = Vector{elem_type(G)}(undef, d)
+      else
+        A.base_to_group = Vector{elem_type(G)}(undef, 1)
+      end
 
       if A.sparse
         if G isa FinGenAbGroup
@@ -218,7 +233,7 @@ end
   end
 end
 
-const GroupAlgebraID = AbstractAlgebra.CacheDictType{Tuple{Ring, Any, Any}, GroupAlgebra}()
+const GroupAlgebraID = AbstractAlgebra.CacheDictType{Tuple{Ring, Any, Any, Bool}, GroupAlgebra}()
 
 mutable struct GroupAlgebraElem{T, S} <: AbstractAssociativeAlgebraElem{T}
   parent::S
@@ -241,11 +256,7 @@ mutable struct GroupAlgebraElem{T, S} <: AbstractAssociativeAlgebraElem{T}
 
   function GroupAlgebraElem{T, S}(A::S, g::U) where {T, S, U}
     if A.sparse
-      i = get!(A.group_to_base, g) do
-        A.ind += 1
-        A.base_to_group[A.ind] = g
-        return A.ind
-      end
+      i = __elem_index(A, g)
       a = GroupAlgebraElem{T, S}(A)
       a.coeffs_sparse = sparse_row(base_ring(A), [i], [one(base_ring(A))])
       return a
@@ -272,11 +283,11 @@ end
 
 __elem_index(A, g) = get!(A.group_to_base, g) do
         A.ind += 1
+        resize!(A.base_to_group, max(A.ind, length(A.base_to_group)))
         A.base_to_group[A.ind] = g
         return A.ind
       end
 
-
 ################################################################################
 #
 #  AbsAlgAssIdl

src/Grp/GenGrp.jl

@@ -199,6 +199,8 @@ end
 #
 ################################################################################
 
+is_finite(::MultTableGroup) = true
+
 elem_type(::Type{MultTableGroup}) = MultTableGroupElem
 
 Base.hash(G::MultTableGroupElem, h::UInt) = Base.hash(G.i, h)
@@ -256,10 +258,12 @@ end
 #
 ################################################################################
 
-function order(G::MultTableGroup)
+function order(::Type{Int}, G::MultTableGroup)
   return size(G.mult_table, 1)
 end
 
+order(G::MultTableGroup) = order(Int, G)
+
 length(G::MultTableGroup) = order(G)
 
 ################################################################################

src/GrpAb/GrpAbFinGen.jl

@@ -586,6 +586,8 @@ function order(A::FinGenAbGroup)
   return prod(elementary_divisors(A))
 end
 
+order(::Type{Int}, A::FinGenAbGroup) = Int(order(A))
+
 ################################################################################
 #
 #  Exponent

test/AlgAss/AlgGrp.jl

@@ -73,4 +73,15 @@
       end
     end
   end
+
+  # abelian groups
+
+  QG = group_algebra(QQ, abelian_group([2, 2]))
+  @test QG isa GroupAlgebra
+  @test QG !== group_algebra(QQ, abelian_group([2, 2]); cached = false)
+  @test QG !== group_algebra(QQ, abelian_group([2, 2]); sparse = true)
+
+  QG = group_algebra(QQ, abelian_group([2 for i in 1:10]); sparse = true)
+  @test QG isa GroupAlgebra
+  @test QG !== group_algebra(QQ, abelian_group([2 for i in 1:10]); sparse = true, cached = false)
 end

test/AlgAss/Elem.jl

@@ -85,4 +85,11 @@
     Hecke.add!(b,b)
     @test b == A(matrix(QQ, [6 8; 10 12]))
   end
+
+  # fancy group algebra element constructor
+  G = abelian_group([2, 2]); a = G([0, 1]);
+  QG = group_algebra(QQ, G);
+  @test QG(Dict(a => 2, zero(G) => 1)) == 2 * QG(a) + 1 * QG(zero(G))
+  QG = group_algebra(QQ, G; sparse = true);
+  @test QG(Dict(a => 2, zero(G) => 1)) == 2 * QG(a) + 1 * QG(zero(G))
 end