diff --git a/src/auxiliary.jl b/src/auxiliary.jl
index 6ed53605..3d7f0ec5 100644
--- a/src/auxiliary.jl
+++ b/src/auxiliary.jl
@@ -13,10 +13,23 @@ function _kron(A, B)
     end
     return C
 end
-_kron_promote(A::Number, B::Number, _, _) = A * B
-function _kron_promote(A₁, B₁, sz₁, sz₂)
-    return _kron(A₁ isa Number ? fill(A₁, sz₁) : A₁, B₁ isa Number ? fill(B₁, sz₂) : B₁)
+# combine the symbols `A` and `B` of the factors `a` and `b` of product sectors; a scalar symbol
+# is promoted to an array of the same kind as the other symbol, with the size given by the
+# multiplicities of its sectors, unless these are all one: then it just scales the other symbol,
+# which is returned as is for a scalar one (symbols are not mutated, and may already be shared
+# with a cache); the type of the result thus only depends on the types of the symbols
+_kron_promote(A::Number, B::Number, a, b) = A * B
+function _kron_promote(A::Number, B::AbstractArray, a, b)
+    sz = _symbol_size(a)
+    all(isone, sz) || return _kron(fill!(similar(B, typeof(A), sz), A), B)
+    return isone(A) && promote_type(typeof(A), eltype(B)) === eltype(B) ? B : A * B
 end
+function _kron_promote(A::AbstractArray, B::Number, a, b)
+    sz = _symbol_size(b)
+    all(isone, sz) || return _kron(A, fill!(similar(A, typeof(B), sz), B))
+    return isone(B) && promote_type(eltype(A), typeof(B)) === eltype(A) ? A : A * B
+end
+_kron_promote(A::AbstractArray, B::AbstractArray, a, b) = _kron(A, B)
 
 # Manhattan based distance enumeration: I is supposed to be one-based index
 # TODO: is there any way to make this faster?
diff --git a/src/product.jl b/src/product.jl
index b0ed5554..6848b617 100644
--- a/src/product.jl
+++ b/src/product.jl
@@ -96,82 +96,80 @@ _symbol_size((a, b, c, d, e, f)::NTuple{6, Sector}) = (Nsymbol(a, b, e), Nsymbol
 @inline function _kron_promote_inputs(sectors)
     heads = map(_firstsector, sectors)
     tails = map(_tailsector, sectors)
-    sz₁ = _symbol_size(heads)
-    sz₂ = _symbol_size(tails)
-    return heads, tails, sz₁, sz₂
+    return heads, tails
 end
 
 function Fsymbol(a::I, b::I, c::I, d::I, e::I, f::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c, d, e, f))
+    heads, tails = _kron_promote_inputs((a, b, c, d, e, f))
     V₁ = Fsymbol(heads...)
     V₂ = Fsymbol(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Fsymbol(a::I, b::I, c::I, d::I, e::I, f::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Fsymbol(map(_firstsector, (a, b, c, d, e, f))...)
 end
 function Fsymbol_from_fusiontensor(a::I, b::I, c::I, d::I, e::I, f::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c, d, e, f))
+    heads, tails = _kron_promote_inputs((a, b, c, d, e, f))
     V₁ = Fsymbol_from_fusiontensor(heads...)
     V₂ = Fsymbol_from_fusiontensor(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Fsymbol_from_fusiontensor(a::I, b::I, c::I, d::I, e::I, f::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Fsymbol_from_fusiontensor(map(_firstsector, (a, b, c, d, e, f))...)
 end
 
 function Rsymbol(a::I, b::I, c::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c))
+    heads, tails = _kron_promote_inputs((a, b, c))
     V₁ = Rsymbol(heads...)
     V₂ = Rsymbol(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Rsymbol(a::I, b::I, c::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Rsymbol(map(_firstsector, (a, b, c))...)
 end
 function Rsymbol_from_fusiontensor(a::I, b::I, c::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c))
+    heads, tails = _kron_promote_inputs((a, b, c))
     V₁ = Rsymbol_from_fusiontensor(heads...)
     V₂ = Rsymbol_from_fusiontensor(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Rsymbol_from_fusiontensor(a::I, b::I, c::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Rsymbol_from_fusiontensor(map(_firstsector, (a, b, c))...)
 end
 
 function Bsymbol(a::I, b::I, c::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c))
+    heads, tails = _kron_promote_inputs((a, b, c))
     V₁ = Bsymbol(heads...)
     V₂ = Bsymbol(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Bsymbol(a::I, b::I, c::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Bsymbol(map(_firstsector, (a, b, c))...)
 end
 function Bsymbol_from_fusiontensor(a::I, b::I, c::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c))
+    heads, tails = _kron_promote_inputs((a, b, c))
     V₁ = Bsymbol_from_fusiontensor(heads...)
     V₂ = Bsymbol_from_fusiontensor(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Bsymbol_from_fusiontensor(a::I, b::I, c::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Bsymbol_from_fusiontensor(map(_firstsector, (a, b, c))...)
 end
 
 function Asymbol(a::I, b::I, c::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c))
+    heads, tails = _kron_promote_inputs((a, b, c))
     V₁ = Asymbol(heads...)
     V₂ = Asymbol(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Asymbol(a::I, b::I, c::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Asymbol(map(_firstsector, (a, b, c))...)
 end
 function Asymbol_from_fusiontensor(a::I, b::I, c::I) where {I <: ProductSector}
-    heads, tails, sz₁, sz₂ = _kron_promote_inputs((a, b, c))
+    heads, tails = _kron_promote_inputs((a, b, c))
     V₁ = Asymbol_from_fusiontensor(heads...)
     V₂ = Asymbol_from_fusiontensor(tails...)
-    return _kron_promote(V₁, V₂, sz₁, sz₂)
+    return _kron_promote(V₁, V₂, heads, tails)
 end
 function Asymbol_from_fusiontensor(a::I, b::I, c::I) where {I <: ProductSector{<:Tuple{Sector}}}
     return Asymbol_from_fusiontensor(map(_firstsector, (a, b, c))...)