Code from BasisLieHighestWeight, Pull request oscar-system#2115

experimental/BasisLieHighestWeight/docs/doc.main

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+[
+   "basisLieHighestWeight.md"
+]

experimental/BasisLieHighestWeight/docs/src/basisLieHighestWeight.md

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+```@meta
+CurrentModule = Oscar.BasisLieHighestWeight
+```
+
+```@setup oscar
+using Oscar.BasisLieHighestWeight
+```
+
+```@contents
+Pages = ["basisLieHighestWeight.md"]
+```
+
+# Monomial bases for Lie algebras
+```@docs
+basisLieHighestWeight2
+```

experimental/BasisLieHighestWeight/src/LieAlgebras.jl

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+using Oscar
+
+fromGap = Oscar.GAP.gap_to_julia
+
+
+function create_lie_lgebra(type::String, rank::Int)::Tuple{GAP.Obj, GAP.Obj}
+    """
+    Creates the Lie-algebra as a GAP object that gets used for a lot of other computations with GAP
+    """
+    lie_algebra = GAP.Globals.SimpleLieAlgebra(GAP.Obj(type), rank, GAP.Globals.Rationals)
+    return lie_algebra, GAP.Globals.ChevalleyBasis(lie_algebra)
+end
+
+
+gapReshape(A) = sparse_matrix(QQ, hcat(A...))
+
+# temporary workaround for issue 2128
+function multiply_scalar(A::SMat{T}, d) where T
+    for i in 1:nrows(A)
+        scale_row!(A, i, T(d))
+    end
+    return A
+end
+
+function matricesForOperators(lie_algebra::GAP.Obj, highest_weight::Vector{Int}, 
+                              ops::GAP.Obj)::Vector{SMat{ZZRingElem}}
+    """
+    used to create tensorMatricesForOperators
+    """
+    M = Oscar.GAP.Globals.HighestWeightModule(lie_algebra, Oscar.GAP.julia_to_gap(highest_weight))
+    matrices_of_operators = Oscar.GAP.Globals.List(ops, o -> Oscar.GAP.Globals.MatrixOfAction(GAP.Globals.Basis(M), o))
+    matrices_of_operators = gapReshape.( Oscar.GAP.gap_to_julia(matrices_of_operators))
+    denominators = map(y->denominator(y[2]), union(union(matrices_of_operators...)...))
+    common_denominator = lcm(denominators)# // 1
+    matrices_of_operators = (A->change_base_ring(ZZ, multiply_scalar(A, common_denominator))).(matrices_of_operators)
+    return matrices_of_operators
+end
+
+
+function weights_for_operators(lie_algebra::GAP.Obj, cartan::GAP.Obj, operators::GAP.Obj)::Vector{Vector{Int}}
+    """
+    Calculates the weight wts[i] for each operator ops[i]
+    """
+    cartan = fromGap(cartan, recursive=false)
+    operators = fromGap(operators, recursive=false)
+    asVec(v) = fromGap(GAP.Globals.ExtRepOfObj(v))
+    if any(iszero.(asVec.(operators)))
+        error("ops should be non-zero")
+    end
+    nzi(v) = findfirst(asVec(v) .!= 0)
+    return [
+        [asVec(h*v)[nzi(v)] / asVec(v)[nzi(v)] for h in cartan] for v in operators
+    ]
+end

experimental/BasisLieHighestWeight/src/MonomialOrder.jl

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+function get_monomial_order_lt(monomial_order::Union{String, Function}, ZZx::ZZMPolyRing, 
+    x::Vector{ZZMPolyRingElem})::Function
+    """
+    Returns the desired monomial_order function less than
+    """
+    #if isa(monomial_order, Function)
+    #   choosen_monomial_order = monomial_order
+    if monomial_order == "GRevLex"
+        choosen_monomial_order = degrevlex(x)
+    elseif monomial_order == "RevLex"
+        choosen_monomial_order = revlex(x)
+    elseif monomial_order == "Lex"
+        choosen_monomial_order = lex(x)
+    else
+        println("no suitable order picked")
+    end
+        return (mon1, mon2) -> (cmp(choosen_monomial_order, mon1, mon2) == -1)
+end

experimental/BasisLieHighestWeight/src/NewMonomial.jl

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+include("./VectorSpaceBases.jl")
+include("./TensorModels.jl")
+include("./LieAlgebras.jl")
+include("./MonomialOrder.jl")
+include("./WeylPolytope.jl")
+
+fromGap = Oscar.GAP.gap_to_julia
+
+
+function calc_weight(mon::ZZMPolyRingElem, weights::Vector{Vector{Int}})::Vector{Int}
+    """
+    calculates weight associated with monomial mon
+    """
+    degree_mon = degrees(mon)
+    weight = [0 for i in 1:length(weights[1])]
+    for i in 1:length(degree_mon)
+        weight .+= degree_mon[i] * weights[i]
+    end
+    return weight
+end
+
+function calc_vec(v0::SRow{ZZRingElem}, mon::ZZMPolyRingElem, 
+                  matrices_of_operators::Vector{SMat{ZZRingElem}})::SRow{ZZRingElem}
+    """
+    calculates vector associated with monomial mon
+    """
+    vec = v0
+    degree_mon = degrees(mon)
+    for i in length(degree_mon):-1:1
+        for j in 1:degree_mon[i]
+            vec = mul(vec, transpose(matrices_of_operators[i])) # currently there is no sparse matrix * vector mult
+        end
+    end
+    return vec
+end
+
+function highest_calc_sub_monomial(x::Vector{ZZMPolyRingElem}, mon::ZZMPolyRingElem, 
+                                    calc_monomials::Dict{ZZMPolyRingElem, Tuple{TVec, Vector{Int}}},
+                                    number_of_operators::Int)::ZZMPolyRingElem
+    """
+    returns the key in calc_monomials that can be extended by the least amount of left-operations to mon
+    """
+    sub_mon = copy(mon)
+
+    for i in 1:number_of_operators
+        while is_divisible_by(sub_mon, x[i])
+            if haskey(calc_monomials, sub_mon)
+                return sub_mon
+            else
+                sub_mon /= x[i]
+            end
+        end
+    end
+    return sub_mon
+end
+
+function calc_new_mon!(x::Vector{ZZMPolyRingElem}, mon::ZZMPolyRingElem, weights::Vector{Vector{Int}}, 
+                        matrices_of_operators::Vector{SMat{ZZRingElem}}, number_of_operators::Int,
+                        calc_monomials::Dict{ZZMPolyRingElem, Tuple{TVec, Vector{Int}}}, 
+                        space::Dict{Vector{Int64}, Oscar.BasisLieHighestWeight.VSBasis}, e::Vector{Vector{Int}}, 
+                        cache_size::Int)::SRow{ZZRingElem}
+    # calculate vector of mon by extending a previous calculated vector to a
+    # monom that differs only by left-multiplication, save results in calc_monomials
+    sub_mon = highest_calc_sub_monomial(x, mon, calc_monomials, number_of_operators)
+    #println("sub_mon: ", sub_mon)
+    sub_mon_cur = copy(sub_mon)
+    (vec, weight) = calc_monomials[sub_mon]
+    for i in number_of_operators:-1:1
+        for k in degrees(sub_mon)[i]:(degrees(mon)[i]-1)
+            sub_mon_cur *= x[i]
+            weight += weights[i]
+            if !haskey(space, weight)
+                space[weight] = nullSpace()
+            end
+
+            vec = mul(vec, transpose(matrices_of_operators[i])) # currently there is no sparse matrix * vector mult
+            if length(calc_monomials) < cache_size
+                calc_monomials[sub_mon_cur] = (vec, weight)
+            end
+
+            # check if the extended monomial can be deleted from calculated_monomials, i.e. the other possible 
+            # extensions by left multiplication with some x[i] are already contained
+            can_be_deleted = true
+            k = number_of_operators
+            for l in 1:number_of_operators
+                if degrees(sub_mon_cur - x[i])[l] != 0
+                    k = l
+                end
+            end
+            for l in 1:k
+                can_be_deleted = can_be_deleted && haskey(calc_monomials, sub_mon_cur - x[i] + x[l])
+            end
+            if can_be_deleted && sub_mon_cur != x[i]
+                delete!(calc_monomials, sub_mon_cur - x[i])
+            end
+        end
+    end
+    #println(length(calc_monomials))
+    return vec
+end
+