first commit of basisLieHighestWeight

Project.toml

@@ -6,6 +6,7 @@ version = "0.12.0-DEV"
 [deps]
 AbstractAlgebra = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
 AlgebraicSolving = "66b61cbe-0446-4d5d-9090-1ff510639f9d"
+Distributed = "8ba89e20-285c-5b6f-9357-94700520ee1b"
 DocStringExtensions = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
 GAP = "c863536a-3901-11e9-33e7-d5cd0df7b904"
 Hecke = "3e1990a7-5d81-5526-99ce-9ba3ff248f21"
@@ -19,6 +20,7 @@ Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 RandomExtensions = "fb686558-2515-59ef-acaa-46db3789a887"
 RecipesBase = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
 Singular = "bcd08a7b-43d2-5ff7-b6d4-c458787f915c"
+SparseArrays = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"
 TOPCOM_jll = "36f60fef-b880-50dc-9289-4aaecee93cc3"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 UUIDs = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

experimental/Experimental.jl

@@ -34,3 +34,4 @@ include("Schemes/BlowupMorphism.jl")
 include("Schemes/ToricSchemes/include.jl")
 
 include("ExteriorAlgebra/ExteriorAlgebra.jl")
+include("basisLieHighestWeight/main.jl")

experimental/basisLieHighestWeight/LieAlgebras.jl

@@ -0,0 +1,50 @@
+# ?
+
+using Oscar
+using SparseArrays
+
+G = Oscar.GAP.Globals
+forGap = Oscar.GAP.julia_to_gap
+fromGap = Oscar.GAP.gap_to_julia
+
+
+function lieAlgebra(t::String, n::Int)
+    """
+    Creates the Lie-algebra as a GAP object that gets used for a lot other computations with GAP
+    """
+    L = G.SimpleLieAlgebra(forGap(t), n, G.Rationals)
+    return L, G.ChevalleyBasis(L)
+end
+
+
+gapReshape(A) = sparse(hcat(A...))
+
+
+function matricesForOperators(L, hw, ops)
+    """
+    used to create tensorMatricesForOperators
+    """
+    M = G.HighestWeightModule(L, forGap(hw))
+    mats = G.List(ops, o -> G.MatrixOfAction(G.Basis(M), o))
+    mats = gapReshape.(fromGap(mats))
+    d = lcm(denominator.(union(mats...)))
+    mats = (A->ZZ.(A*d)).(mats)
+    return mats
+end
+
+
+function weightsForOperators(L, cartan, ops)
+    """
+    Calculates the weight wts[i] for each operator ops[i]
+    """
+    cartan = fromGap(cartan, recursive=false)
+    ops = fromGap(ops, recursive=false)
+    asVec(v) = fromGap(G.ExtRepOfObj(v))
+    if any(iszero.(asVec.(ops)))
+        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 ops
+    ]
+end

experimental/basisLieHighestWeight/LongestWord.jl

@@ -0,0 +1,50 @@
+#using Gapjm
+
+G = Oscar.GAP.Globals
+forGap = Oscar.GAP.julia_to_gap
+fromGap = Oscar.GAP.gap_to_julia
+
+#function longest_weyl_word(t, n)
+#    """
+#    generates a reduced expression of the longest weylword of type (t,n) by choosing uniformly a random reflection that is not leftdescending
+#    the resulting longest words are not uniformly distributed
+#    """
+#    W = Gapjm.coxgroup(Symbol(t),n) # Weyl-group
+#    S = Gapjm.gens(W) # generators of W (simple reflections)
+#    m = length(S)
+#    p = W() # id
+#    word = [] # generated word
+#
+#    # extend p with reflection that are not leftdescending until not possible (i.e. reached longest word)
+#    while true
+#        not_desc = [i for i=1:m if !(i in Gapjm.leftdescents(W,p))] # set of i s.t. length(S[i]*p) > length(p)
+#        if length(not_desc) >= 1
+#            i = rand(not_desc)
+#            push!(word, i)
+#            p = S[i]*p
+#        else
+#            break
+#        end
+#    end
+#    return word
+#end
+
+#function is_longest_weyl_word(t,n,word)
+#    """
+#    returns if word is a reduced expression of the longest weyl word of type (t,n) 
+#    is_longest_weyl_word(t,n,longest_weyl_word(t, n)) is always true
+#    """
+#    W = Gapjm.coxgroup(Symbol(t),n) # Weyl-group
+#    p = W(word ...) # group element of word
+#    return p == longest(W) # is word longest word?
+#end
+
+function sub_simple_refl(word, L, n)
+    """
+    substitute simple reflections (i,i+1), saved in dec by i, with E_{i,i+1}  
+    """
+    R = G.RootSystem(L)
+    CG = fromGap(G.CanonicalGenerators(R)[1], recursive = false)
+    ops = forGap([CG[i] for i in word], recursive = false)
+    return ops
+end