more cartanType for crg

src/ComplexR.jl

@@ -56,28 +56,55 @@ julia> crg(4) # there is also a short alias
 G₄
 ```
 """
-function complex_reflection_group(i::Integer)
+function complex_reflection_group(i::Integer,cartanType=1)
   if i==23     coxgroup(:H,3)
-  elseif i==28 coxgroup(:F,4)
+  elseif i==28 coxgroup(:F,4,cartanType)
   elseif i==30 coxgroup(:H,4)
   elseif i==35 coxgroup(:E,6)
   elseif i==36 coxgroup(:E,7)
   elseif i==37 coxgroup(:E,8)
-  else t=TypeIrred(series=:ST,ST=Int(i))
+  elseif cartanType==1
+    t=TypeIrred(series=:ST,ST=Int(i))
     PRG(simpleroots(t),simplecoroots(t))
+  else
+    if i in (4,8,12,16,20,22) error("G",i," has no cartanType") end
+    t=TypeIrred(;series=:ST,ST=Int(i))
+    r=simpleroots(t)
+    c=cartan(t)
+    t.cartanType=cartanType
+    if i in (5,6,9,10,14,17,18,21)
+      D=Diagonal([cartanType,1])
+      cr=solutionmat(transpose(r),D*c*inv(D//1))
+      if nothing in cr error("unexpected") end
+      PRG(r,cr)
+    else error("cartanType!=1 not implemented for G",i)
+    end
   end
 end
 
-function complex_reflection_group(p,q,r)
+function complex_reflection_group(p,q,r,cartanType=1)
   if !iszero(p%q) || p<=0 || r<=0 || (r==1 && q!=1)
    error("complex_reflection_group(p,q,r) must satisfy: q|p, r>0, and r=1 => q=1")
   end
   if p==1 return r==1 ? coxgroup() : rootdatum(:gl,r)
   elseif p==2 return rootdatum(:so,q==2 ? 2r : 2r+1)
   elseif p==q && r==2 return coxgroup(:I,2,p)
   end
-  t=TypeIrred(;series=:ST,p,q,rank=r)
-  PRG(simpleroots(t),simplecoroots(t))
+  if cartanType!=1
+#   if r>2 error("cartanType!=1 not implemented for G",(p,q,r)) end
+    if r==1 || p==q  error("G",(p,q,r)," has no cartanType") end
+    t=TypeIrred(;series=:ST,p=Int(p),q=Int(q),r=Int(r),rank=r)
+    rr=simpleroots(t)
+    c=cartan(t)
+    t.cartanType=cartanType
+    D=Diagonal(vcat([cartanType],fill(1,r-1)))
+    cr=solutionmat(transpose(rr),D*c*inv(D//1))
+    if nothing in cr error("unexpected") end
+    PRG(rr,cr)
+  else
+    t=TypeIrred(;series=:ST,p,q,rank=r)
+    PRG(simpleroots(t),simplecoroots(t))
+  end
 end
 
 const crg=complex_reflection_group

src/PermRoot.jl

@@ -401,8 +401,8 @@ function Base.show(io::IO, t::TypeIrred)
         if haskey(t,:ST) n*="($(t.ST)"
         else n*="($(t.p),$(t.q),$(t.rank)"
         end
-        if haskey(t,:cartanType)
-          n*=","*xrepr(io,t.cartanType)*")"
+        if haskey(t,:cartanType) n*=","*xrepr(io,t.cartanType)*")"
+        else n*=")"
         end
       end
     else
@@ -964,10 +964,10 @@ function findgensDiagCartan(H,C,p)
     cc=unique.(cc)
     if any(x->length(x)>1,cc) return nothing end
     cc=only.(cc)
-    best=findfirst(==(1),cc)
+    best=findfirst(==(d[n-1]),cc)
     if !isnothing(best)
       p[n]=l[best]
-      d[n]=1
+      d[n]=d[n-1]
     else
       d[n]=cc[1]
       p[n]=l[1]
@@ -1032,9 +1032,10 @@ function refltype(W::PermRootGroup)
       if verbose print("#4") end
       if !isnothing(better) good=better[1]
         rd=diagconj_elt(C,cartan(R,good))
-        if length(rd)==2 && isone(rd[1])
-          d.cartanType=improve_type(rd[2])
-        end
+        if rd[1]!=1 error("hah non") end
+        if !allequal(rd[2:end])
+        println("##### Warning: cartantypes for ",d,"=",rd[3:end].//rd[2]) end
+        d.cartanType=improve_type(rd[2])
       else better=findgensDiagCartan2(R,C)
         if verbose print("#5") end
         if !isnothing(better) good=better[1] 

src/Weyl.jl

@@ -811,7 +811,7 @@ julia> reflrep(W)
  [1 3; 0 -1]
 ```
 """
-CoxGroups.coxeter_group(t::Symbol,r::Int=0,b::Int...;sc=false)=iszero(r) ? coxgroup() :
+CoxGroups.coxeter_group(t::Symbol,r::Int=0,b::Number...;sc=false)=iszero(r) ? coxgroup() :
 sc ? rootdatum(permutedims(cartan(t,r,b...)),Matrix{Int}(I,r,r)) : rootdatum(cartan(t,r,b...))
 
 """