# The computation presented here are related to the article: # M. Bulois, On the normality of the null-fiber of the moment map for theta- and tori representations, https://arxiv.org/abs/1706.05456 # #They were made using GAP (version 4.8.3) and W. deGraaf's GAP package "sla" (http://www.science.unitn.it/~degraaf/sla.html, version 1.2) # # More precisely, it contains the computations justifying the claims made # -in Proposition 4.10 (list of locally free stable theta-representations of rank 1 in the exceptional cases), see function "Stable_rk1_Inner" for the inner cases and "Stable_rk1_Outer" for the outer cases # -in Lemma 4.11 (classification of the locally free stable theta-representations of rank 1 where property (P) holds), see functions "Has_regs_check" and "Really_Non_reg" # -in Lemma 4.12 (the non-existence of locally free theta-rep having a locally free stable proper Levi of rank 1 where property (P) does not hold.), see function "ListOfpossLocfreec" # # The file is divided in 3 parts. In the first part "Functions" we define the functions used in the second part "Computations" # # #------------------------------------------- # A. Functions #------------------------------------------- LoadPackage("sla"); #Loading the "sla" package prodscal:=function(l1,l2) #computes the scalar product of two lists local i,k; k:=0; for i in [1..Length(l1)] do k:=k+(l1[i]*l2[i]); od; return k; end;; deuxn:=function(n) #returns a list of all non-zero (n+1)-uples with value in {0,1} local l,i,s; l:=[List([1..n],x->0)]; Add(l[1],1); for i in [2..(2^(n+1)-1)] do s:=n+1; Add(l,StructuralCopy(l[i-1])); while l[i-1][s]=1 do l[i][s]:=0; s:=s-1; od; l[i][s]:=l[i-1][s]+1; od; return l; end;; ListOfTheta:=function(R) #R is a Root system, the inner gradings on the corresponding Lie algebra g are classified by labellings of the extended Dynkin diagrams. #The function computes the list of labellings in {0,1} (see above Lemma 4.9 in the article to see why restricting to such labellings) giving rise to a grading on g such that $dim g_1-dim g_0=1$. local S,P,n,l,s,i,g0,g1,k,m,wp,p,hir,res,lindep; n:=Length(SimpleSystem(R)); #rank of the root system P:=PositiveRootsNF(R); #list of positive roots, written as linear combinations of the simple roots. hir:=ShallowCopy(P[Length(P)]); #coordinates of the highest root lindep:=[1]; Append(lindep,hir); #1 added on the extended node of the affine Dynkin diagram (arbitrarily sent at the start of the list), #thus the coefficients in lindep are those of the linear dependency among simple roots of the roots in the extended Dynkin diagram. l:=deuxn(n); #list of all possible labellings (for untwisted Kac diagram giving rise to an action with positive rank). res:=[]; #the selected gradings will be stored in res for s in l do g1:=0; #g1 will store the dimension of g_1 g0:=n; #g0 will store the dimension of g_0. Since g_0 contains a Cartan subspace, g0 is initialized to the rank before looking at root_spaces. m:=prodscal(lindep,s); #m stores the order of the grading for p in P do wp:=(prodscal(s{[2..(n+1)]},p) mod m); #the root p lies (modulo m) in the wp-th part of the grading induced by the labelling s. if wp=1 then g1:=g1+1; fi; #1 dimension added to g1 if the root space corresponding to p lies in g_1. if wp=m-1 then g1:=g1+1; fi; #1 dimension added to g1 if the root space corresponding to p lies in g_{-1} (the root spaces for the corresponding negative root lie in g_1). if wp=0 then g0:=g0+2; fi; #2 dimension added to g0 if the root space corresponding to p lies in g_0 (since the opposite root space then also lies in g0). od; if (g1-g0=1) then Add(res,s); fi; #[s,g0,g1,m] is stored if and only if $dim g_1-dim g_0=1$. od; return(res); end;; basesg0g1:=function(g) #g is a Simple Lie algebra. #For each grading selected by the above function, a basis of g_0 and g_1 is computed. #The output is a list of quadruples (g0,g1,s,m) where g0 (resp. g1) is a basis of g_0 (resp. g_1), s is the labelling and m the order of the automorphism. #The name of the variables and the structure of the program is globally the same as in ListOfTheta, except that g0 and g1 store basis instead of dimension. #The advantage of lauching "ListOfTheta" firstly is that it deals quickly with many labellings, thus allowing the remaining of the program to deal with few. local R,S,n,res,P,hir,Pv,Pvm,B,Car,r,s,i,wp,g0,g1,bases,m,p,lindep; R:=RootSystem(g); n:=Length(SimpleSystem(R)); res:=ListOfTheta(R); P:=PositiveRootsNF(R); hir:=ShallowCopy(P[Length(P)]); lindep:=[1]; Append(lindep,hir); Pv:=PositiveRootVectors(R); Pvm:=NegativeRootVectors(R); B:=Basis(g); Car:=B{[(Length(B)-(n-1))..Length(B)]}; #Basis of the Cartan subspace of g bases:=[]; for s in res do #s is a labelling giving rise to a grading with $dim g_1-dim g_0=1$ g0:=StructuralCopy(Car); g1:=[]; m:=prodscal(lindep,s); for i in [1..Length(P)] do p:=P[i]; wp:=(prodscal(s{[2..n+1]},p) mod m); if (wp = 1) then Add(g1,Pv[i]); fi; if (wp = m-1) then Add(g1,Pvm[i]); fi; if (wp = 0) then Add(g0,Pv[i]); Add(g0,Pvm[i]); fi; od; Add(bases,[g0,g1,s,m]); od; return bases; end;; Matad:=function(x,baseg) #x is an element of a Lie algebra g, baseg is a precomputed basis of g. #the function returns the matrix of the endomorphism ad_x of g. local y,vy,l; l:=[]; for y in baseg do vy:=x*y; Add(l,Coefficients(baseg,vy)); od; return l; end;; kern:=function(g,baseg,M) #g is a Lie algebra, baseg is a precomputed basis of g, M is the matrix (in baseg) of an endomorphism of g. #The function computes the kernel of the endomorphism, as a subspace of g. local N,K,i; N:=NullspaceMat(M); #N is a list of list. each sublist consists of the coordinates of an element of the basis of the (right) kernel. K:=[]; for i in [1..Length(N)] do Add(K,LinearCombination(baseg,N[i])); od; return(Subspace(g,K)); end;; Info_perso:=NewInfoClass("Info_perso"); SetInfoLevel(Info_perso,2); Stable_rk1_Inner:=function(g) #g is a simple Lie algebra. #for each labelling of the extended Dynkin diagram of g, the function returns the pairs (g0,g1) each followed by additional information where #g0 and g1 are bases of the homogenous spaces g_0 and g_1 given by the (inner) grading corresponding to the labelling #If the first additional information is a 1 then the pair correspond to a locally free stable theta-rep of rank 1 #If the first additional information is a 0 then no conclusion is made on the pair; local bases, g0,g1,Vg1,x,M,Vp,Vt,i,j,Mj,baseg,Vpg1,test; bases:=basesg0g1(g); baseg:=Basis(g); for i in [1..Length(bases)] do Info(Info_perso, 1, "#Testing labelling ",bases[i][3]," ", i,"/",Length(bases)); test:=0; while test=0 do; g0:=bases[i][1]; #g0 and g1 contains basis of g_0 and g_1 such that dim g_1-dim g_0=1. In particular, the rank of the action is at least 1 and a generic element in g1 has non-zero semsimple part. g1:=bases[i][2]; Vg1:=Subspace(g,g1); #Vg1 is the subspace of g generated by g1 x:=Random(Vg1); #x is a (supposedly) generic element of g1 M:=Matad(x,baseg); Vp:=kern(g,baseg,M); #Vp is the kernel of ad_x (i.e., the centralizer of x in g) j:=1; Mj:=M*M; Vt:=kern(g,baseg,Mj); #Vt is the kernel of (ad_x)^2 while (Vp<>Vt) do j:=j+1; Vp:=Vt; Mj:=Mj*Mj; Vt:=kern(g,baseg,Mj); od; #in the end of this loop, Vp is the kernel of (ad_x)^n for assymptotically large n. In other words, Vp is the centralizer of the semisimple part s of x. #x is semisimple if and only if j=1 if Vp<>g then test:=1; fi; #this is the case when the semisimple part of x is 0. In these cases, x is not as generic as wished. od; Vpg1:=Intersection(Vp,Vg1); #stores the centralizer of s in g_1. if (j=1) and (Dimension(Vpg1)=1) then Add(bases[i],"lf stable of rk 1"); #This happens when x is semisimple and its centralizer is 1-dimensional. #This situation corresponds to locally free stable theta-representations of rank 1, see Lemma 4.5 of the paper #If x is not semisimple (j<>1), x has a nilpotent part so the action cannot be stable of rank 1 as soon as the semisimple part of x is non-zero. #If x is semisimple (j=1) and Dimension(Vpg1)>1, either the rank is greater than 1 or there exists an element x'\neq x in g_1 with semisimple part x and non-zero nilpotent part. else Add(bases[i],"NOT lfs rk1"); fi; od; return bases; end;; ListofOuterThetatoM:=function(a,b,m,d) #a is a type "A", "D" or "E", b is a rank, m and d are integers #the function returns a list of the outer finite order automorphism of a simple Lie algebra of type and rank a and b, with order at most m, inducing a diagram automotphism of order d (2 or 3) local l,i; l:=[]; for i in [1..m] do l:=Concatenation(l,FiniteOrderOuterAutomorphisms(a,b,i,d)); od; return l; end;; gooddimOut:=function(l) #l is a list of (possibly outer) finite order automorphisms of a Lie algebra g #the function returns those which induce a grading with dim g_1- dim g_0=1. #Due to the slow function "Grading", ListOfTheta is much more efficient for inner grading. local f,res,Gradf; res:=[]; for f in l do Gradf:=Grading(f); if (Length(Gradf[2])=Length(Gradf[1])+1) then Add(res,f); fi; od; return res; end;; Stable_rk1_Outer:=function(l) #works globally as the function "Stable_rk1_Inner" except that l is a list of finite order (possibly outer) automorphisms instead of just a Lie algebra g #l will usually be taken as the output of gooddimOut(ListofOuterThetaoM(...)) local bases,Gradf, g,g0,g1,Vg1,x,M,Vp,Vt,i,j,Mj,baseg,Vpg1,res,test; res:=[]; for i in [1..Length(l)] do g:=Source(l[i]); Gradf:=Grading(l[i]); baseg:=Basis(g); g0:=Gradf[1]; g1:=Gradf[2]; Vg1:=Subspace(g,g1); test:=0; while test=0 do; x:=Random(Vg1); M:=Matad(x,baseg); Vp:=kern(g,baseg,M); j:=1; Mj:=M*M; Vt:=kern(g,baseg,Mj); while (Vp<>Vt) do j:=j+1; Vp:=Vt; Mj:=Mj*Mj; Vt:=kern(g,baseg,Mj); od; if Vp<>g then test:=1; fi; od; Vpg1:=Intersection(Vp,Vg1); if (j=1) and (Dimension(Vpg1)=1) then Add(res,[l[i],KacDiagram(l[i]).weights,"lf stable of rk 1"]); else Add(res,[l[i],KacDiagram(l[i]).weights,"NOT lfs rk1"]); fi; od; return res; end;; Has_reg:=function(n,f) #f is a finite order automorphism of a Lie algebra g, n is a normal sl_2-triple (f,h,e) for the corresponding grading (e\in g_1, h\in g_0, f\in g_{-1}) . #the function returns a triple (e,p,q) where p (resp. q) is the dimension of the centralizer of (e,y) in g_0 (resp. in g) for some supposedly generic element y in the centraliser of e in g_{-1}. local g,basegi,m,g0,g1,gm1,gn,gnm1,y,gny,g0ny; g:=Source(f); basegi:=Grading(f); m:=Length(basegi); g0:=Subspace(g,basegi[1]); g1:=Subspace(g,basegi[2]); gm1:=Subspace(g,basegi[m]); gn:=LieCentralizer(g,Subspace(g,[n[3]])); gnm1:=Intersection(gn,gm1); y:=Random(gnm1); gny:=LieCentralizer(g,Subspace(g,[y,n[3]])); g0ny:=Intersection(gny,g0); return([n[3],Dimension(g0ny),Dimension(gny)]); end;; Has_regs:=function(f) #f is a finite order automorphism of a simple Lie algebra g #the function returns a list of triples (e,p,q) as above where e ranges over a list of representatives of G_0-orbits in g_1. local N,v,n; N:=NilpotentOrbitsOfThetaRepresentation(f); v:=[]; for n in N do Add(v,Has_reg(n,f)); od; return v; end;; Has_regs_check:=function(f) #f is a finite order automorphism of a simple Lie algebra g #the function returns the maximum dimension of the centralizer of (e,y) in g_0 for e ranging in a list of representatives of the G_0-orbits in g_1 #and y being a supposedly generic element in the centralizer of e in g_{-1}. #if the function returns 0, then the theta-rep satisfies property p #if it does not returns 0, no conclusion can be made since y might not be as generic as wished. local centmax,Hr,x,num; Hr:=Has_regs(f); centmax:=0; for x in Hr do centmax:=Maximum(centmax,x[2]); od; return centmax; end;; Non_regs:=function(f) #f is a finite order automorphism of a simple Lie algebra g #the function returns a list of triples (e,p,q) as in Has_regs but selects only the cases where p is non-zero. local N,v,n,q; N:=NilpotentOrbitsOfThetaRepresentation(f); v:=[]; for n in N do q:=Has_reg(n,f); if (q[2]<>0) then Add(v,q); fi; od; return v; end;; Really_Non_reg:=function(l,f) #f is a finite order automorphism of a simple Lie algebra g, l should be the list given by the function "Non_regs" #for each nilpotent representative e appearing in l, the centralizer gxm1 of e in g_{-1} is computed. Then the simultaneous centralizer g0xygen in g_0 of x and gxm1 is computed #The function returns the list of (e,m,p) where (e,p,q) appears in l and m is the dimension of g0xygen. #If there exists e with a non-zero m then the corresponding theta-rep fails to satify property (P), see proof of Lemma 4.11 #(and an equality between m and p indicates that the supposedly generic centralizer computed in Has_reg was indeed generic) local Gradf, g, g0,g1,gm1,v,x,gx,gxm1, gygen, gxygen, g0xygen,m; Gradf:=Grading(f); g:=Source(f); g0:=Subspace(g, Gradf[1]); g1:=Subspace(g, Gradf[2]); m:=Length(Gradf); gm1:=Subspace(g, Gradf[m]); v:=[]; for x in l do gx:=LieCentralizer(g, Subspace(g,[x[1]])); gxm1:=Intersection(gx,gm1); gygen:=LieCentralizer(g,gxm1); gxygen:=Intersection(gx,gygen); g0xygen:=Intersection(g0,gxygen); Add(v,[g0xygen,Dimension(g0xygen),x[2]]); od; return v; end;; Inner_Automorphism_By_Kac_diagram:=function(s,type,rank) #An automorphism of a Lie algebra of type "type" and of rank "rank" over some cyclotomic field is constructed. #This automorphism has a Kac diagram given by the labelling s (which is assumed to have length of rank+1). local R,n,Pr,d,hir,m,P,Pn,B,w,ImB,f,cc,lindep,g; R:=RootSystem(type,rank); n:=Length(SimpleSystem(R)); Pr:=PositiveRootsNF(R); d:=Length(Pr); hir:=ShallowCopy(Pr[d]); lindep:=[1]; Append(lindep,hir); m:=prodscal(lindep,s); #m is the order g:=SimpleLieAlgebra(type,rank,CF(m)); #The Lie algebra g is constructed over a cyclotomic field of order m w:=E(m); R:=RootSystem(g); P:=PositiveRootVectors(R); Pn:=NegativeRootVectors(R); B:=[Pn[d]]; Append(B,P{[1..n]}); #The system of simple roots in the affine Dynkin diagram starts with the extended root. ImB:=List( [1..n+1], i -> w^s[i]*B[i] ); f:=AlgebraHomomorphismByImagesNC(g,g,B,ImB); #Construction of the automorphism SetOrder(f,m); cc:= ExtendedCartanMatrix( R ); SetKacDiagram( f, rec( CM:= cc.ECM, labels:= cc.labels, weights:= s)); #additional informations attached to f so that the SLA-routines can work. return f; end;; ListOfpossLocfreec:=function(R,c) #R is a root system and c is an integer #The function returns the labellings giving rise to a grading for which dim g_1-dim g_0=c. The order of the grading is attached with the labelling. #it works similarly as ListOfTheta local S,P,n,l,s,i,g0,g1,k,m,wp,p,hir,res,lindep; S:=SimpleSystem(R); P:=PositiveRootsNF(R); hir:=ShallowCopy(P[Length(P)]); lindep:=[1]; Append(lindep,hir); n:=Length(S); l:=deuxn(n); res:=[]; for s in l do g1:=0; g0:=n; m:=prodscal(lindep,s); for p in P do wp:=(prodscal(s{[2..(n+1)]},p) mod m); if wp=1 then g1:=g1+1; fi; if wp=m-1 then g1:=g1+1; fi; if wp=0 then g0:=g0+2; fi; od; if (g1-g0=c) then Add(res,[s,m]); fi; od; return(res); end;; #-------------------------------------------- # B. Computations #-------------------------------------------- Q:=Rationals ###### TYPE G2 ####### g:=SimpleLieAlgebra("G",2,Q); Stable_rk1_Inner(g); #[ [ [ v.13, v.14, v.1, v.7 ], [ v.2, v.3, v.4, v.5, v.12 ], [ 1, 0, 1 ], 3, # "lf stable of rk 1" ], # [ [ v.13, v.14 ], [ v.1, v.2, v.12 ], [ 1, 1, 1 ], 6, "lf stable of rk 1" ] # ] #In other words, there are two labelling of the extended Dynkin diagram ([1,0,1] and [1,1,1], see the article, Table 2 for the numbering of simple roots) such that the corresponding grading satisfies dim g_1-dimg_0=1. #Both correspond to locally free stable theta-representation of rank 1. #The order of the first grading is 3, with dim g_0=4 and dim g_1=5. #The last labelling [1,1,1], correspond to G_2^{(1)}(\underline{1}) in the notation of Lemma 4.9 in the paper. We don't need to make any computation in such case. Moreover some routine in SLA do not work when the semisimple part of g_0 is trivial. f101:=Inner_Automorphism_By_Kac_diagram([1,0,1],"G",2); #[ v.12, v.1, v.2 ] -> [ (E(3))*v.12, v.1, (E(3))*v.2 ] Grading(f101); #[ [ v.1, v.7, v.13, v.14 ], [ v.2, v.3, v.4, v.5, v.12 ], # [ v.6, v.8, v.9, v.10, v.11 ] ] #We recover the corresponding automorphism (and check that the grading on g is the same) Has_regs(f101); #[ [ v.12, 0, 4 ], [ v.5+v.12, 0, 3 ], [ v.5, 0, 2 ], [ v.4+v.12, 0, 2 ], [ v.4, 0, 3 ], [ v.3+v.5, 0, 4 ] ] #In other words, there are 6 non-zero G_0 orbits in g_1. #For each, one computation of a centralizer in g_0 of a pair (e,y) with e in the orbit and y in the centralizer in g_{-1} of e, turned out to be zero #In particular the corresponding theta-representation satisies property (P) #Also, we see that the centralizer in g ranges from 2 (the rank in the minimal possible centralizer of a commuting pair in g) to 4 #Another way to check it: Has_regs_check(f101); #0 ###### TYPE F4 ####### g:=SimpleLieAlgebra("F",4,Q); Stable_rk1_Inner(g); #[ [ [ v.49, v.50, v.51, v.52, v.3, v.27 ], [ v.1, v.2, v.4, v.5, v.7, v.10, v.48 ], [ 1, 1, 1, 0, 1 ], 8, "lf stable of rk 1" ], # [ [ v.49, v.50, v.51, v.52 ], [ v.1, v.2, v.3, v.4, v.48 ], [ 1, 1, 1, 1, 1 ], 12, "lf stable of rk 1" ] ] #Again 2 grading giving rise to stable locally free representations of rank 1. The first one satisfies dim g_0=6, dim g_1=7, and the order is 8. f11101:=Inner_Automorphism_By_Kac_diagram([1,1,1,0,1],"F",4); Has_regs_check(f11101); #0 #So, the theta-respresentaiton satisfies property (P) again in this case. ###### TYPE E6 INNER ####### g:=SimpleLieAlgebra("E",6,Q); Stable_rk1_Inner(g); #[ [ [ v.73, v.74, v.75, v.76, v.77, v.78, v.4, v.40 ], [ v.1, v.2, v.3, v.5, v.6, v.8, v.9, v.10, v.72 ], [ 1, 1, 1, 1, 0, 1, 1 ], 9, "lf stable of rk 1" ], # [ [ v.73, v.74, v.75, v.76, v.77, v.78 ], [ v.1, v.2, v.3, v.4, v.5, v.6, v.72 ], [ 1, 1, 1, 1, 1, 1, 1 ], 12, "lf stable of rk 1" ] ] #Again 2 grading giving rise to stable locally free representations of rank 1. f1111011:=Inner_Automorphism_By_Kac_diagram([1,1,1,1,0,1,1],"E",6); Has_regs_check(f1111011); #1 #So, the theta-respresentation may not satisfy property (P) in this case. l:=Non_regs(f1111011); #[ [ v.1+v.6+v.8+v.9+v.10, 1, 12 ], [ v.1+v.6+v.8+v.9+v.10+v.72, 1, 16 ], [ v.1+v.8+v.9+v.10, 1, 12 ], [ v.1+v.8+v.9+v.10+v.72, 1, 12 ], [ v.6+v.8+v.9+v.10, 1, 12 ], [ v.6+v.8+v.9+v.10+v.72, 1, 12 ], # [ v.8+v.9+v.10, 1, 16 ], [ v.8+v.9+v.10+v.72, 1, 12 ] ] #In other words there are 8 orbits likely to contradict property (P) l1:=Really_Non_reg(l,f1111011); #[ [ , 1, 1 ], [ , 1, 1 ], [ , 1, 1 ], # [ , 1, 1 ], [ , 1, 1 ], [ , 1, 1 ], # [ , 1, 1 ], [ , 1, 1 ] ] #So it turns that in each of the 8 cases, the 1-dimensional centralizer found was indeed minimal amond pairs (e,y) with e in each selected orbit. [l[7][1],GeneratorsOfLeftModule(l1[7][1])]; #[ v.8+v.9+v.10, [ v.4 ] ] #v.8+v.9+v.10 lies in a nilpotent orbit and the considered 1-dim centralizer is generated by v.4. This is the example given in Table 1 of the paper. ###### TYPE E7 ####### g:=SimpleLieAlgebra("E",7,Q); Stable_rk1_Inner(g); #[ [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.1, v.64, v.2, v.65, v.3, v.66, v.5, v.68, v.8, v.71 ], [ v.4, v.6, v.7, v.9, v.10, v.11, v.12, v.14, v.15, v.16, v.17, v.20, v.21, v.22, v.26, # v.124, v.125, v.126 ], [ 1, 0, 0, 0, 1, 0, 1, 1 ], 8, "NOT lfs rk1" ], [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.1, v.64, v.4, v.67, v.6, v.69 ], # [ v.2, v.3, v.5, v.7, v.8, v.9, v.10, v.11, v.12, v.13, v.14, v.18, v.125, v.126 ], [ 1, 0, 1, 1, 0, 1, 0, 1 ], 10, "NOT lfs rk1" ], # [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.1, v.64, v.4, v.67 ], [ v.2, v.3, v.5, v.6, v.7, v.8, v.9, v.10, v.11, v.14, v.125, v.126 ], [ 1, 0, 1, 1, 0, 1, 1, 1 ], 12, "NOT lfs rk1" ], # [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.2, v.65, v.3, v.66, v.5, v.68, v.6, v.69, v.12, v.75 ], [ v.1, v.4, v.7, v.8, v.9, v.10, v.11, v.13, v.15, v.16, v.17, v.18, v.19, v.22, v.23, # v.24, v.29, v.126 ], [ 1, 1, 0, 0, 1, 0, 0, 1 ], 8, "NOT lfs rk1" ], [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.2, v.65, v.3, v.66, v.5, v.68 ], # [ v.1, v.4, v.6, v.7, v.8, v.9, v.10, v.11, v.12, v.15, v.16, v.17, v.22, v.126 ], [ 1, 1, 0, 0, 1, 0, 1, 1 ], 10, "NOT lfs rk1" ], # [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.4, v.67, v.6, v.69 ], [ v.1, v.2, v.3, v.5, v.7, v.9, v.10, v.11, v.12, v.13, v.18, v.126 ], [ 1, 1, 1, 1, 0, 1, 0, 1 ], 12, "NOT lfs rk1" ], # [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133, v.4, v.67 ], [ v.1, v.2, v.3, v.5, v.6, v.7, v.9, v.10, v.11, v.126 ], [ 1, 1, 1, 1, 0, 1, 1, 1 ], 14, "lf stable of rk 1" ], # [ [ v.127, v.128, v.129, v.130, v.131, v.132, v.133 ], [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.126 ], [ 1, 1, 1, 1, 1, 1, 1, 1 ], 18, "lf stable of rk 1" ] ] #Among the 8 grading satisfying dim g_1-dim g_0=1, only 2 correspond to stable locally free representations of rank 1. [1,1,1,1,0,1,1,1] of order 14 and [1,1,1,1,1,1,1,1]. f11110111:=Inner_Automorphism_By_Kac_diagram([1,1,1,1,0,1,1,1],"E",7); Has_regs_check(f11110111); #1 #So, the theta-respresentation may not satisfy property (P) in this case. l:=Non_regs(f11110111); l1:=Really_Non_reg(l,f11110111); #64 orbits yield a centralizer of a generic pair of dimension 1. This contradicts property (P) in this case. [l[15][1],GeneratorsOfLeftModule(l1[15][1])]; [ v.9+v.10+v.11, [ v.4 ] ] #Hence v.9+v.10+v.11 lies in a nilpotent orbit and the considered 1-dim centralizer is generated by v.4. This is the example given in Table 1 of the paper. ###### TYPE E8 ####### g:=SimpleLieAlgebra("E",8,Q); Stable_rk1_Inner(g); # [ [ [ 1, 0, 0, 0, 1, 0, 0, 0, 1 ], 9, "NOT lfs rk1" ], [ [ 1, 0, 0, 0, 1, 0, 0, 1, 1 ], 12, "NOT lfs rk1" ], [ [ 1, 1, 0, 0, 1, 0, 0, 1, 1 ], 14, "NOT lfs rk1" ], # [ [ 1, 1, 0, 0, 1, 0, 1, 0, 1 ], 15, "lf stable of rk 1" ], [ [ 1, 1, 0, 0, 1, 0, 1, 1, 1 ], 18, "NOT lfs rk1" ], [ [ 1, 1, 1, 1, 0, 1, 0, 1, 0 ], 18, "NOT lfs rk1" ], # [ [ 1, 1, 1, 1, 0, 1, 0, 1, 1 ], 20, "lf stable of rk 1" ], [ [ 1, 1, 1, 1, 0, 1, 1, 1, 1 ], 24, "lf stable of rk 1" ], [ [ 1, 1, 1, 1, 1, 1, 1, 1, 1 ], 30, "lf stable of rk 1" ] ] #Among the 9 grading satisfying dim g_1-dim g_0=1, only 4 correspond to stable locally free representations of rank 1: #[1,1,0,0,1,0,1,0,1] of order 15; [1,1,1,1,0,1,0,1,1] of order 20; [1,1,1,1,0,1,1,1,1] of order 24 and [1,1,1,1,1,1,1,1,1] of order 30. f110010101:=Inner_Automorphism_By_Kac_diagram([1,1,0,0,1,0,1,0,1],"E",8); f111101011:=Inner_Automorphism_By_Kac_diagram([1,1,1,1,0,1,0,1,1],"E",8); f111101111:=Inner_Automorphism_By_Kac_diagram([1,1,1,1,0,1,1,1,1],"E",8); Has_regs_check(f110010101); #2 Has_regs_check(f111101011); #1 Has_regs_check(f111101111); #1 #So, the theta-respresentation may not satisfy property (P) in this case. l:=Non_regs(f110010101); l1:=Really_Non_reg(l,f110010101); #84 orbits yield a centralizer of a generic pair of dimension at least 1. This contradicts property (P) in this case. #6 orbits even yield a centralizer of dimension 2. [l[17][1],GeneratorsOfLeftModule(l1[17][1])]; #[ v.15+v.17+v.18+v.21, [ v.2, v.7 ] ] #Hence v.15+v.17+v.18+v.21 lies in a nilpotent orbit and the considered 2-dim centralizer is generated by v.2 and v.7. This is the example given in Table 1 of the paper. l:=Non_regs(f111101011); l1:=Really_Non_reg(l,f111101011); #64 orbits yield a centralizer of a generic pair of dimension 1. This contradicts property (P) in this case. [l[55][1],GeneratorsOfLeftModule(l1[55][1])]; #[ v.14+v.20, [ v.6 ] ] #Hence v.14+v.20 lies in a nilpotent orbit and the considered 1-dim centralizer is generated by v.6. This is the example given in Table 1 of the paper. l:=Non_regs(f111101111); l1:=Really_Non_reg(l,f111101111); #32 orbits yield a centralizer of a generic pair of dimension 1. This contradicts property (P) in this case. [l[31][1],GeneratorsOfLeftModule(l1[31][1])]; #[ v.10+v.11+v.12, [ v.4 ] ] #Hence v.10+v.11+v.12 lies in a nilpotent orbit and the considered 1-dim centralizer is generated by v.4. This is the example given in Table 1 of the paper. ###### TYPE D4 Outer ####### Aut3_D4:=ListofOuterThetatoM("D",4,12,3); #This is the list of finite order outer automorphisms in the case D4 with involution of Dynkin diagram of order 3. #Given labellings of Kac diagrams in {0,1}, we reach a maximum possible order of automorphism of 12. #We consider here ALL outer automorphisms of order at most 12. gooddimD4:=gooddimOut(Aut3_D4); #[ [ v.17+(-E(3)^2)*v.18+(-E(3))*v.19, v.23, v.8+v.9+(-1)*v.10 ] -> [ (-E(12)^7)*v.17+(E(4))*v.18+(E(12)^11)*v.19, (-E(12)^7)*v.23, (-E(12)^7)*v.8+(-E(12)^7)*v.9+(E(12)^7)*v.10 ] ] #So we get only one automorphism giving rise to a grading with dim g_1-dim g_0=1. KacDiagram(gooddimD4[1]).weights; #[ 1, 1, 1 ] #This automorphism arises from the grading with only 1s. Hence we do not have to consider it (see Lemma 4.9 in the paper). ###### TYPE E6 Outer ####### Aut2_E6:=ListofOuterThetatoM("E",6,18,2); #This is the list of finite order outer automorphisms in the case E6 with involution of Dynkin diagram of order 2. #Given labellings of Kac diagrams in {0,1}, we reach a maximum possible order of automorphism of 18. #We consider here ALL outer automorphisms of order at most 18. gooddimE6:=gooddimOut(Aut2_E6); l:=Stable_rk1_Outer(gooddimE6); #[ [ [ v.49+v.50, v.40, v.70, v.37+v.42, v.32+(-1)*v.33 ] -> [ (-E(5)^3)*v.49+(-E(5)^3)*v.50, v.40, (-E(5)^3)*v.70, (-E(5)^3)*v.37+(-E(5)^3)*v.42, v.32+(-1)*v.33 ], [ 1, 0, 1, 1, 0 ], "NOT lfs rk1" ], # [ [ v.49+v.50, v.40, v.70, v.37+v.42, v.32+(-1)*v.33 ] -> [ (-E(12)^7)*v.49+(-E(12)^7)*v.50, (-E(12)^7)*v.40, (-E(12)^7)*v.70, (-E(12)^7)*v.37+(-E(12)^7)*v.42, v.32+(-1)*v.33 ], [ 1, 1, 1, 1, 0 ], # "lf stable of rk 1" ], [ [ v.49+v.50, v.40, v.70, v.37+v.42, v.32+(-1)*v.33 ] -> [ (-E(9)^5)*v.49+(-E(9)^5)*v.50, (-E(9)^5)*v.40, (-E(9)^5)*v.70, (-E(9)^5)*v.37+(-E(9)^5)*v.42, # (-E(9)^5)*v.32+(E(9)^5)*v.33 ], [ 1, 1, 1, 1, 1 ], "lf stable of rk 1" ] ] #So we have a total of 3 automorphisms giving rise to a grading with dim g_1-dim g_0=1. #The last 2 correspond to stable locally free theta-representations of rank 1 f11110:=l[2][1] #We consider the one with Kac diagram labelled by [1,1,1,1,0] Has_regs_check(f11110); #0 #So, the theta-respresentaiton satisfies property (P) in this case. ###### Levi subalgebras ####### R:=RootSystem("E",7); ListOfpossLocfreec(R,2); #[ [ [ 0, 0, 0, 0, 0, 1, 0, 0 ], 3 ], [ [ 0, 0, 0, 1, 0, 0, 0, 0 ], 3 ], [ [ 0, 0, 0, 1, 0, 0, 0, 1 ], 4 ], [ [ 1, 0, 0, 0, 0, 1, 0, 0 ], 4 ] ] #Any locally free stable theta-representation of rank 2 of type E7 must appear in the above list #But the order of the gradings are 3 or 4, hence not a multiple of 9 #As a consequence, in the above cases, no Levi arising from g_1 of rank 1 can coincide with the E6 type with Kac diagram [1,1,1,1,0,1,1] R:=RootSystem("E",8); ListOfpossLocfreec(R,2); #[ [ [ 0, 0, 0, 0, 0, 1, 0, 0, 0 ], 5 ], [ [ 0, 0, 0, 0, 1, 0, 0, 0, 1 ], 8 ], [ [ 1, 0, 0, 0, 1, 0, 0, 1, 0 ], 10 ], [ [ 1, 1, 0, 0, 1, 0, 0, 1, 0 ], 12 ] ] #Any locally free stable theta-representation of rank 2 of type E8 must appear in the above list #But the order of the gradings are not multiple of 9 or 14. #As a consequence, in the above cases, no Levi arising from g_1 of rank 1 can coincide with the E6 type with Kac diagram [1,1,1,1,0,1,1] or the E7 type [1,1,1,1,0,1,1,1] ListOfpossLocfreec(R,3); #[ ] #There are no locally free stable theta-rep of rank 3 of type E8.