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Identifier
Values
=>
Cc0005;cc-rep
[1,0]=>2 [1,0,1,0]=>3 [1,1,0,0]=>3 [1,0,1,0,1,0]=>4 [1,0,1,1,0,0]=>4 [1,1,0,0,1,0]=>4 [1,1,0,1,0,0]=>4 [1,1,1,0,0,0]=>4 [1,0,1,0,1,0,1,0]=>5 [1,0,1,0,1,1,0,0]=>5 [1,0,1,1,0,0,1,0]=>5 [1,0,1,1,0,1,0,0]=>5 [1,0,1,1,1,0,0,0]=>5 [1,1,0,0,1,0,1,0]=>5 [1,1,0,0,1,1,0,0]=>5 [1,1,0,1,0,0,1,0]=>5 [1,1,0,1,0,1,0,0]=>6 [1,1,0,1,1,0,0,0]=>5 [1,1,1,0,0,0,1,0]=>5 [1,1,1,0,0,1,0,0]=>5 [1,1,1,0,1,0,0,0]=>5 [1,1,1,1,0,0,0,0]=>5 [1,0,1,0,1,0,1,0,1,0]=>6 [1,0,1,0,1,0,1,1,0,0]=>6 [1,0,1,0,1,1,0,0,1,0]=>6 [1,0,1,0,1,1,0,1,0,0]=>6 [1,0,1,0,1,1,1,0,0,0]=>6 [1,0,1,1,0,0,1,0,1,0]=>6 [1,0,1,1,0,0,1,1,0,0]=>6 [1,0,1,1,0,1,0,0,1,0]=>6 [1,0,1,1,0,1,0,1,0,0]=>4 [1,0,1,1,0,1,1,0,0,0]=>6 [1,0,1,1,1,0,0,0,1,0]=>6 [1,0,1,1,1,0,0,1,0,0]=>6 [1,0,1,1,1,0,1,0,0,0]=>6 [1,0,1,1,1,1,0,0,0,0]=>6 [1,1,0,0,1,0,1,0,1,0]=>6 [1,1,0,0,1,0,1,1,0,0]=>6 [1,1,0,0,1,1,0,0,1,0]=>6 [1,1,0,0,1,1,0,1,0,0]=>6 [1,1,0,0,1,1,1,0,0,0]=>6 [1,1,0,1,0,0,1,0,1,0]=>6 [1,1,0,1,0,0,1,1,0,0]=>6 [1,1,0,1,0,1,0,0,1,0]=>4 [1,1,0,1,0,1,0,1,0,0]=>3 [1,1,0,1,0,1,1,0,0,0]=>4 [1,1,0,1,1,0,0,0,1,0]=>6 [1,1,0,1,1,0,0,1,0,0]=>6 [1,1,0,1,1,0,1,0,0,0]=>3 [1,1,0,1,1,1,0,0,0,0]=>6 [1,1,1,0,0,0,1,0,1,0]=>6 [1,1,1,0,0,0,1,1,0,0]=>6 [1,1,1,0,0,1,0,0,1,0]=>6 [1,1,1,0,0,1,0,1,0,0]=>4 [1,1,1,0,0,1,1,0,0,0]=>6 [1,1,1,0,1,0,0,0,1,0]=>6 [1,1,1,0,1,0,0,1,0,0]=>3 [1,1,1,0,1,0,1,0,0,0]=>4 [1,1,1,0,1,1,0,0,0,0]=>6 [1,1,1,1,0,0,0,0,1,0]=>6 [1,1,1,1,0,0,0,1,0,0]=>6 [1,1,1,1,0,0,1,0,0,0]=>6 [1,1,1,1,0,1,0,0,0,0]=>6 [1,1,1,1,1,0,0,0,0,0]=>6 [1,0,1,0,1,0,1,0,1,0,1,0]=>7 [1,0,1,0,1,0,1,0,1,1,0,0]=>7 [1,0,1,0,1,0,1,1,0,0,1,0]=>7 [1,0,1,0,1,0,1,1,0,1,0,0]=>7 [1,0,1,0,1,0,1,1,1,0,0,0]=>7 [1,0,1,0,1,1,0,0,1,0,1,0]=>7 [1,0,1,0,1,1,0,0,1,1,0,0]=>7 [1,0,1,0,1,1,0,1,0,0,1,0]=>7 [1,0,1,0,1,1,0,1,0,1,0,0]=>10 [1,0,1,0,1,1,0,1,1,0,0,0]=>7 [1,0,1,0,1,1,1,0,0,0,1,0]=>7 [1,0,1,0,1,1,1,0,0,1,0,0]=>7 [1,0,1,0,1,1,1,0,1,0,0,0]=>7 [1,0,1,0,1,1,1,1,0,0,0,0]=>7 [1,0,1,1,0,0,1,0,1,0,1,0]=>7 [1,0,1,1,0,0,1,0,1,1,0,0]=>7 [1,0,1,1,0,0,1,1,0,0,1,0]=>7 [1,0,1,1,0,0,1,1,0,1,0,0]=>7 [1,0,1,1,0,0,1,1,1,0,0,0]=>7 [1,0,1,1,0,1,0,0,1,0,1,0]=>7 [1,0,1,1,0,1,0,0,1,1,0,0]=>7 [1,0,1,1,0,1,0,1,0,0,1,0]=>10 [1,0,1,1,0,1,0,1,0,1,0,0]=>12 [1,0,1,1,0,1,0,1,1,0,0,0]=>10 [1,0,1,1,0,1,1,0,0,0,1,0]=>7 [1,0,1,1,0,1,1,0,0,1,0,0]=>7 [1,0,1,1,0,1,1,0,1,0,0,0]=>12 [1,0,1,1,0,1,1,1,0,0,0,0]=>7 [1,0,1,1,1,0,0,0,1,0,1,0]=>7 [1,0,1,1,1,0,0,0,1,1,0,0]=>7 [1,0,1,1,1,0,0,1,0,0,1,0]=>7 [1,0,1,1,1,0,0,1,0,1,0,0]=>10 [1,0,1,1,1,0,0,1,1,0,0,0]=>7 [1,0,1,1,1,0,1,0,0,0,1,0]=>7 [1,0,1,1,1,0,1,0,0,1,0,0]=>12 [1,0,1,1,1,0,1,0,1,0,0,0]=>10 [1,0,1,1,1,0,1,1,0,0,0,0]=>7 [1,0,1,1,1,1,0,0,0,0,1,0]=>7 [1,0,1,1,1,1,0,0,0,1,0,0]=>7 [1,0,1,1,1,1,0,0,1,0,0,0]=>7 [1,0,1,1,1,1,0,1,0,0,0,0]=>7 [1,0,1,1,1,1,1,0,0,0,0,0]=>7 [1,1,0,0,1,0,1,0,1,0,1,0]=>7 [1,1,0,0,1,0,1,0,1,1,0,0]=>7 [1,1,0,0,1,0,1,1,0,0,1,0]=>7 [1,1,0,0,1,0,1,1,0,1,0,0]=>7 [1,1,0,0,1,0,1,1,1,0,0,0]=>7 [1,1,0,0,1,1,0,0,1,0,1,0]=>7 [1,1,0,0,1,1,0,0,1,1,0,0]=>7 [1,1,0,0,1,1,0,1,0,0,1,0]=>7 [1,1,0,0,1,1,0,1,0,1,0,0]=>10 [1,1,0,0,1,1,0,1,1,0,0,0]=>7 [1,1,0,0,1,1,1,0,0,0,1,0]=>7 [1,1,0,0,1,1,1,0,0,1,0,0]=>7 [1,1,0,0,1,1,1,0,1,0,0,0]=>7 [1,1,0,0,1,1,1,1,0,0,0,0]=>7 [1,1,0,1,0,0,1,0,1,0,1,0]=>7 [1,1,0,1,0,0,1,0,1,1,0,0]=>7 [1,1,0,1,0,0,1,1,0,0,1,0]=>7 [1,1,0,1,0,0,1,1,0,1,0,0]=>7 [1,1,0,1,0,0,1,1,1,0,0,0]=>7 [1,1,0,1,0,1,0,0,1,0,1,0]=>10 [1,1,0,1,0,1,0,0,1,1,0,0]=>10 [1,1,0,1,0,1,0,1,0,0,1,0]=>12 [1,1,0,1,0,1,0,1,0,1,0,0]=>12 [1,1,0,1,0,1,0,1,1,0,0,0]=>12 [1,1,0,1,0,1,1,0,0,0,1,0]=>10 [1,1,0,1,0,1,1,0,0,1,0,0]=>10 [1,1,0,1,0,1,1,0,1,0,0,0]=>12 [1,1,0,1,0,1,1,1,0,0,0,0]=>10 [1,1,0,1,1,0,0,0,1,0,1,0]=>7 [1,1,0,1,1,0,0,0,1,1,0,0]=>7 [1,1,0,1,1,0,0,1,0,0,1,0]=>7 [1,1,0,1,1,0,0,1,0,1,0,0]=>10 [1,1,0,1,1,0,0,1,1,0,0,0]=>7 [1,1,0,1,1,0,1,0,0,0,1,0]=>12 [1,1,0,1,1,0,1,0,0,1,0,0]=>7 [1,1,0,1,1,0,1,0,1,0,0,0]=>12 [1,1,0,1,1,0,1,1,0,0,0,0]=>12 [1,1,0,1,1,1,0,0,0,0,1,0]=>7 [1,1,0,1,1,1,0,0,0,1,0,0]=>7 [1,1,0,1,1,1,0,0,1,0,0,0]=>7 [1,1,0,1,1,1,0,1,0,0,0,0]=>12 [1,1,0,1,1,1,1,0,0,0,0,0]=>7 [1,1,1,0,0,0,1,0,1,0,1,0]=>7 [1,1,1,0,0,0,1,0,1,1,0,0]=>7 [1,1,1,0,0,0,1,1,0,0,1,0]=>7 [1,1,1,0,0,0,1,1,0,1,0,0]=>7 [1,1,1,0,0,0,1,1,1,0,0,0]=>7 [1,1,1,0,0,1,0,0,1,0,1,0]=>7 [1,1,1,0,0,1,0,0,1,1,0,0]=>7 [1,1,1,0,0,1,0,1,0,0,1,0]=>10 [1,1,1,0,0,1,0,1,0,1,0,0]=>12 [1,1,1,0,0,1,0,1,1,0,0,0]=>10 [1,1,1,0,0,1,1,0,0,0,1,0]=>7 [1,1,1,0,0,1,1,0,0,1,0,0]=>7 [1,1,1,0,0,1,1,0,1,0,0,0]=>12 [1,1,1,0,0,1,1,1,0,0,0,0]=>7 [1,1,1,0,1,0,0,0,1,0,1,0]=>7 [1,1,1,0,1,0,0,0,1,1,0,0]=>7 [1,1,1,0,1,0,0,1,0,0,1,0]=>12 [1,1,1,0,1,0,0,1,0,1,0,0]=>12 [1,1,1,0,1,0,0,1,1,0,0,0]=>12 [1,1,1,0,1,0,1,0,0,0,1,0]=>10 [1,1,1,0,1,0,1,0,0,1,0,0]=>12 [1,1,1,0,1,0,1,0,1,0,0,0]=>7 [1,1,1,0,1,0,1,1,0,0,0,0]=>10 [1,1,1,0,1,1,0,0,0,0,1,0]=>7 [1,1,1,0,1,1,0,0,0,1,0,0]=>7 [1,1,1,0,1,1,0,0,1,0,0,0]=>12 [1,1,1,0,1,1,0,1,0,0,0,0]=>12 [1,1,1,0,1,1,1,0,0,0,0,0]=>7 [1,1,1,1,0,0,0,0,1,0,1,0]=>7 [1,1,1,1,0,0,0,0,1,1,0,0]=>7 [1,1,1,1,0,0,0,1,0,0,1,0]=>7 [1,1,1,1,0,0,0,1,0,1,0,0]=>10 [1,1,1,1,0,0,0,1,1,0,0,0]=>7 [1,1,1,1,0,0,1,0,0,0,1,0]=>7 [1,1,1,1,0,0,1,0,0,1,0,0]=>12 [1,1,1,1,0,0,1,0,1,0,0,0]=>10 [1,1,1,1,0,0,1,1,0,0,0,0]=>7 [1,1,1,1,0,1,0,0,0,0,1,0]=>7 [1,1,1,1,0,1,0,0,0,1,0,0]=>12 [1,1,1,1,0,1,0,0,1,0,0,0]=>12 [1,1,1,1,0,1,0,1,0,0,0,0]=>10 [1,1,1,1,0,1,1,0,0,0,0,0]=>7 [1,1,1,1,1,0,0,0,0,0,1,0]=>7 [1,1,1,1,1,0,0,0,0,1,0,0]=>7 [1,1,1,1,1,0,0,0,1,0,0,0]=>7 [1,1,1,1,1,0,0,1,0,0,0,0]=>7 [1,1,1,1,1,0,1,0,0,0,0,0]=>7 [1,1,1,1,1,1,0,0,0,0,0,0]=>7
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Description
The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path.
References
[1] Ringel, C. M. The finitistic dimension of a Nakayama algebra arXiv:2008.10044
Code

DeclareOperation("coordinateofsimple", [IsList]);

InstallMethod(coordinateofsimple, "for a representation of a quiver", [IsList],0,function(L)

local A,S,simA,n,U,UU;

A:=L[1];
S:=L[2];
simA:=SimpleModules(A);
n:=Size(simA);
U:=DimensionVector(S);
UU:=Filtered([1..n],x->U[x]>0);
return(Minimum(UU));
end
);


DeclareOperation("eofsimplemodule", [IsList]);

InstallMethod(eofsimplemodule, "for a representation of a quiver", [IsList],0,function(L)


local S,t1,U,t2;
S:=L[1];
t1:=ProjDimensionOfModule(S,33);
U:=Range(InjectiveEnvelope(S));
t2:=ProjDimensionOfModule(U,33);
return(Minimum(t1,t2));
end
);


DeclareOperation("Ringelbijectionforgivensimple", [IsList]);

InstallMethod(Ringelbijectionforgivensimple, "for a representation of a quiver", [IsList],0,function(L)


local S,t1,U,t2,e;
S:=L[1];
e:=eofsimplemodule([S]);
U:=Range(InjectiveEnvelope(S));
if IsOddInt(e)=true then return(NthSyzygy(S,e));
else return(NthSyzygy(U,e));fi;
end
);





DeclareOperation("Ringelbijection",[IsList]);

InstallMethod(Ringelbijection, "for a representation of a quiver", [IsList],0,function(LIST)

local A,simA,n,W,i,WW,WW2;

A:=LIST[1];
simA:=SimpleModules(A);
n:=Size(simA);
W:=[];for i in simA do Append(W,[Ringelbijectionforgivensimple([i])]);od;
WW:=[];for i in [1..n] do Append(WW,[[i,coordinateofsimple([A,W[i]])]]);od;
WW2:=[];for i in [1..n] do Append(WW2,[WW[i][2]]);od;
return([WW,Size(Set(WW2))=n]);
#first entry is map, second entry is whether it is a bijection.

end);



DeclareOperation("Ringelpermutationnormalform",[IsList]);

InstallMethod(Ringelpermutationnormalform, "for a representation of a quiver", [IsList],0,function(LIST)

local n,l,i,W,k,WW,WW2,O,f;

A:=LIST[1];
f:=Ringelbijection([A])[1];
W:=[];for i in f do Append(W,[i[2]]);od;
return(W);
end);


DeclareOperation("ringelmapaspermutation",[IsList]);

InstallMethod(ringelmapaspermutation, "for a representation of a quiver", [IsList],0,function(LIST)

local A,W,n,U,D;

A:=LIST[1];
W:=Ringelpermutationnormalform([A]);n:=Size(W);U:=PermList(W);
return(U);
end);

#example: A:=NakayamaAlgebra([3,3,2,1],GF(3));W:=Ringelpermutationnormalform([A]);

#z:=5;R:=[];for i in [2..z] do Append(R,[BuildSequencesLNak(i)]);od;R:=Union(R);RR:=[];for i in R do Append(RR,[[i,Ringelpermutationnormalform([NakayamaAlgebra(i,GF(3))])]]);od;RR;


DeclareOperation("OrderRingelpermutation",[IsList]);

InstallMethod(OrderRingelpermutation, "for a representation of a quiver", [IsList],0,function(LIST)

local n,l,i,W,k,WW,WW2,O,f;

A:=LIST[1];
f:=ringelmapaspermutation([A]);
return(Order(f));
end);

#example: A:=NakayamaAlgebra([3,3,2,1],GF(3));W:=OrderRingelpermutation([A]);
Created
Nov 29, 2020 at 19:57 by Rene Marczinzik
Updated
Nov 29, 2020 at 19:57 by Rene Marczinzik