Identifier
Identifier
Values
[1,0] generating graphics... => 0
[1,0,1,0] generating graphics... => 1
[1,1,0,0] generating graphics... => 0
[1,0,1,0,1,0] generating graphics... => 2
[1,0,1,1,0,0] generating graphics... => 1
[1,1,0,0,1,0] generating graphics... => 1
[1,1,0,1,0,0] generating graphics... => 2
[1,1,1,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0] generating graphics... => 3
[1,0,1,0,1,1,0,0] generating graphics... => 2
[1,0,1,1,0,0,1,0] generating graphics... => 2
[1,0,1,1,0,1,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0] generating graphics... => 2
[1,1,0,0,1,1,0,0] generating graphics... => 1
[1,1,0,1,0,0,1,0] generating graphics... => 3
[1,1,0,1,0,1,0,0] generating graphics... => 4
[1,1,0,1,1,0,0,0] generating graphics... => 2
[1,1,1,0,0,0,1,0] generating graphics... => 1
[1,1,1,0,0,1,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0] generating graphics... => 3
[1,1,1,1,0,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0,1,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,0,0] generating graphics... => 3
[1,0,1,0,1,1,0,0,1,0] generating graphics... => 3
[1,0,1,0,1,1,0,1,0,0] generating graphics... => 4
[1,0,1,0,1,1,1,0,0,0] generating graphics... => 2
[1,0,1,1,0,0,1,0,1,0] generating graphics... => 3
[1,0,1,1,0,0,1,1,0,0] generating graphics... => 2
[1,0,1,1,0,1,0,0,1,0] generating graphics... => 4
[1,0,1,1,0,1,0,1,0,0] generating graphics... => 5
[1,0,1,1,0,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,0,1,0] generating graphics... => 2
[1,0,1,1,1,0,0,1,0,0] generating graphics... => 3
[1,0,1,1,1,0,1,0,0,0] generating graphics... => 4
[1,0,1,1,1,1,0,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0,1,0] generating graphics... => 3
[1,1,0,0,1,0,1,1,0,0] generating graphics... => 2
[1,1,0,0,1,1,0,0,1,0] generating graphics... => 2
[1,1,0,0,1,1,0,1,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,0,0] generating graphics... => 1
[1,1,0,1,0,0,1,0,1,0] generating graphics... => 4
[1,1,0,1,0,0,1,1,0,0] generating graphics... => 3
[1,1,0,1,0,1,0,0,1,0] generating graphics... => 5
[1,1,0,1,0,1,0,1,0,0] generating graphics... => 6
[1,1,0,1,0,1,1,0,0,0] generating graphics... => 4
[1,1,0,1,1,0,0,0,1,0] generating graphics... => 3
[1,1,0,1,1,0,0,1,0,0] generating graphics... => 4
[1,1,0,1,1,0,1,0,0,0] generating graphics... => 5
[1,1,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,0,0,1,0,1,0] generating graphics... => 2
[1,1,1,0,0,0,1,1,0,0] generating graphics... => 1
[1,1,1,0,0,1,0,0,1,0] generating graphics... => 3
[1,1,1,0,0,1,0,1,0,0] generating graphics... => 4
[1,1,1,0,0,1,1,0,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0,1,0] generating graphics... => 4
[1,1,1,0,1,0,0,1,0,0] generating graphics... => 5
[1,1,1,0,1,0,1,0,0,0] generating graphics... => 6
[1,1,1,0,1,1,0,0,0,0] generating graphics... => 3
[1,1,1,1,0,0,0,0,1,0] generating graphics... => 1
[1,1,1,1,0,0,0,1,0,0] generating graphics... => 2
[1,1,1,1,0,0,1,0,0,0] generating graphics... => 3
[1,1,1,1,0,1,0,0,0,0] generating graphics... => 4
[1,1,1,1,1,0,0,0,0,0] generating graphics... => 0
[1,0,1,0,1,0,1,0,1,0,1,0] generating graphics... => 5
[1,0,1,0,1,0,1,0,1,1,0,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,0,0,1,0] generating graphics... => 4
[1,0,1,0,1,0,1,1,0,1,0,0] generating graphics... => 5
[1,0,1,0,1,0,1,1,1,0,0,0] generating graphics... => 3
[1,0,1,0,1,1,0,0,1,0,1,0] generating graphics... => 4
[1,0,1,0,1,1,0,0,1,1,0,0] generating graphics... => 3
[1,0,1,0,1,1,0,1,0,0,1,0] generating graphics... => 5
[1,0,1,0,1,1,0,1,0,1,0,0] generating graphics... => 6
[1,0,1,0,1,1,0,1,1,0,0,0] generating graphics... => 4
[1,0,1,0,1,1,1,0,0,0,1,0] generating graphics... => 3
[1,0,1,0,1,1,1,0,0,1,0,0] generating graphics... => 4
[1,0,1,0,1,1,1,0,1,0,0,0] generating graphics... => 5
[1,0,1,0,1,1,1,1,0,0,0,0] generating graphics... => 2
[1,0,1,1,0,0,1,0,1,0,1,0] generating graphics... => 4
[1,0,1,1,0,0,1,0,1,1,0,0] generating graphics... => 3
[1,0,1,1,0,0,1,1,0,0,1,0] generating graphics... => 3
[1,0,1,1,0,0,1,1,0,1,0,0] generating graphics... => 4
[1,0,1,1,0,0,1,1,1,0,0,0] generating graphics... => 2
[1,0,1,1,0,1,0,0,1,0,1,0] generating graphics... => 5
[1,0,1,1,0,1,0,0,1,1,0,0] generating graphics... => 4
[1,0,1,1,0,1,0,1,0,0,1,0] generating graphics... => 6
[1,0,1,1,0,1,0,1,0,1,0,0] generating graphics... => 7
[1,0,1,1,0,1,0,1,1,0,0,0] generating graphics... => 5
[1,0,1,1,0,1,1,0,0,0,1,0] generating graphics... => 4
[1,0,1,1,0,1,1,0,0,1,0,0] generating graphics... => 5
[1,0,1,1,0,1,1,0,1,0,0,0] generating graphics... => 6
[1,0,1,1,0,1,1,1,0,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,0,0,1,0,1,0] generating graphics... => 3
[1,0,1,1,1,0,0,0,1,1,0,0] generating graphics... => 2
[1,0,1,1,1,0,0,1,0,0,1,0] generating graphics... => 4
[1,0,1,1,1,0,0,1,0,1,0,0] generating graphics... => 5
[1,0,1,1,1,0,0,1,1,0,0,0] generating graphics... => 3
[1,0,1,1,1,0,1,0,0,0,1,0] generating graphics... => 5
[1,0,1,1,1,0,1,0,0,1,0,0] generating graphics... => 6
[1,0,1,1,1,0,1,0,1,0,0,0] generating graphics... => 7
[1,0,1,1,1,0,1,1,0,0,0,0] generating graphics... => 4
[1,0,1,1,1,1,0,0,0,0,1,0] generating graphics... => 2
[1,0,1,1,1,1,0,0,0,1,0,0] generating graphics... => 3
[1,0,1,1,1,1,0,0,1,0,0,0] generating graphics... => 4
[1,0,1,1,1,1,0,1,0,0,0,0] generating graphics... => 5
[1,0,1,1,1,1,1,0,0,0,0,0] generating graphics... => 1
[1,1,0,0,1,0,1,0,1,0,1,0] generating graphics... => 4
[1,1,0,0,1,0,1,0,1,1,0,0] generating graphics... => 3
[1,1,0,0,1,0,1,1,0,0,1,0] generating graphics... => 3
[1,1,0,0,1,0,1,1,0,1,0,0] generating graphics... => 4
[1,1,0,0,1,0,1,1,1,0,0,0] generating graphics... => 2
[1,1,0,0,1,1,0,0,1,0,1,0] generating graphics... => 3
[1,1,0,0,1,1,0,0,1,1,0,0] generating graphics... => 2
[1,1,0,0,1,1,0,1,0,0,1,0] generating graphics... => 4
[1,1,0,0,1,1,0,1,0,1,0,0] generating graphics... => 5
[1,1,0,0,1,1,0,1,1,0,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,0,0,1,0] generating graphics... => 2
[1,1,0,0,1,1,1,0,0,1,0,0] generating graphics... => 3
[1,1,0,0,1,1,1,0,1,0,0,0] generating graphics... => 4
[1,1,0,0,1,1,1,1,0,0,0,0] generating graphics... => 1
[1,1,0,1,0,0,1,0,1,0,1,0] generating graphics... => 5
[1,1,0,1,0,0,1,0,1,1,0,0] generating graphics... => 4
[1,1,0,1,0,0,1,1,0,0,1,0] generating graphics... => 4
[1,1,0,1,0,0,1,1,0,1,0,0] generating graphics... => 5
[1,1,0,1,0,0,1,1,1,0,0,0] generating graphics... => 3
[1,1,0,1,0,1,0,0,1,0,1,0] generating graphics... => 6
[1,1,0,1,0,1,0,0,1,1,0,0] generating graphics... => 5
[1,1,0,1,0,1,0,1,0,0,1,0] generating graphics... => 7
[1,1,0,1,0,1,0,1,0,1,0,0] generating graphics... => 8
[1,1,0,1,0,1,0,1,1,0,0,0] generating graphics... => 6
[1,1,0,1,0,1,1,0,0,0,1,0] generating graphics... => 5
[1,1,0,1,0,1,1,0,0,1,0,0] generating graphics... => 6
[1,1,0,1,0,1,1,0,1,0,0,0] generating graphics... => 7
[1,1,0,1,0,1,1,1,0,0,0,0] generating graphics... => 4
[1,1,0,1,1,0,0,0,1,0,1,0] generating graphics... => 4
[1,1,0,1,1,0,0,0,1,1,0,0] generating graphics... => 3
[1,1,0,1,1,0,0,1,0,0,1,0] generating graphics... => 5
[1,1,0,1,1,0,0,1,0,1,0,0] generating graphics... => 6
[1,1,0,1,1,0,0,1,1,0,0,0] generating graphics... => 4
[1,1,0,1,1,0,1,0,0,0,1,0] generating graphics... => 6
[1,1,0,1,1,0,1,0,0,1,0,0] generating graphics... => 7
[1,1,0,1,1,0,1,0,1,0,0,0] generating graphics... => 8
[1,1,0,1,1,0,1,1,0,0,0,0] generating graphics... => 5
[1,1,0,1,1,1,0,0,0,0,1,0] generating graphics... => 3
[1,1,0,1,1,1,0,0,0,1,0,0] generating graphics... => 4
[1,1,0,1,1,1,0,0,1,0,0,0] generating graphics... => 5
[1,1,0,1,1,1,0,1,0,0,0,0] generating graphics... => 6
[1,1,0,1,1,1,1,0,0,0,0,0] generating graphics... => 2
[1,1,1,0,0,0,1,0,1,0,1,0] generating graphics... => 3
[1,1,1,0,0,0,1,0,1,1,0,0] generating graphics... => 2
[1,1,1,0,0,0,1,1,0,0,1,0] generating graphics... => 2
[1,1,1,0,0,0,1,1,0,1,0,0] generating graphics... => 3
[1,1,1,0,0,0,1,1,1,0,0,0] generating graphics... => 1
[1,1,1,0,0,1,0,0,1,0,1,0] generating graphics... => 4
[1,1,1,0,0,1,0,0,1,1,0,0] generating graphics... => 3
[1,1,1,0,0,1,0,1,0,0,1,0] generating graphics... => 5
[1,1,1,0,0,1,0,1,0,1,0,0] generating graphics... => 6
[1,1,1,0,0,1,0,1,1,0,0,0] generating graphics... => 4
[1,1,1,0,0,1,1,0,0,0,1,0] generating graphics... => 3
[1,1,1,0,0,1,1,0,0,1,0,0] generating graphics... => 4
[1,1,1,0,0,1,1,0,1,0,0,0] generating graphics... => 5
[1,1,1,0,0,1,1,1,0,0,0,0] generating graphics... => 2
[1,1,1,0,1,0,0,0,1,0,1,0] generating graphics... => 5
[1,1,1,0,1,0,0,0,1,1,0,0] generating graphics... => 4
[1,1,1,0,1,0,0,1,0,0,1,0] generating graphics... => 6
[1,1,1,0,1,0,0,1,0,1,0,0] generating graphics... => 7
[1,1,1,0,1,0,0,1,1,0,0,0] generating graphics... => 5
[1,1,1,0,1,0,1,0,0,0,1,0] generating graphics... => 7
[1,1,1,0,1,0,1,0,0,1,0,0] generating graphics... => 8
[1,1,1,0,1,0,1,0,1,0,0,0] generating graphics... => 9
[1,1,1,0,1,0,1,1,0,0,0,0] generating graphics... => 6
[1,1,1,0,1,1,0,0,0,0,1,0] generating graphics... => 4
[1,1,1,0,1,1,0,0,0,1,0,0] generating graphics... => 5
[1,1,1,0,1,1,0,0,1,0,0,0] generating graphics... => 6
[1,1,1,0,1,1,0,1,0,0,0,0] generating graphics... => 7
[1,1,1,0,1,1,1,0,0,0,0,0] generating graphics... => 3
[1,1,1,1,0,0,0,0,1,0,1,0] generating graphics... => 2
[1,1,1,1,0,0,0,0,1,1,0,0] generating graphics... => 1
[1,1,1,1,0,0,0,1,0,0,1,0] generating graphics... => 3
[1,1,1,1,0,0,0,1,0,1,0,0] generating graphics... => 4
[1,1,1,1,0,0,0,1,1,0,0,0] generating graphics... => 2
[1,1,1,1,0,0,1,0,0,0,1,0] generating graphics... => 4
[1,1,1,1,0,0,1,0,0,1,0,0] generating graphics... => 5
[1,1,1,1,0,0,1,0,1,0,0,0] generating graphics... => 6
[1,1,1,1,0,0,1,1,0,0,0,0] generating graphics... => 3
[1,1,1,1,0,1,0,0,0,0,1,0] generating graphics... => 5
[1,1,1,1,0,1,0,0,0,1,0,0] generating graphics... => 6
[1,1,1,1,0,1,0,0,1,0,0,0] generating graphics... => 7
[1,1,1,1,0,1,0,1,0,0,0,0] generating graphics... => 8
[1,1,1,1,0,1,1,0,0,0,0,0] generating graphics... => 4
[1,1,1,1,1,0,0,0,0,0,1,0] generating graphics... => 1
[1,1,1,1,1,0,0,0,0,1,0,0] generating graphics... => 2
[1,1,1,1,1,0,0,0,1,0,0,0] generating graphics... => 3
[1,1,1,1,1,0,0,1,0,0,0,0] generating graphics... => 4
[1,1,1,1,1,0,1,0,0,0,0,0] generating graphics... => 5
[1,1,1,1,1,1,0,0,0,0,0,0] generating graphics... => 0
click to show generating function       
Description
The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra.
The statistic is also equal to the number of non-projective torsionless indecomposable modules in the corresponding Nakayama algebra.
See theorem 5.8. in the reference for a motivation.
References
[1] Iyama, O., Solberg, √ėyvind Auslander-Gorenstein algebras and precluster tilting. zbMATH:06833443
Code
DeclareOperation("highertaufixall",[IsList]);

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

local A,RegA,CoRegA,t,simA,U,L;

A:=LIST[1];
t:=LIST[2];
L:=ARQuiverNak([A]);
U:=Filtered(L,x->  IsomorphicModules(highertauinverse([DTr(NthSyzygy(x,t)),t]),x)=true);
return(Size(U));

end);
Created
Oct 20, 2018 at 20:09 by Rene Marczinzik
Updated
Nov 26, 2018 at 21:48 by Rene Marczinzik