Identifier
St001066: Dyck paths ⟶ ℤ
Values
Modified entries:

[1,0]=>1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Diff Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Code
DeclareOperation("IsNthSyzygy",[IsList]);

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

local M, n, f, N, i, h,W;

M:=LIST[1];

n:=LIST[2];

N:=DualOfModule(NthSyzygy(DualOfModule(M),n));

W:=NthSyzygy(N,n);

if IsDirectSummand(M,W)=true
then
return(1);
else return(0);
fi;
end);


DeclareOperation("NumberOfReflexiveSimpleModules",[IsList]);

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

local M, n, f, N, i, h,W;

L:=LIST[1];

A:=NakayamaAlgebra(L,GF(3));
simA:=SimpleModules(A);
U:=Filtered(simA,x->IsNthSyzygy([x,2])=1);
return(Size(U));


end);


Created
Dec 30, 2017 at 14:30 by Rene Marczinzik
Updated
Apr 30, 2025 at 18:16 by Rene Marczinzik
3 Comments (hide)

Martin Rubey
23 May 13:45
Hi René! I don't understand the edit, could you please explain?
Rene marczinzik
23 May 13:48
Not so important but: Deleting this entry or adding the entry for the statistic https://www.findstat.org/StatisticsDatabase/St000931/ (probably adding it would have been a better idea...) should be done so that one find more results with "search for individual values". (I wanted to show this to students as an interesting result and how findstat works, but the entry [1,0]=>1 "ruined it" since then the result is not found).
Martin Rubey
23 May 14:09
Actually, the statistic *is* found, but the ordering on the webpage is very strange. I opened an issue, but currently there is nobody doing any programming :-(

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