Given a connected quiver algebra $A$ over a finite field $K$.

Question : Is there an effective/quick method to obtain all $d$-dimensional indecomposable representations for a fixed $d$ with a computer algebra system such as the GAP package QPA?

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Given a connected quiver algebra $A$ over a finite field $K$.

Question : Is there an effective/quick method to obtain all $d$-dimensional indecomposable representations for a fixed $d$ with a computer algebra system such as the GAP package QPA?

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Now this is implemented in QPA version 1. There is an error above, one needs to consider both `ExtOverAlgebra(S[i],iso[j])`

and `ExtOverAlgebra(iso[j],S[i])`

in order to get all modules of the given length. It looks like this in QPA:

```
gap> A := KroneckerAlgebra(GF(3),2);;
gap> AllModulesOfLengthAtMost( A, 4 );
[ <[ 0, 0 ]>, <[ 1, 0 ]>, <[ 0, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>,
<[ 1, 1 ]>, <[ 1, 1 ]>, <[ 2, 1 ]>, <[ 2, 1 ]>, <[ 2, 1 ]>, <[ 2, 1 ]>,
<[ 2, 1 ]>, <[ 2, 1 ]>, <[ 1, 2 ]>, <[ 1, 2 ]>, <[ 1, 2 ]>, <[ 1, 2 ]>,
<[ 1, 2 ]>, <[ 1, 2 ]>, <[ 3, 1 ]>, <[ 3, 1 ]>, <[ 3, 1 ]>, <[ 3, 1 ]>,
<[ 3, 1 ]>, <[ 2, 2 ]>, <[ 3, 1 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>,
<[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>,
<[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>,
<[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>,
<[ 2, 2 ]>, <[ 2, 2 ]>, <[ 1, 3 ]>, <[ 1, 3 ]>, <[ 1, 3 ]>, <[ 1, 3 ]>,
<[ 1, 3 ]>, <[ 1, 3 ]> ]
gap> AllIndecModulesOfLengthAtMost( A, 4 );
[ <[ 1, 0 ]>, <[ 0, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>,
<[ 2, 1 ]>, <[ 1, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]>,
<[ 2, 2 ]>, <[ 2, 2 ]>, <[ 2, 2 ]> ]
```

The first one finds all non-isomorphic modules of length at most 4, and the second one finds all non-isomorphic indecomposable modules of length at most 4.

I hope that these comments are helpful.

Best regards, the QPA-team.

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I don't know if it would be quick, but you can inductively construct all modules of dimension $d$ using QPA. First you can construct all modules of dimension $2$, then all modules of dimension $3$, and so on. The command `ExtOverAlgebra(S[i],S[j])`

would compute a basis for all the extensions between the different simple modules `S[i]`

and `S[j]`

. The basis vectors are maps from the first syzygy of `S[i]`

to `S[j]`

. Taking all linear combinations of these and then taking the pushout along the inclusion of the first syzygy of `S[i]`

to the projective cover of `S[i]`

, would produce all modules of dimension $2$. It would possibly be a lot of duplicates on this list of modules, but those can be first reduced by applying the command `Unique(list)`

and then doing actually isomorphism test with the command `IsomorphicModules`

.

For the Kronecker algebra over a field of three elements it would look like something like this:

```
gap> A := KroneckerAlgebra(GF(3),2);;
gap> S := SimpleModules(A);;
gap> len2 := List( [1..2], i -> List( [1..2], j -> ExtOverAlgebra(S[i],S[j])));;
gap> len2[1][1];len2[2][1];len2[2][2];
[ [ ], [ ], [ ] ]
[ [ ], [ ], [ ] ]
[ [ ], [ ], [ ] ]
gap> len2 := len2[1][2];
[ <<[ 0, 2 ]> ---> <[ 1, 2 ]>>, [ <<[ 0, 2 ]> ---> <[ 0, 1 ]>>, <<[ 0, 2 ]> ---> <[ 0, 1 ]>> ], function( map ) ... end ]
gap> test := [];
[ ]
gap> for a in Elements(GF(3)) do for b in Elements(GF(3)) do Add(test, a*len2[2][1] + b*len2[2][2]); od; od;
gap> temp := List( test, t -> Range(PushOut(t,len2[1])[2]));
[ <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]> ]
gap> temp := Unique(temp);
[ <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]>, <[ 1, 1 ]> ]
gap> iso := [];;
gap> for i in [1..5] do for j in [i+1..5] do Add(iso, [i,j,IsomorphicModules( temp[i], temp[j])]); od;od;
gap> iso;
[ [ 1, 2, false ], [ 1, 3, false ], [ 1, 4, false ], [ 1, 5, false ], [ 2, 3, false ], [ 2, 4, false ], [ 2, 5, false ], [ 3, 4, false ], [ 3, 5, false ],
[ 4, 5, false ] ]
```

The variable `iso`

now contain all the five non-isomorphic modules of dimension $2$. Repeating the above procedure for all elements on the list `iso`

and each simple, that is, do `ExtOverAlgebra(S[i], iso[j])`

for all $i$ and $j$, constructing all extensions: Then you would have a list of all modules of length $3$, and so on.

I hope that these comments are helpful.

Best regards, the QPA-team.