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9 Auslander-Reiten theory
9.1 Almost split sequences and AR-quivers
9.1-1 AlmostSplitSequence
9.1-2 IsTauPeriodic
9.1-3 PredecessorOfModule

### 9 Auslander-Reiten theory

This chapter describes the functions implemented for almost split sequences and Auslander-Reiten theory in QPA.

#### 9.1 Almost split sequences and AR-quivers

##### 9.1-1 AlmostSplitSequence

`‣ AlmostSplitSequence`( M ) | ( attribute )

Arguments: M - an indecomposable non-projective module.

Returns: the almost split sequence ending in the module M if it is indecomposable and not projective. It returns the almost split sequence in terms of two maps, a left minimal almost split map and a right minimal almost split map.

The range of the right minimal almost split map is not necessarily equal to the module M one started with, but isomorphic. The function assumes that the module M is indecomposable.

##### 9.1-2 IsTauPeriodic

`‣ IsTauPeriodic`( M, n ) | ( operation )

Arguments: M -- a path algebra module (`PathAlgebraMatModule`), n -- be a positive integer.

Returns: `i`, where `i` is the smallest positive integer less or equal `n` such that the representation M is isomorphic to the τ^i(M), and false otherwise.

##### 9.1-3 PredecessorOfModule

`‣ PredecessorOfModule`( M, n ) | ( operation )

Arguments: M - an indecomposable non-projective module and n - a positive integer.

Returns: the predecessors of the module M in the AR-quiver of the algebra M is given over of distance less or equal to n.

It returns two lists, the first is the indecomposable modules in the different layers and the second is the valuations for the arrows in the AR-quiver. The different entries in the first list are the modules at distance zero, one, two, three, and so on, until layer n. The `m`-th entry in the second list is the valuations of the irreducible morphism from indecomposable module number `i` in layer `m+1` to indecomposable module number `j` in layer `m` for the values of `i` and `j` there is an irreducible morphism. Whenever `false` occur in the output, it means that this valuation has not been computed. The function assumes that the module M is indecomposable and that the quotient of the path algebra is given over a finite field.

    gap> A := KroneckerAlgebra(GF(4),2);
    <GF(2^2)[<quiver with 2 vertices and 2 arrows>]>
    gap> S := SimpleModules(A)[1];
    <[ 1, 0 ]>
    gap> ass := AlmostSplitSequence(S);
    [ <<[ 3, 2 ]> ---> <[ 4, 2 ]>>
        , <<[ 4, 2 ]> ---> <[ 1, 0 ]>>
         ]
    gap> DecomposeModule(Range(ass[1]));
    [ <[ 2, 1 ]>, <[ 2, 1 ]> ]
    gap> PredecessorsOfModule(S,5);
    [ [ [ <[ 1, 0 ]> ], [ <[ 2, 1 ]> ], [ <[ 3, 2 ]> ], [ <[ 4, 3 ]> ],
          [ <[ 5, 4 ]> ], [ <[ 6, 5 ]> ] ],
      [ [ [ 1, 1, [ 2, false ] ] ], [ [ 1, 1, [ 2, 2 ] ] ],
          [ [ 1, 1, [ 2, 2 ] ] ], [ [ 1, 1, [ 2, 2 ] ] ],
          [ [ 1, 1, [ false, 2 ] ] ] ] ]
    gap> A:=NakayamaAlgebra([5,4,3,2,1],GF(4));
    <GF(2^2)[<quiver with 5 vertices and 4 arrows>]>
    gap> S := SimpleModules(A)[1];
    <[ 1, 0, 0, 0, 0 ]>
    gap> PredecessorsOfModule(S,5);
    [ [ [ <[ 1, 0, 0, 0, 0 ]> ], [ <[ 1, 1, 0, 0, 0 ]> ],
          [ <[ 0, 1, 0, 0, 0 ]>, <[ 1, 1, 1, 0, 0 ]> ],
          [ <[ 0, 1, 1, 0, 0 ]>, <[ 1, 1, 1, 1, 0 ]> ],
          [ <[ 0, 0, 1, 0, 0 ]>, <[ 0, 1, 1, 1, 0 ]>, <[ 1, 1, 1, 1, 1 ]>
             ], [ <[ 0, 0, 1, 1, 0 ]>, <[ 0, 1, 1, 1, 1 ]> ] ],
      [ [ [ 1, 1, [ 1, false ] ] ],
          [ [ 1, 1, [ 1, 1 ] ], [ 2, 1, [ 1, false ] ] ],
          [ [ 1, 1, [ 1, 1 ] ], [ 1, 2, [ 1, 1 ] ],
              [ 2, 2, [ 1, false ] ] ],
          [ [ 1, 1, [ 1, 1 ] ], [ 2, 1, [ 1, 1 ] ], [ 2, 2, [ 1, 1 ] ],
              [ 3, 2, [ 1, false ] ] ],
          [ [ 1, 1, [ false, 1 ] ], [ 1, 2, [ false, 1 ] ],
              [ 2, 2, [ false, 1 ] ], [ 2, 3, [ false, 1 ] ] ] ] ]

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