GAP (QPA) - Chapter 9: Auslander-Reiten theory

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9 Auslander-Reiten theory
9.1 Almost split sequences and AR-quivers
9.1-1 AlmostSplitSequence
9.1-2 AlmostSplitSequenceInPerpT
9.1-3 IrreducibleMorphismsEndingIn
9.1-4 IsTauPeriodic
9.1-5 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 )

`‣ AlmostSplitSequence`( M, e ) | ( attribute )

Arguments: M - an indecomposable non-projective module, e - either l = left or r = right

Returns: the almost split sequence ending in the module M if it is indecomposable and not projective, for the first variant. The second variant finds the almost split sequence starting or ending in the module M depending on whether the second argument e is l or r (l = almost split sequence starting with M, or r = almost split sequence ending in M), if the module is indecomposable and not injective or not projective, respectively. It returns fail if the module is injective (l) or projective (r).

The almost split sequence is returned as a pair of maps, the monomorphism and the epimorphism. The function assumes that the module M is indecomposable, and the source of the monomorphism (l) or the range of the epimorphism (r) is a module that is isomorphic to M, not necessarily identical.

##### 9.1-2 AlmostSplitSequenceInPerpT

`‣ AlmostSplitSequenceInPerpT`( T, M ) | ( operation )

Arguments: T - a cotilting module, M - an indecomposable non-projective module

Returns: the almost split sequence in ^\perp T ending in the module M, if the module is indecomposable. It returns fail if the module is in \add T projective. The almost split sequence is returned as a pair of maps, the monomorphism and the epimorphism.

The function assumes that the module M is indecomposable and in ^\perp T, and the range of the epimorphism is a module that is isomorphic to the input, not necessarily identical.

##### 9.1-3 IrreducibleMorphismsEndingIn

`‣ IrreducibleMorphismsEndingIn`( M ) | ( attribute )

`‣ IrreducibleMorphismsStartingIn`( M ) | ( attribute )

Arguments: M - an indecomposable module

Returns: the collection of irreducible morphisms ending and starting in the module M, respectively. The argument is assumed to be an indecomposable module.

The irreducible morphisms are returned as a list of maps. Even in the case of only one irreducible morphism, it is returned as a list. The function assumes that the module M is indecomposable over a quiver algebra with a finite field as the ground ring.

##### 9.1-4 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.

##### 9.1-5 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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