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

`‣ 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.

`‣ 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 and not projective (that is, not projective object in \(^\perp T\)). It returns fail if the module `M` is in \(\add T\) projective. The almost split sequence is returned as a pair of maps, the monomorphism and the epimorphism, and the range of the epimorphism is a module that is isomorphic to the input, not necessarily identical.

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.

`‣ 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.

`‣ 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 \(\tau^i(M)\), and false otherwise.

`‣ 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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