<Chapter Label="AR-theory"><Heading>Auslander-Reiten theory</Heading>

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


<Section><Heading>Almost split sequences and AR-quivers</Heading>

<ManSection>
   <Attr Name="AlmostSplitSequence" Arg="M" Comm=""/>
   <Description>
     Arguments: <Arg>M</Arg> - an indecomposable non-projective module.<Br />
   </Description>
   <Returns>the almost split sequence ending in the
   module <Arg>M</Arg> 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.
   </Returns>
   <Description>
    The range of the right minimal almost split map is not necessarily
    equal to the module <Arg>M</Arg> one started with, but isomorphic. 
    The function assumes that the module <Arg>M</Arg> is indecomposable. 
   </Description>
</ManSection>

<ManSection>
     <Oper Name="IsTauPeriodic" Arg="M, n" Comm="for a
   PathAlgebraMatModule and a positive integer"/>
   <Description> 
    Arguments: <Arg>M</Arg> -- a path algebra module
    (<C>PathAlgebraMatModule</C>), <Arg>n</Arg> -- be a positive integer.
   <Br /></Description>
   <Returns><C>i</C>, where <C>i</C> is the smallest positive integer
     less or equal <C>n</C> such that the representation <Arg>M</Arg> is isomorphic
     to the <M>\tau^i(M)</M>, and false otherwise.
   </Returns>
</ManSection>

<ManSection>
   <Oper Name="PredecessorOfModule" Arg="M, n" Comm=""/>
   <Description>
     Arguments: <Arg>M</Arg> - an indecomposable non-projective module
     and <Arg>n</Arg> - a positive integer.<Br />
   </Description>
   <Returns>the predecessors of the module <Arg>M</Arg> in the
    AR-quiver of the algebra <Arg>M</Arg> is given over of distance
    less or equal to <Arg>n</Arg>. 
   </Returns>
   <Description>
    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  <Arg>n</Arg>. The <C>m</C>-th entry in the second list is 
    the valuations of the irreducible morphism from indecomposable 
    module number <C>i</C> in layer <C>m+1</C> to indecomposable module
    number <C>j</C> in layer <C>m</C> for the values of <C>i</C> and 
    <C>j</C> there is an irreducible morphism. Whenever <C>false</C> 
    occur in the output, it means that this valuation has not been 
    computed. 

    The function assumes that the module <Arg>M</Arg> is indecomposable 
    and that the quotient of the path algebra is given over a finite field. 
   </Description>
</ManSection>

<Example><![CDATA[
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 ] ] ] ] ] ]]>
</Example>
</Section>
</Chapter>
