Merge branch 'pr-13' into devel

* pr-13:
  bumped Tibor's version
  recode ibmpc..lat1 examples/DecomposeOnceByRandomEndomorphism.g
  finished example DecomposeOnceByRandomEndomorphism and solved the documentation problem.
  Tests run, no errors. Included example in documentation, but it will not appear as example between C3C3 and C4C4 in the printed manual. Maybe need to change CatRepsWithCAP.gd ?
  newline at end
  Example for one step of decomposition of fortyone. Compare the block diagonal structure of matrices fortyone(kq.a) with S(kq.a). Some tests needed.

PackageInfo.g

@@ -11,9 +11,9 @@ SetPackageInfo( rec(
 PackageName := "CatReps",
 Subtitle := "Representations and cohomology of finite categories",
 Version := Maximum( [
-                   "2020.10.01", ## Mohamed's version
+                   "2020.10-01", ## Mohamed's version
                    ## this line prevents merge conflicts
-                   "2020.07.04", ## Tibor's version
+                   "2020.10-02", ## Tibor's version
                    ## this line prevents merge conflicts
                    "2020.01.01", ## Peter's version
                    ] ),

doc/Doc.autodoc

@@ -5,6 +5,9 @@
 @Subsection A category of module homomorphisms
 @InsertChunk CategoryOfRepresentations
 
+@Subsection One step in the direct sum decomposition
+@InsertChunk DecomposeOnce
+
 @Subsection Another category of module homomorphisms
 @InsertChunk RepresentingC4C4
 

examples/DecomposeOnceByRandomEndomorphism.g

@@ -0,0 +1,300 @@
+#! @BeginChunk DecomposeOnce
+
+LoadPackage( "CatReps" );
+
+#! @BeginExample
+c3c3 := ConcreteCategoryForCAP( [ [2,3,1], [4,5,6], [,,,5,6,4] ] );
+#! A finite concrete category
+GF3 := HomalgRingOfIntegers( 3 );
+#! GF(3)
+kq := Algebroid( GF3, c3c3 );
+#! Algebroid generated by the right quiver q(2)[a:1->1,b:1->2,c:2->2]
+SetIsLinearClosureOfACategory( kq, true );
+CatReps := Hom( kq, GF3 );
+#! The category of functors: Algebroid generated by the right quiver
+#! q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices over GF(3)
+d := [[1,1,0,0,0],[0,1,1,0,0],[0,0,1,0,0],[0,0,0,1,1],[0,0,0,0,1]];;
+e := [[0,1,0,0],[0,0,1,0],[0,0,0,0],[0,1,0,1],[0,0,1,0]];;
+f := [[1,1,0,0],[0,1,1,0],[0,0,1,0],[0,0,0,1]];;
+nine := AsObjectInHomCategory( kq, [ 5, 4 ], [ d, e, f ] );
+#! <(1)->5, (2)->4; (a)->5x5, (b)->5x4, (c)->4x4>
+DecomposeOnceByRandomEndomorphism( nine );
+#! fail
+#! @EndExample
+
+#! The above shows that our representation <C>nine</C> is
+#! indecomposable (with a high probability).
+#! We use the tensor product to generate another representation
+#! <C>fortyone</C>, that is hopefully decomposable, and
+#! inspect the two resulting embeddings <C>iota</C> and
+#! <C>kappa</C>.
+
+#! @BeginExample
+fortyone := TensorProductOnObjects( nine, nine );
+#! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
+result := DecomposeOnceByRandomEndomorphism( fortyone );
+#! [ <(1)->3x25, (2)->1x16>, <(1)->22x25, (2)->15x16> ]
+iota := result[1];
+#! <(1)->3x25, (2)->1x16>
+kappa := result[2];
+#! <(1)->22x25, (2)->15x16>
+Display( fortyone );
+#! An object in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 25
+#! 
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 16
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  1 1 . . . 1 1 . . . . . . . . . . . . . . . . . .
+#!  . 1 1 . . . 1 1 . . . . . . . . . . . . . . . . .
+#!  . . 1 . . . . 1 . . . . . . . . . . . . . . . . .
+#!  . . . 1 1 . . . 1 1 . . . . . . . . . . . . . . .
+#!  . . . . 1 . . . . 1 . . . . . . . . . . . . . . .
+#!  . . . . . 1 1 . . . 1 1 . . . . . . . . . . . . .
+#!  . . . . . . 1 1 . . . 1 1 . . . . . . . . . . . .
+#!  . . . . . . . 1 . . . . 1 . . . . . . . . . . . .
+#!  . . . . . . . . 1 1 . . . 1 1 . . . . . . . . . .
+#!  . . . . . . . . . 1 . . . . 1 . . . . . . . . . .
+#!  . . . . . . . . . . 1 1 . . . . . . . . . . . . .
+#!  . . . . . . . . . . . 1 1 . . . . . . . . . . . .
+#!  . . . . . . . . . . . . 1 . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . 1 1 . . . . . . . . . .
+#!  . . . . . . . . . . . . . . 1 . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . 1 1 . . . 1 1 . . .
+#!  . . . . . . . . . . . . . . . . 1 1 . . . 1 1 . .
+#!  . . . . . . . . . . . . . . . . . 1 . . . . 1 . .
+#!  . . . . . . . . . . . . . . . . . . 1 1 . . . 1 1
+#!  . . . . . . . . . . . . . . . . . . . 1 . . . . 1
+#!  . . . . . . . . . . . . . . . . . . . . 1 1 . . .
+#!  . . . . . . . . . . . . . . . . . . . . . 1 1 . .
+#!  . . . . . . . . . . . . . . . . . . . . . . 1 . .
+#!  . . . . . . . . . . . . . . . . . . . . . . . 1 1
+#!  . . . . . . . . . . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  . . . . . 1 . . . . . . . . . .
+#!  . . . . . . 1 . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . 1 . 1 . . . . . . . .
+#!  . . . . . . 1 . . . . . . . . .
+#!  . . . . . . . . . 1 . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . 1 . 1 . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . 1 . . . . . . . 1 . .
+#!  . . . . . . 1 . . . . . . . 1 .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . 1 . 1 . . . . . 1 . 1
+#!  . . . . . . 1 . . . . . . . 1 .
+#!  . . . . . . . . . 1 . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . 1 . 1 . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  1 1 . . 1 1 . . . . . . . . . .
+#!  . 1 1 . . 1 1 . . . . . . . . .
+#!  . . 1 . . . 1 . . . . . . . . .
+#!  . . . 1 . . . 1 . . . . . . . .
+#!  . . . . 1 1 . . 1 1 . . . . . .
+#!  . . . . . 1 1 . . 1 1 . . . . .
+#!  . . . . . . 1 . . . 1 . . . . .
+#!  . . . . . . . 1 . . . 1 . . . .
+#!  . . . . . . . . 1 1 . . . . . .
+#!  . . . . . . . . . 1 1 . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . 1 1 . .
+#!  . . . . . . . . . . . . . 1 1 .
+#!  . . . . . . . . . . . . . . 1 .
+#!  . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+S := DirectSum( [ Source( iota ), Source( kappa ) ] );
+#! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16>
+Display( S );
+#! An object in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 25
+#! 
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 16
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  . 2 . . . . . . . . . . . . . . . . . . . . . . .
+#!  1 2 2 . . . . . . . . . . . . . . . . . . . . . .
+#!  . . 1 . . . . . . . . . . . . . . . . . . . . . .
+#!  . . . 1 1 . . . 1 1 . . . . . . . . . . . . . . .
+#!  . . . . 1 1 . . . 1 1 . . . . . . . . . . . . . .
+#!  . . . . . 1 . . . . 1 . . . . . . . . . . . . . .
+#!  . . . . . . 1 1 . . . 1 1 . . . . . . . . . . . .
+#!  . . . . . . . 1 . . . . 1 . . . . . . . . . . . .
+#!  . . . . . . . . 1 1 . . . 1 1 . . . . . . . . . .
+#! . . . . . . . . . 1 1 . . . 1 1 . . . . . . . . .
+#! . . . . . . . . . . 1 . . . . 1 . . . . . . . . .
+#! . . . . . . . . . . . 1 1 . . . 1 1 . . . . . . .
+#! . . . . . . . . . . . . 1 . . . . 1 . . . . . . .
+#! . . . . . . . . . . . . . 1 1 . . . . . . . . . .
+#! . . . . . . . . . . . . . . 1 1 . . . . . . . . .
+#! . . . . . . . . . . . . . . . 1 . . . . . . . . .
+#! . . . . . . . . . . . . . . . . 1 1 . . . . . . .
+#! . . . . . . . . . . . . . . . . . 1 . . . . . . .
+#! . . . . . . . . . . . . . . . . . . 1 1 . 1 1 . .
+#! . . . . . . . . . . . . . . . . . . . 1 1 . 1 1 .
+#! . . . . . . . . . . . . . . . . . . . . 1 . . 1 .
+#! . . . . . . . . . . . . . . . . . . . . . 1 1 . .
+#! . . . . . . . . . . . . . . . . . . . . . . 1 1 .
+#! . . . . . . . . . . . . . . . . . . . . . . . 1 .
+#! . . . . . . . . . . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  2 . . . . . . . . . . . . . . .
+#!  1 . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . 1 . . . . . . . . .
+#!  . . . . . . . 1 . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . 1 . 1 . . . . . . .
+#!  . . . . . . . 1 . . . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . 1 . 1 . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . 1 . . . . . . . 1 .
+#!  . . . . . . . 1 . . . . . . . 1
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . . . . 1 . . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . . . . .
+#!  . . . . . . . 2 . . 1 . 1 . . 2
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  1 . . . . . . . . . . . . . . .
+#!  . 1 1 . . 1 1 . . . . . . . . .
+#!  . . 1 1 . . 1 1 . . . . . . . .
+#!  . . . 1 . . . 1 . . . . . . . .
+#!  . . . . 1 . . . 1 . . . . . . .
+#!  . . . . . 1 1 . . 1 1 . . . . .
+#!  . . . . . . 1 1 . . 1 1 . . . .
+#!  . . . . . . . 1 . . . 1 . . . .
+#!  . . . . . . . . 1 . . . 1 . . .
+#!  . . . . . . . . . 1 1 . . . . .
+#!  . . . . . . . . . . 1 1 . . . .
+#!  . . . . . . . . . . . 1 . . . .
+#!  . . . . . . . . . . . . 1 . . .
+#!  . . . . . . . . . . . . . 1 1 .
+#!  . . . . . . . . . . . . . . 1 1
+#!  . . . . . . . . . . . . . . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! @EndExample
+
+#! Comparing the matrices of <C>fortyone</C> with those of 
+#! <C>S</C>, we see
+#! that after decomposing once, we have separated one small
+#! matrix on the diagonal: A $3\times 3$-matrix from <C>S(kq.a)</C>,
+#! a $3 \times 1$-matrix from <C>S(kq.b)</C> and a $1\times 1$-matrix
+#! from <C>S(kq.c)</C>. This matches with the source of the 
+#! embedding <C>iota</C>.
+
+#! @BeginExample
+Display( iota );
+#! A morphism in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#! 
+#! 
+#! Image of <(1)>:
+#!  2 2 . 1 1 . . . . . . . . . . 1 2 1 2 1 . . . . .
+#!  1 2 1 2 1 1 2 1 2 1 . . . . . 2 . . 1 . 2 . . 1 .
+#!  . . 2 . . . 1 2 . 2 2 . . 1 . . . 1 . . . 2 . . 1
+#! 
+#! A split monomorphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of <(2)>:
+#!  . . . . . . . . . . . . . . . 1
+#! 
+#! A split monomorphism in Category of matrices over GF(3)
+Display( Source( iota) );
+#! An object in The category of functors: Algebroid generated by the
+#! right quiver q(2)[a:1->1,b:1->2,c:2->2] -> Category of matrices
+#! over GF(3) defined by the following data:
+#!
+#! Image of <(1)>:
+#! A vector space object over GF(3) of dimension 3
+#! 
+#! Image of <(2)>:
+#! A vector space object over GF(3) of dimension 1
+#! 
+#! Image of (1)-[{ Z(3)^0*(a) }]->(1):
+#!  . 2 .
+#!  1 2 2
+#!  . . 1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (1)-[{ Z(3)^0*(b) }]->(2):
+#!  2
+#!  1
+#!  .
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! 
+#! 
+#! Image of (2)-[{ Z(3)^0*(c) }]->(2):
+#!  1
+#! 
+#! A morphism in Category of matrices over GF(3)
+#! @EndExample
+
+#! We can then look at the other embedding of the direct sum 
+#! decomposition, <C>kappa</C>. The iteration of 
+#! <C>WeakDirectSumDecomposition</C> will continue then
+#! with <C>Source( kappa )</C>. Each time the random
+#! endomorphism will decompose the representation,
+#! lowering the dimensions of each object at most by $3$.
+
+#! @Example
+Source( kappa );
+#! <(1)->22, (2)->15; (a)->22x22, (b)->22x15, (c)->15x15>
+result2 := DecomposeOnceByRandomEndomorphism( Source( kappa ) );
+#! [ <(1)->3x22, (2)->3x15>, <(1)->19x22, (2)->12x15> ]
+#! @EndExample
+#! @EndChunk