From 83891cfd5716013117695b90d8bce29f2e077869 Mon Sep 17 00:00:00 2001 From: Tibor Gruen Date: Fri, 16 Oct 2020 12:09:00 +0200 Subject: [PATCH 1/6] 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. --- examples/DecomposeOnceByRandomEndomorphism.g | 252 +++++++++++++++++++ 1 file changed, 252 insertions(+) create mode 100644 examples/DecomposeOnceByRandomEndomorphism.g diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g new file mode 100644 index 0000000..a820fcb --- /dev/null +++ b/examples/DecomposeOnceByRandomEndomorphism.g @@ -0,0 +1,252 @@ +#! @Chunk DecomposeOnceByRandomEndomorphism + +LoadPackage( "CatReps" ); + +#! @Example +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 nine is indecomposable. +#! We use the tensor product to generate another representation +#! fortyone, that is hopefully decomposable. + +#! @Example +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 fortyone with those of S, we see +#! that after decomposing once, we have seperated one small +#! matrix on the diagonal: A $3\times 3$-matrix from $S(kq.a)$, +#! a $3 \times 1$-matrix from $S(kq.b)$ and a $1\times 1$-matrix +#! from S(kq.c). This matches with the sources of the +#! natural transformation $\iota$. + +#! @Example +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) +Source( iota(kq.1) ); +#! +Source( iota(kq.2) ); +#! \ No newline at end of file From 39978fb2c6d2c2a2da5ebfb3439f4571b4638cef Mon Sep 17 00:00:00 2001 From: Tibor Gruen Date: Fri, 16 Oct 2020 12:15:14 +0200 Subject: [PATCH 2/6] newline at end --- examples/DecomposeOnceByRandomEndomorphism.g | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g index a820fcb..3488798 100644 --- a/examples/DecomposeOnceByRandomEndomorphism.g +++ b/examples/DecomposeOnceByRandomEndomorphism.g @@ -249,4 +249,4 @@ Display( iota ); Source( iota(kq.1) ); #! Source( iota(kq.2) ); -#! \ No newline at end of file +#! From 45d023bfcf9bd784adaad2b3eae7f2fee373e703 Mon Sep 17 00:00:00 2001 From: Tibor Gruen Date: Fri, 16 Oct 2020 13:24:18 +0200 Subject: [PATCH 3/6] 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 ? --- doc/Doc.autodoc | 3 + examples/DecomposeOnceByRandomEndomorphism.g | 68 ++++++++++++++++---- 2 files changed, 59 insertions(+), 12 deletions(-) diff --git a/doc/Doc.autodoc b/doc/Doc.autodoc index 611e187..684ecfe 100644 --- a/doc/Doc.autodoc +++ b/doc/Doc.autodoc @@ -5,6 +5,9 @@ @Subsection A category of module homomorphisms @InsertChunk CategoryOfRepresentations +@Subsection One step in the direct sum decomposition +@InsertChunk DecomposeOnceByRandomEndomorphism + @Subsection Another category of module homomorphisms @InsertChunk RepresentingC4C4 diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g index 3488798..05bd498 100644 --- a/examples/DecomposeOnceByRandomEndomorphism.g +++ b/examples/DecomposeOnceByRandomEndomorphism.g @@ -1,4 +1,4 @@ -#! @Chunk DecomposeOnceByRandomEndomorphism +#! @BeginChunk DecomposeOnceByRandomEndomorphism LoadPackage( "CatReps" ); @@ -22,9 +22,9 @@ DecomposeOnceByRandomEndomorphism( nine ); #! fail #! @EndExample -#! The above shows that our representation nine is indecomposable. +#! The above shows that our representation nine is indecomposable. #! We use the tensor product to generate another representation -#! fortyone, that is hopefully decomposable. +#! fortyone, that is hopefully decomposable. #! @Example fortyone := TensorProductOnObjects( nine, nine ); @@ -221,11 +221,12 @@ Display( S ); #! A morphism in Category of matrices over GF(3) #! @EndExample -#! Comparing the matrices of fortyone with those of S, we see -#! that after decomposing once, we have seperated one small -#! matrix on the diagonal: A $3\times 3$-matrix from $S(kq.a)$, -#! a $3 \times 1$-matrix from $S(kq.b)$ and a $1\times 1$-matrix -#! from S(kq.c). This matches with the sources of the +#! Comparing the matrices of fortyone with those of +#! S, we see +#! that after decomposing once, we have separated one small +#! matrix on the diagonal: A $3\times 3$-matrix from S(kq.a), +#! a $3 \times 1$-matrix from S(kq.b) and a $1\times 1$-matrix +#! from S(kq.c). This matches with the source of the #! natural transformation $\iota$. #! @Example @@ -246,7 +247,50 @@ Display( iota ); #! . . . . . . . . . . . . . . . 1 #! #! A split monomorphism in Category of matrices over GF(3) -Source( iota(kq.1) ); -#! -Source( iota(kq.2) ); -#! +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 factor of the direct sum +#! decomposition, i.e. $\kappa$. The iteration of +#! WeakDirectSumDecomposition will continue then +#! with Source( kappa ). Each time the random +#! endomorphism will decompose the representation by +#! at most a dimension of $3$. + +#! @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 From 9713da12b8b1d8ad67562b8c1a02362e1a4ca241 Mon Sep 17 00:00:00 2001 From: Tibor Gruen Date: Fri, 16 Oct 2020 14:49:04 +0200 Subject: [PATCH 4/6] finished example DecomposeOnceByRandomEndomorphism and solved the documentation problem. --- doc/Doc.autodoc | 2 +- examples/DecomposeOnceByRandomEndomorphism.g | 30 +++++++++++--------- 2 files changed, 18 insertions(+), 14 deletions(-) diff --git a/doc/Doc.autodoc b/doc/Doc.autodoc index 684ecfe..07ba7b5 100644 --- a/doc/Doc.autodoc +++ b/doc/Doc.autodoc @@ -6,7 +6,7 @@ @InsertChunk CategoryOfRepresentations @Subsection One step in the direct sum decomposition -@InsertChunk DecomposeOnceByRandomEndomorphism +@InsertChunk DecomposeOnce @Subsection Another category of module homomorphisms @InsertChunk RepresentingC4C4 diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g index 05bd498..add2fb7 100644 --- a/examples/DecomposeOnceByRandomEndomorphism.g +++ b/examples/DecomposeOnceByRandomEndomorphism.g @@ -1,8 +1,8 @@ -#! @BeginChunk DecomposeOnceByRandomEndomorphism +#! @BeginChunk DecomposeOnce LoadPackage( "CatReps" ); -#! @Example +#! @BeginExample c3c3 := ConcreteCategoryForCAP( [ [2,3,1], [4,5,6], [,,,5,6,4] ] ); #! A finite concrete category GF3 := HomalgRingOfIntegers( 3 ); @@ -22,11 +22,14 @@ DecomposeOnceByRandomEndomorphism( nine ); #! fail #! @EndExample -#! The above shows that our representation nine is indecomposable. +#! The above shows that our representation nine is +#! indecomposable (with a high probability). #! We use the tensor product to generate another representation -#! fortyone, that is hopefully decomposable. +#! fortyone, that is hopefully decomposable, and +#! inspect the two resulting embeddings iota and +#! kappa. -#! @Example +#! @BeginExample fortyone := TensorProductOnObjects( nine, nine ); #! <(1)->25, (2)->16; (a)->25x25, (b)->25x16, (c)->16x16> result := DecomposeOnceByRandomEndomorphism( fortyone ); @@ -227,12 +230,13 @@ Display( S ); #! matrix on the diagonal: A $3\times 3$-matrix from S(kq.a), #! a $3 \times 1$-matrix from S(kq.b) and a $1\times 1$-matrix #! from S(kq.c). This matches with the source of the -#! natural transformation $\iota$. +#! embedding iota. -#! @Example +#! @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: +#! 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)>: @@ -280,12 +284,12 @@ Display( Source( iota) ); #! A morphism in Category of matrices over GF(3) #! @EndExample -#! We can then look at the other factor of the direct sum -#! decomposition, i.e. $\kappa$. The iteration of +#! We can then look at the other embedding of the direct sum +#! decomposition, kappa. The iteration of #! WeakDirectSumDecomposition will continue then #! with Source( kappa ). Each time the random -#! endomorphism will decompose the representation by -#! at most a dimension of $3$. +#! endomorphism will decompose the representation, +#! lowering the dimensions of each object at most by $3$. #! @Example Source( kappa ); From 4ad619f8d0164d681262de91e882607559bc8636 Mon Sep 17 00:00:00 2001 From: Mohamed Barakat Date: Sat, 17 Oct 2020 17:56:01 +0200 Subject: [PATCH 5/6] recode ibmpc..lat1 examples/DecomposeOnceByRandomEndomorphism.g --- examples/DecomposeOnceByRandomEndomorphism.g | 600 +++++++++---------- 1 file changed, 300 insertions(+), 300 deletions(-) diff --git a/examples/DecomposeOnceByRandomEndomorphism.g b/examples/DecomposeOnceByRandomEndomorphism.g index add2fb7..12ce041 100644 --- a/examples/DecomposeOnceByRandomEndomorphism.g +++ b/examples/DecomposeOnceByRandomEndomorphism.g @@ -1,300 +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 nine is -#! indecomposable (with a high probability). -#! We use the tensor product to generate another representation -#! fortyone, that is hopefully decomposable, and -#! inspect the two resulting embeddings iota and -#! kappa. - -#! @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 fortyone with those of -#! S, we see -#! that after decomposing once, we have separated one small -#! matrix on the diagonal: A $3\times 3$-matrix from S(kq.a), -#! a $3 \times 1$-matrix from S(kq.b) and a $1\times 1$-matrix -#! from S(kq.c). This matches with the source of the -#! embedding iota. - -#! @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, kappa. The iteration of -#! WeakDirectSumDecomposition will continue then -#! with Source( kappa ). 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 +#! @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 nine is +#! indecomposable (with a high probability). +#! We use the tensor product to generate another representation +#! fortyone, that is hopefully decomposable, and +#! inspect the two resulting embeddings iota and +#! kappa. + +#! @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 fortyone with those of +#! S, we see +#! that after decomposing once, we have separated one small +#! matrix on the diagonal: A $3\times 3$-matrix from S(kq.a), +#! a $3 \times 1$-matrix from S(kq.b) and a $1\times 1$-matrix +#! from S(kq.c). This matches with the source of the +#! embedding iota. + +#! @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, kappa. The iteration of +#! WeakDirectSumDecomposition will continue then +#! with Source( kappa ). 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 From f2e72d50ece66223257efb6741dad41c0e443371 Mon Sep 17 00:00:00 2001 From: Mohamed Barakat Date: Sat, 17 Oct 2020 17:57:43 +0200 Subject: [PATCH 6/6] bumped Tibor's version --- PackageInfo.g | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/PackageInfo.g b/PackageInfo.g index f990e74..8eb3002 100644 --- a/PackageInfo.g +++ b/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 ] ),