\begin{patch}{RepA6PagePatch1} \begin{paste}{RepA6PageFull1}{RepA6PageEmpty1} \pastebutton{RepA6PageFull1}{\hidepaste} \tab{5}\spadcommand{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]]\bound{genA6 }} \indentrel{3}\begin{verbatim} (1) [(1 2 3),(2 3 4 5 6)] Type: List Permutation Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty1} \begin{paste}{RepA6PageEmpty1}{RepA6PagePatch1} \pastebutton{RepA6PageEmpty1}{\showpaste} \tab{5}\spadcommand{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]]\bound{genA6 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch2} \begin{paste}{RepA6PageFull2}{RepA6PageEmpty2} \pastebutton{RepA6PageFull2}{\hidepaste} \tab{5}\spadcommand{pRA6 := permutationRepresentation (genA6, 6)\bound{pRA6 }\free{genA6 }} \indentrel{3}\begin{verbatim} Ú0 0 1 0 0 0¿ Ú1 0 0 0 0 0¿ ³ ³ ³ ³ ³1 0 0 0 0 0³ ³0 0 0 0 0 1³ ³ ³ ³ ³ ³0 1 0 0 0 0³ ³0 1 0 0 0 0³ (2) [³ ³,³ ³] ³0 0 0 1 0 0³ ³0 0 1 0 0 0³ ³ ³ ³ ³ ³0 0 0 0 1 0³ ³0 0 0 1 0 0³ ³ ³ ³ ³ À0 0 0 0 0 1Ù À0 0 0 0 1 0Ù Type: List Matrix Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty2} \begin{paste}{RepA6PageEmpty2}{RepA6PagePatch2} \pastebutton{RepA6PageEmpty2}{\showpaste} \tab{5}\spadcommand{pRA6 := permutationRepresentation (genA6, 6)\bound{pRA6 }\free{genA6 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch3} \begin{paste}{RepA6PageFull3}{RepA6PageEmpty3} \pastebutton{RepA6PageFull3}{\hidepaste} \tab{5}\spadcommand{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2))\free{pRA6 }\bound{sp0 }} \indentrel{3}\begin{verbatim} Fingerprint element in generated algebra is singular A proper cyclic submodule is found. Transition matrix computed The inverse of the transition matrix computed Now transform the matrices Ú0 0 1 0 0¿ Ú1 0 0 0 0¿ ³ ³ ³ ³ ³1 0 0 0 0³ ³1 1 1 1 1³ ³ ³ ³ ³ (3) [[³0 1 0 0 0³,³0 1 0 0 0³],[[1],[1]]] ³ ³ ³ ³ ³0 0 0 1 0³ ³0 0 1 0 0³ ³ ³ ³ ³ À0 0 0 0 1Ù À0 0 0 1 0Ù Type: List List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty3} \begin{paste}{RepA6PageEmpty3}{RepA6PagePatch3} \pastebutton{RepA6PageEmpty3}{\showpaste} \tab{5}\spadcommand{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2))\free{pRA6 }\bound{sp0 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch4} \begin{paste}{RepA6PageFull4}{RepA6PageEmpty4} \pastebutton{RepA6PageFull4}{\hidepaste} \tab{5}\spadcommand{sp1 := meatAxe sp0.1\bound{sp1 }} \indentrel{3}\begin{verbatim} Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper We know that all the cyclic submodules generated by a ll non-trivial element of the singular matrix under vi ew are not proper, hence Norton's irreducibility test can be done: A proper cyclic submodule is found. Transition matrix computed The inverse of the transition matrix computed Now transform the matrices Representation is not irreducible and it will be spli t: Ú0 1 0 0¿ Ú0 1 1 1¿ ³ ³ ³ ³ ³0 0 1 0³ ³1 1 0 1³ (4) [[[1],[1]],[³ ³,³ ³]] ³1 0 0 0³ ³1 1 1 0³ ³ ³ ³ ³ À0 0 0 1Ù À1 1 1 1Ù Type: List List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty4} \begin{paste}{RepA6PageEmpty4}{RepA6PagePatch4} \pastebutton{RepA6PageEmpty4}{\showpaste} \tab{5}\spadcommand{sp1 := meatAxe sp0.1\bound{sp1 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch5} \begin{paste}{RepA6PageFull5}{RepA6PageEmpty5} \pastebutton{RepA6PageFull5}{\hidepaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp1.2} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra has one-dimensional kernel We know that all the cyclic submodules generated by a ll non-trivial element of the singular matrix under vi ew are not proper, hence Norton's irreducibility test can be done: The generated cyclic submodule was not proper Representation is absolutely irreducible (5) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty5} \begin{paste}{RepA6PageEmpty5}{RepA6PagePatch5} \pastebutton{RepA6PageEmpty5}{\showpaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp1.2} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch6} \begin{paste}{RepA6PageFull6}{RepA6PageEmpty6} \pastebutton{RepA6PageFull6}{\hidepaste} \tab{5}\spadcommand{d2211 := irreducibleRepresentation ([2,2,1,1],genA6)\bound{d2211 }} \indentrel{3}\begin{verbatim} (6) Ú1 0 0 - 1 1 0 0 0 0 ¿ ³ ³ ³0 1 0 1 0 1 0 0 0 ³ ³ ³ ³0 0 1 0 1 - 1 0 0 0 ³ ³ ³ ³0 0 0 - 1 0 0 - 1 0 0 ³ ³ ³ [³0 0 0 0 - 1 0 0 - 1 0 ³, ³ ³ ³0 0 0 0 0 - 1 0 0 - 1³ ³ ³ ³0 0 0 1 0 0 0 0 0 ³ ³ ³ ³0 0 0 0 1 0 0 0 0 ³ ³ ³ À0 0 0 0 0 1 0 0 0 Ù Ú 0 0 1 0 0 0 1 0 0¿ ³ ³ ³ 0 0 0 0 1 0 - 1 0 0³ ³ ³ ³ 0 0 0 0 0 1 1 0 0³ ³ ³ ³ 0 0 0 0 0 0 1 1 0³ ³ ³ ³ 0 0 0 0 0 0 - 1 0 1³] ³ ³ ³ 0 0 0 0 0 0 1 0 0³ ³ ³ ³- 1 0 0 0 0 0 - 1 0 0³ ³ ³ ³ 0 - 1 0 0 0 0 1 0 0³ ³ ³ À 0 0 0 - 1 0 0 - 1 0 0Ù Type: List Matrix Integer \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty6} \begin{paste}{RepA6PageEmpty6}{RepA6PagePatch6} \pastebutton{RepA6PageEmpty6}{\showpaste} \tab{5}\spadcommand{d2211 := irreducibleRepresentation ([2,2,1,1],genA6)\bound{d2211 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch7} \begin{paste}{RepA6PageFull7}{RepA6PageEmpty7} \pastebutton{RepA6PageFull7}{\hidepaste} \tab{5}\spadcommand{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2\free{d2211 }\bound{sp2 }} \indentrel{3}\begin{verbatim} Fingerprint element in generated algebra is singular A proper cyclic submodule is found. Transition matrix computed The inverse of the transition matrix computed Now transform the matrices (7) Ú1 0 1 1¿ Ú0 0 1 0¿ ³ ³ ³ ³ ³0 1 0 1³ ³1 1 1 1³ [[³ ³,³ ³], ³1 1 0 0³ ³1 0 1 1³ ³ ³ ³ ³ À0 1 0 0Ù À0 1 0 1Ù Ú1 0 0 0 0¿ Ú1 1 1 0 0¿ ³ ³ ³ ³ ³0 1 1 1 1³ ³0 0 1 1 1³ ³ ³ ³ ³ [³0 1 1 0 0³,³1 0 0 1 0³]] ³ ³ ³ ³ ³0 1 0 1 0³ ³0 0 1 0 1³ ³ ³ ³ ³ À0 1 1 1 0Ù À1 0 0 1 1Ù Type: List List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty7} \begin{paste}{RepA6PageEmpty7}{RepA6PagePatch7} \pastebutton{RepA6PageEmpty7}{\showpaste} \tab{5}\spadcommand{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2\free{d2211 }\bound{sp2 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch8} \begin{paste}{RepA6PageFull8}{RepA6PageEmpty8} \pastebutton{RepA6PageFull8}{\hidepaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp2.1} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra has one-dimensional kernel We know that all the cyclic submodules generated by a ll non-trivial element of the singular matrix under vi ew are not proper, hence Norton's irreducibility test can be done: The generated cyclic submodule was not proper Representation is absolutely irreducible (8) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty8} \begin{paste}{RepA6PageEmpty8}{RepA6PagePatch8} \pastebutton{RepA6PageEmpty8}{\showpaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp2.1} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch9} \begin{paste}{RepA6PageFull9}{RepA6PageEmpty9} \pastebutton{RepA6PageFull9}{\hidepaste} \tab{5}\spadcommand{areEquivalent? (sp1.2, sp2.1)} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra has one-dimensional kernel There is no isomorphism, as the only possible one fails to do the necessary base change Representations are not equivalent. (9) [0] Type: Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty9} \begin{paste}{RepA6PageEmpty9}{RepA6PagePatch9} \pastebutton{RepA6PageEmpty9}{\showpaste} \tab{5}\spadcommand{areEquivalent? (sp1.2, sp2.1)} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch10} \begin{paste}{RepA6PageFull10}{RepA6PageEmpty10} \pastebutton{RepA6PageFull10}{\hidepaste} \tab{5}\spadcommand{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16\bound{dA6d16 }} \indentrel{3}\begin{verbatim} Fingerprint element in generated algebra is non-singula r Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper Fingerprint element in generated algebra is non-singula r Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper We know that all the cyclic submodules generated by a ll non-trivial element of the singular matrix under vi ew are not proper, hence Norton's irreducibility test can be done: The generated cyclic submodule was not proper Representation is irreducible, but we don't know whether it is absolutely irreducible (10) [ Ú0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0¿ ³ ³ ³0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0³ ³ ³ ³1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0³ [³ ³, ³0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0³ ³ ³ ³0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1³ ³ ³ ³0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0³ ³ ³ À0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0Ù Ú0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0¿ ³ ³ ³0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1³ ³ ³ ³0 0 0 0 1 1 1 0 1 1 1 0 1 1 1 0³ ³ ³ ³0 0 0 0 0 1 1 1 0 1 1 1 0 1 1 1³ ³ ³ ³0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0³ ³ ³ ³0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1³ ³ ³ ³1 1 1 0 1 1 1 0 1 1 1 0 1 1 1 0³ ³ ³ ³0 1 1 1 0 1 1 1 0 1 1 1 0 1 1 1³ ³ ³] ³0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0³ ³ ³ ³0 1 0 1 0 0 0 0 0 1 0 1 0 1 0 1³ ³ ³ ³1 1 1 0 0 0 0 0 1 1 1 0 1 1 1 0³ ³ ³ ³0 1 1 1 0 0 0 0 0 1 1 1 0 1 1 1³ ³ ³ ³0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0³ ³ ³ ³0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1³ ³ ³ ³1 1 1 0 1 1 1 0 0 0 0 0 1 1 1 0³ ³ ³ À0 1 1 1 0 1 1 1 0 0 0 0 0 1 1 1Ù ] Type: List List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty10} \begin{paste}{RepA6PageEmpty10}{RepA6PagePatch10} \pastebutton{RepA6PageEmpty10}{\showpaste} \tab{5}\spadcommand{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16\bound{dA6d16 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch11} \begin{paste}{RepA6PageFull11}{RepA6PageEmpty11} \pastebutton{RepA6PageFull11}{\hidepaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? dA6d16} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel We have not found a one-dimensional kernel so far, as we do a random search you could try again (11) false Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty11} \begin{paste}{RepA6PageEmpty11}{RepA6PagePatch11} \pastebutton{RepA6PageEmpty11}{\showpaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? dA6d16} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch12} \begin{paste}{RepA6PageFull12}{RepA6PageEmpty12} \pastebutton{RepA6PageFull12}{\hidepaste} \tab{5}\spadcommand{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2)))\bound{sp3 }} \indentrel{3}\begin{verbatim} Fingerprint element in generated algebra is non-singula r Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper Fingerprint element in generated algebra is non-singula r Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper The generated cyclic submodule was not proper Fingerprint element in generated algebra is singular The generated cyclic submodule was not proper The generated cyclic submodule was not proper A proper cyclic submodule is found. Transition matrix computed The inverse of the transition matrix computed Now transform the matrices (12) [ [ [[%A,%A + 1,0,%A,1,%A + 1,0,0], [0,0,%A,%A + 1,%A,%A,0,0], [%A,%A + 1,%A,1,%A + 1,0,0,0], [%A,%A + 1,%A,1,%A,0,0,0], [%A + 1,1,1,1,0,0,%A + 1,%A], [0,0,%A + 1,1,0,0,%A,0], [1,0,1,1,0,0,0,0], [1,1,0,0,0,0,0,0]] , [[1,0,%A,0,1,1,%A,%A + 1], [1,%A + 1,0,0,0,%A + 1,1,%A + 1], [%A,1,%A + 1,%A + 1,%A + 1,1,%A,0], [%A + 1,%A + 1,0,0,1,%A + 1,1,1], [1,0,%A + 1,0,1,1,%A,%A], [0,0,%A + 1,%A + 1,%A + 1,1,1,%A], [0,0,1,0,0,1,0,1], [0,%A,0,%A,1,%A + 1,%A + 1,%A]] ] , Ú0 1 1 %A + 1 0 0 0 0¿ ³ ³ ³1 1 %A + 1 0 0 0 0 0³ ³ ³ ³%A 0 0 0 0 0 0 0³ ³ ³ ³1 %A 0 0 0 0 0 0³ [³ ³, ³%A %A + 1 1 1 1 0 1 1³ ³ ³ ³0 0 %A 1 0 1 0 1³ ³ ³ ³%A 1 0 1 1 1 0 0³ ³ ³ À1 %A %A + 1 %A 0 1 0 0Ù [[%A + 1,1,%A,0,0,%A + 1,0,1], [0,%A,1,1,1,0,%A + 1,%A], [0,%A + 1,0,%A + 1,%A + 1,1,%A + 1,%A], [1,%A + 1,1,%A + 1,0,0,%A + 1,1], [0,%A,0,%A + 1,%A + 1,0,0,%A + 1], [%A + 1,0,%A + 1,%A,0,%A + 1,0,%A + 1], [0,1,0,1,%A + 1,0,%A + 1,%A + 1], [%A,%A,%A,1,%A,%A,1,%A + 1]] ] ] Type: List List Matrix FiniteField(2,2) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty12} \begin{paste}{RepA6PageEmpty12}{RepA6PagePatch12} \pastebutton{RepA6PageEmpty12}{\showpaste} \tab{5}\spadcommand{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2)))\bound{sp3 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch13} \begin{paste}{RepA6PageFull13}{RepA6PageEmpty13} \pastebutton{RepA6PageFull13}{\hidepaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.1} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra has one-dimensional kernel We know that all the cyclic submodules generated by a ll non-trivial element of the singular matrix under vi ew are not proper, hence Norton's irreducibility test can be done: The generated cyclic submodule was not proper Representation is absolutely irreducible (13) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty13} \begin{paste}{RepA6PageEmpty13}{RepA6PagePatch13} \pastebutton{RepA6PageEmpty13}{\showpaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.1} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch14} \begin{paste}{RepA6PageFull14}{RepA6PageEmpty14} \pastebutton{RepA6PageFull14}{\hidepaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.2} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra has one-dimensional kernel We know that all the cyclic submodules generated by a ll non-trivial element of the singular matrix under vi ew are not proper, hence Norton's irreducibility test can be done: The generated cyclic submodule was not proper Representation is absolutely irreducible (14) true Type: Boolean \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty14} \begin{paste}{RepA6PageEmpty14}{RepA6PagePatch14} \pastebutton{RepA6PageEmpty14}{\showpaste} \tab{5}\spadcommand{isAbsolutelyIrreducible? sp3.2} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch15} \begin{paste}{RepA6PageFull15}{RepA6PageEmpty15} \pastebutton{RepA6PageFull15}{\hidepaste} \tab{5}\spadcommand{areEquivalent? (sp3.1,sp3.2)} \indentrel{3}\begin{verbatim} Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra does not have a one-dimensional kernel Random element in generated algebra has one-dimensional kernel There is no isomorphism, as the only possible one fails to do the necessary base change Representations are not equivalent. (15) [0] Type: Matrix FiniteField(2,2) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty15} \begin{paste}{RepA6PageEmpty15}{RepA6PagePatch15} \pastebutton{RepA6PageEmpty15}{\showpaste} \tab{5}\spadcommand{areEquivalent? (sp3.1,sp3.2)} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch16} \begin{paste}{RepA6PageFull16}{RepA6PageEmpty16} \pastebutton{RepA6PageFull16}{\hidepaste} \tab{5}\spadcommand{sp0.2\free{sp0 }} \indentrel{3}\begin{verbatim} (16) [[1],[1]] Type: List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty16} \begin{paste}{RepA6PageEmpty16}{RepA6PagePatch16} \pastebutton{RepA6PageEmpty16}{\showpaste} \tab{5}\spadcommand{sp0.2\free{sp0 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch17} \begin{paste}{RepA6PageFull17}{RepA6PageEmpty17} \pastebutton{RepA6PageFull17}{\hidepaste} \tab{5}\spadcommand{sp1.2\free{sp1 }} \indentrel{3}\begin{verbatim} Ú0 1 0 0¿ Ú0 1 1 1¿ ³ ³ ³ ³ ³0 0 1 0³ ³1 1 0 1³ (17) [³ ³,³ ³] ³1 0 0 0³ ³1 1 1 0³ ³ ³ ³ ³ À0 0 0 1Ù À1 1 1 1Ù Type: List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty17} \begin{paste}{RepA6PageEmpty17}{RepA6PagePatch17} \pastebutton{RepA6PageEmpty17}{\showpaste} \tab{5}\spadcommand{sp1.2\free{sp1 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch18} \begin{paste}{RepA6PageFull18}{RepA6PageEmpty18} \pastebutton{RepA6PageFull18}{\hidepaste} \tab{5}\spadcommand{sp2.1\free{sp2 }} \indentrel{3}\begin{verbatim} Ú1 0 1 1¿ Ú0 0 1 0¿ ³ ³ ³ ³ ³0 1 0 1³ ³1 1 1 1³ (18) [³ ³,³ ³] ³1 1 0 0³ ³1 0 1 1³ ³ ³ ³ ³ À0 1 0 0Ù À0 1 0 1Ù Type: List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty18} \begin{paste}{RepA6PageEmpty18}{RepA6PagePatch18} \pastebutton{RepA6PageEmpty18}{\showpaste} \tab{5}\spadcommand{sp2.1\free{sp2 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch19} \begin{paste}{RepA6PageFull19}{RepA6PageEmpty19} \pastebutton{RepA6PageFull19}{\hidepaste} \tab{5}\spadcommand{sp3.1\free{sp3 }} \indentrel{3}\begin{verbatim} (19) [ [[%A,%A + 1,0,%A,1,%A + 1,0,0], [0,0,%A,%A + 1,%A,%A,0,0], [%A,%A + 1,%A,1,%A + 1,0,0,0], [%A,%A + 1,%A,1,%A,0,0,0], [%A + 1,1,1,1,0,0,%A + 1,%A], [0,0,%A + 1,1,0,0,%A,0], [1,0,1,1,0,0,0,0], [1,1,0,0,0,0,0,0]] , [[1,0,%A,0,1,1,%A,%A + 1], [1,%A + 1,0,0,0,%A + 1,1,%A + 1], [%A,1,%A + 1,%A + 1,%A + 1,1,%A,0], [%A + 1,%A + 1,0,0,1,%A + 1,1,1], [1,0,%A + 1,0,1,1,%A,%A], [0,0,%A + 1,%A + 1,%A + 1,1,1,%A], [0,0,1,0,0,1,0,1], [0,%A,0,%A,1,%A + 1,%A + 1,%A]] ] Type: List Matrix FiniteField(2,2) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty19} \begin{paste}{RepA6PageEmpty19}{RepA6PagePatch19} \pastebutton{RepA6PageEmpty19}{\showpaste} \tab{5}\spadcommand{sp3.1\free{sp3 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch20} \begin{paste}{RepA6PageFull20}{RepA6PageEmpty20} \pastebutton{RepA6PageFull20}{\hidepaste} \tab{5}\spadcommand{sp3.2\free{sp3 }} \indentrel{3}\begin{verbatim} (20) Ú0 1 1 %A + 1 0 0 0 0¿ ³ ³ ³1 1 %A + 1 0 0 0 0 0³ ³ ³ ³%A 0 0 0 0 0 0 0³ ³ ³ ³1 %A 0 0 0 0 0 0³ [³ ³, ³%A %A + 1 1 1 1 0 1 1³ ³ ³ ³0 0 %A 1 0 1 0 1³ ³ ³ ³%A 1 0 1 1 1 0 0³ ³ ³ À1 %A %A + 1 %A 0 1 0 0Ù [[%A + 1,1,%A,0,0,%A + 1,0,1], [0,%A,1,1,1,0,%A + 1,%A], [0,%A + 1,0,%A + 1,%A + 1,1,%A + 1,%A], [1,%A + 1,1,%A + 1,0,0,%A + 1,1], [0,%A,0,%A + 1,%A + 1,0,0,%A + 1], [%A + 1,0,%A + 1,%A,0,%A + 1,0,%A + 1], [0,1,0,1,%A + 1,0,%A + 1,%A + 1], [%A,%A,%A,1,%A,%A,1,%A + 1]] ] Type: List Matrix FiniteField(2,2) \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty20} \begin{paste}{RepA6PageEmpty20}{RepA6PagePatch20} \pastebutton{RepA6PageEmpty20}{\showpaste} \tab{5}\spadcommand{sp3.2\free{sp3 }} \end{paste}\end{patch} \begin{patch}{RepA6PagePatch21} \begin{paste}{RepA6PageFull21}{RepA6PageEmpty21} \pastebutton{RepA6PageFull21}{\hidepaste} \tab{5}\spadcommand{dA6d16\free{dA6d16 }} \indentrel{3}\begin{verbatim} (21) Ú0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0¿ ³ ³ ³0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0³ [³ ³, ³1 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0³ ³ ³ ³0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1³ ³ ³ ³0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1³ ³ ³ ³0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0³ ³ ³ À0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0Ù Ú0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0¿ ³ ³ ³0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1³ ³ ³ ³0 0 0 0 1 0 1 1 1 0 1 1 1 0 1 1³ ³ ³ ³0 0 0 0 0 1 0 1 0 1 0 1 0 1 0 1³ ³ ³ ³0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0³ ³ ³ ³1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1³ ³ ³ ³1 0 1 1 1 0 1 1 0 0 0 0 1 0 1 1³ ³ ³ ³0 1 0 1 0 1 0 1 0 0 0 0 0 1 0 1³ ³ ³] ³0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0³ ³ ³ ³1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0³ ³ ³ ³1 0 1 1 1 0 1 1 1 0 1 1 0 0 0 0³ ³ ³ ³0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0³ ³ ³ ³0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0³ ³ ³ ³1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1³ ³ ³ ³1 0 1 1 1 0 1 1 1 0 1 1 1 0 1 1³ ³ ³ À0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1Ù Type: List Matrix PrimeField 2 \end{verbatim} \indentrel{-3}\end{paste}\end{patch} \begin{patch}{RepA6PageEmpty21} \begin{paste}{RepA6PageEmpty21}{RepA6PagePatch21} \pastebutton{RepA6PageEmpty21}{\showpaste} \tab{5}\spadcommand{dA6d16\free{dA6d16 }} \end{paste}\end{patch}