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authordos-reis <gdr@axiomatics.org>2007-08-14 05:14:52 +0000
committerdos-reis <gdr@axiomatics.org>2007-08-14 05:14:52 +0000
commitab8cc85adde879fb963c94d15675783f2cf4b183 (patch)
treec202482327f474583b750b2c45dedfc4e4312b1d /src/hyper/pages/grpthry.pht
downloadopen-axiom-ab8cc85adde879fb963c94d15675783f2cf4b183.tar.gz
Initial population.
Diffstat (limited to 'src/hyper/pages/grpthry.pht')
-rw-r--r--src/hyper/pages/grpthry.pht866
1 files changed, 866 insertions, 0 deletions
diff --git a/src/hyper/pages/grpthry.pht b/src/hyper/pages/grpthry.pht
new file mode 100644
index 00000000..b04349ef
--- /dev/null
+++ b/src/hyper/pages/grpthry.pht
@@ -0,0 +1,866 @@
+\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}
+
generated by cgit v1.2.3 (git 2.39.1) at 2024-07-06 20:01:19 +0000