$SPAD/src/input repa6.input

Holger Gollan, Johannes Grabmeier, and Timothy Daly

December 3, 2016

Abstract

Contents

2

1 License

— license —

--Copyright The Numerical Algorithms Group Limited 1991.

———-

— * —

)set break resume
)spool repa6.output
)set message test on
)set message auto off
)clear all

———-

This file demonstrates Representation Theory in Scratchpad using the packages REP1, REP2, IRSN and SGCF, which are the abbreviations for RepresentationPackage1, RepresentationPackage2 IrrRepSymNatPackage and SymmetricGroupCombinatoricFunctions.

authors: Holger Gollan, Johannes Grabmeier release 1.0 09/30/87 release 2.0 11/10/88: J. Grabmeier: add functions from IRSN release 2.1 08/04/89: J. Grabmeier: adjusting to new PERM and modified REP1 release 2.2 06/05/89: J. Grabmeier: adjusting to new algebra release 2.3 08/20/89: J. Grabmeier: minor adjustments

In the sequel we show how to get all 2-modular irreducible representations of the alternating group A6.

We generate A6 by the permutations threecycle x=(1,2,3) and the 5-cycle y=(2,3,4,5,6) — * —

--S 1 of 33
genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
--R
--R
--R   (1)  [(1 2 3),(2 3 4 5 6)]
--R                                             Type: List(Permutation(Integer))
--E 1

———-

pRA6 is the permutation representation over the Integers...

— * —

--S 2 of 33
pRA6 := permutationRepresentation (genA6, 6)
--R
--R
--R         +0  0  1  0  0  0+ +1  0  0  0  0  0+
--R         |                | |                |

--R         |1  0  0  0  0  0| |0  0  0  0  0  1|
--R         |                | |                |
--R         |0  1  0  0  0  0| |0  1  0  0  0  0|
--R   (2)  [|                |,|                |]
--R         |0  0  0  1  0  0| |0  0  1  0  0  0|
--R         |                | |                |
--R         |0  0  0  0  1  0| |0  0  0  1  0  0|
--R         |                | |                |
--R         +0  0  0  0  0  1+ +0  0  0  0  1  0+
--R                                                  Type: List(Matrix(Integer))
--E 2

———-

And pRA6m2 is the permutation representation over PrimeField 2:

— * —

--S 3 of 33
pRA6m2 : List Matrix PrimeField 2 := pRA6
--R
--R
--R         +0  0  1  0  0  0+ +1  0  0  0  0  0+
--R         |                | |                |
--R         |1  0  0  0  0  0| |0  0  0  0  0  1|
--R         |                | |                |
--R         |0  1  0  0  0  0| |0  1  0  0  0  0|
--R   (3)  [|                |,|                |]
--R         |0  0  0  1  0  0| |0  0  1  0  0  0|
--R         |                | |                |
--R         |0  0  0  0  1  0| |0  0  0  1  0  0|
--R         |                | |                |
--R         +0  0  0  0  0  1+ +0  0  0  0  1  0+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 3

———-

Now try to split pRA6m2:

— * —

--S 4 of 33
sp0 := meatAxe pRA6m2
--R
--R   Fingerprint element in generated algebra is singular
--R     A proper cyclic submodule is found.
--R     Transition matrix computed
--R     The inverse of the transition matrix computed
--R     Now transform the matrices
--R
--R          +0  0  1  0  0+ +1  0  0  0  0+
--R          |             | |             |

--R          |1  0  0  0  0| |1  1  1  1  1|
--R          |             | |             |
--R   (4)  [[|0  1  0  0  0|,|0  1  0  0  0|],[[1],[1]]]
--R          |             | |             |
--R          |0  0  0  1  0| |0  0  1  0  0|
--R          |             | |             |
--R          +0  0  0  0  1+ +0  0  0  1  0+
--R                                      Type: List(List(Matrix(PrimeField(2))))
--E 4

———-

We have found the trivial module as a factormodule and a 5-dimensional submodule. — * —

--S 5 of 33
dA6d1 := sp0.2
--R
--R
--R   (5)  [[1],[1]]
--R                                            Type: List(Matrix(PrimeField(2)))
--E 5

———-

Try to split again...

— * —

--S 6 of 33
sp1 := meatAxe sp0.1
--R
--R   Fingerprint element in generated algebra is singular
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     We know that all the cyclic submodules generated by all
--R       non-trivial element of the singular matrix under view are
--R       not proper, hence Norton's irreducibility test can be done:
--R     A proper cyclic submodule is found.
--R     Transition matrix computed
--R     The inverse of the transition matrix computed
--R     Now transform the matrices
--R     Representation is not irreducible and it will be split:
--R
--R                    +0  1  0  0+ +0  1  1  1+
--R                    |          | |          |
--R                    |0  0  1  0| |1  1  0  1|
--R   (6)  [[[1],[1]],[|          |,|          |]]
--R                    |1  0  0  0| |1  1  1  0|
--R                    |          | |          |
--R                    +0  0  0  1+ +1  1  1  1+

--R                                      Type: List(List(Matrix(PrimeField(2))))
--E 6

———-

And find a 4-dimensional submodule, say dA6d4a, and the trivial one again.

— * —

--S 7 of 33
dA6d4a := sp1.2
--R
--R
--R         +0  1  0  0+ +0  1  1  1+
--R         |          | |          |
--R         |0  0  1  0| |1  1  0  1|
--R   (7)  [|          |,|          |]
--R         |1  0  0  0| |1  1  1  0|
--R         |          | |          |
--R         +0  0  0  1+ +1  1  1  1+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 7

———-

Now we want to test, whether dA6d4a is absolutely irreducible...

— * —

--S 8 of 33 random input, failure ok
isAbsolutelyIrreducible? dA6d4a
--R
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra has
--R     one-dimensional kernel
--R     We know that all the cyclic submodules generated by all
--R       non-trivial element of the singular matrix under view are
--R       not proper, hence Norton's irreducibility test can be done:
--R     The generated cyclic submodule was not proper
--R     Representation is absolutely irreducible
--R
--R   (8)  true
--R                                                                Type: Boolean
--E 8

———-

And see: dA6d4a is absolutely irreducible. So we have found a second irreducible representation.

Now construct a representation from reducing an irreducible one of the symmetric group S_{6} over the integers taken mod 2 What is the degree of the representation belonging to partition [2,2,1,1]?

— * —

-- lambda : PRTITION := partition [2,2,1,1]
--S 9 of 33
lambda := [2,2,1,1]
--R
--R
--R   (9)  [2,2,1,1]
--R                                                  Type: List(PositiveInteger)
--E 9
--S 10 of 33
dimensionOfIrreducibleRepresentation lambda
--R
--R
--R   (10)  9
--R                                                        Type: PositiveInteger
--E 10

———-

Now create the restriction to A6:

— * —

--S 11 of 33
d2211  := irreducibleRepresentation(lambda, genA6)
--R
--R
--R   (11)
--R    +1  0  0  - 1   1    0    0    0    0 + + 0    0   1   0   0  0   1   0  0+
--R    |                                     | |                                 |
--R    |0  1  0   1    0    1    0    0    0 | | 0    0   0   0   1  0  - 1  0  0|
--R    |                                     | |                                 |
--R    |0  0  1   0    1   - 1   0    0    0 | | 0    0   0   0   0  1   1   0  0|
--R    |                                     | |                                 |
--R    |0  0  0  - 1   0    0   - 1   0    0 | | 0    0   0   0   0  0   1   1  0|
--R    |                                     | |                                 |
--R   [|0  0  0   0   - 1   0    0   - 1   0 |,| 0    0   0   0   0  0  - 1  0  1|]
--R    |                                     | |                                 |
--R    |0  0  0   0    0   - 1   0    0   - 1| | 0    0   0   0   0  0   1   0  0|
--R    |                                     | |                                 |
--R    |0  0  0   1    0    0    0    0    0 | |- 1   0   0   0   0  0  - 1  0  0|
--R    |                                     | |                                 |
--R    |0  0  0   0    1    0    0    0    0 | | 0   - 1  0   0   0  0   1   0  0|
--R    |                                     | |                                 |
--R    +0  0  0   0    0    1    0    0    0 + + 0    0   0  - 1  0  0  - 1  0  0+

--R                                                  Type: List(Matrix(Integer))
--E 11

———-

And d2211m2 is the representation over PrimeField 2:

— * —

--S 12 of 33
d2211m2 : List Matrix PrimeField 2 := d2211
--R
--R
--R          +1  0  0  1  1  0  0  0  0+ +0  0  1  0  0  0  1  0  0+
--R          |                         | |                         |
--R          |0  1  0  1  0  1  0  0  0| |0  0  0  0  1  0  1  0  0|
--R          |                         | |                         |
--R          |0  0  1  0  1  1  0  0  0| |0  0  0  0  0  1  1  0  0|
--R          |                         | |                         |
--R          |0  0  0  1  0  0  1  0  0| |0  0  0  0  0  0  1  1  0|
--R          |                         | |                         |
--R   (12)  [|0  0  0  0  1  0  0  1  0|,|0  0  0  0  0  0  1  0  1|]
--R          |                         | |                         |
--R          |0  0  0  0  0  1  0  0  1| |0  0  0  0  0  0  1  0  0|
--R          |                         | |                         |
--R          |0  0  0  1  0  0  0  0  0| |1  0  0  0  0  0  1  0  0|
--R          |                         | |                         |
--R          |0  0  0  0  1  0  0  0  0| |0  1  0  0  0  0  1  0  0|
--R          |                         | |                         |
--R          +0  0  0  0  0  1  0  0  0+ +0  0  0  1  0  0  1  0  0+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 12

———-

And split it:

— * —

--S 13 of 33
sp2 := meatAxe d2211m2
--R
--R   Fingerprint element in generated algebra is singular
--R     A proper cyclic submodule is found.
--R     Transition matrix computed
--R     The inverse of the transition matrix computed
--R     Now transform the matrices
--R
--R                                       +1  0  0  0  0+ +1  1  1  0  0+
--R           +1  0  1  1+ +0  0  1  0+   |             | |             |
--R           |          | |          |   |0  1  1  1  1| |0  0  1  1  1|
--R           |0  1  0  1| |1  1  1  1|   |             | |             |
--R   (13)  [[|          |,|          |],[|0  1  1  0  0|,|1  0  0  1  0|]]

--R           |1  1  0  0| |1  0  1  1|   |             | |             |
--R           |          | |          |   |0  1  0  1  0| |0  0  1  0  1|
--R           +0  1  0  0+ +0  1  0  1+   |             | |             |
--R                                       +0  1  1  1  0+ +1  0  0  1  1+
--R                                      Type: List(List(Matrix(PrimeField(2))))
--E 13

———-

A 5 and a 4-dimensional one.

we take the 4-dimensional one, say dA6d4b: — * —

--S 14 of 33
dA6d4b := sp2.1
--R
--R
--R          +1  0  1  1+ +0  0  1  0+
--R          |          | |          |
--R          |0  1  0  1| |1  1  1  1|
--R   (14)  [|          |,|          |]
--R          |1  1  0  0| |1  0  1  1|
--R          |          | |          |
--R          +0  1  0  0+ +0  1  0  1+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 14

———-

This is absolutely irreducible, too ...

— * —

--S 15 of 33 random generation, failure ok.
isAbsolutelyIrreducible? dA6d4b
--R
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra has
--R     one-dimensional kernel
--R     We know that all the cyclic submodules generated by all
--R       non-trivial element of the singular matrix under view are
--R       not proper, hence Norton's irreducibility test can be done:
--R     The generated cyclic submodule was not proper
--R     Representation is absolutely irreducible
--R
--R   (15)  true
--R                                                                Type: Boolean
--E 15

———- And dA6d4a and dA6d4b are not equivalent:

— * —

--S 16 of 33 random generation, failure ok.
areEquivalent? ( dA6d4a , dA6d4b )
--R
--R   Dimensions of kernels differ
--R
--R   Representations are not equivalent.
--R
--R   (16)  [0]
--R                                                  Type: Matrix(PrimeField(2))
--E 16

———-

So the third irreducible representation is found.

Now construct a new representation with the help of the tensorproduct — * —

--S 17 of 33
dA6d16 := tensorProduct ( dA6d4a , dA6d4b )
--R
--R
--R   (17)
--R    +0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0+
--R    |                                              |
--R    |0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
--R   [|                                              |,
--R    |1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1|

--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0|
--R    |                                              |
--R    +0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0+
--R    +0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0+
--R    |                                              |
--R    |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
--R    |                                              |
--R    |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
--R    |                                              |
--R    |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
--R    |                                              |
--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1|
--R    |                                              |
--R    |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
--R    |                                              |]
--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
--R    |                                              |
--R    |0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0|
--R    |                                              |
--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1|
--R    |                                              |
--R    +0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 17

———-

And try to split it...

— * —

--S 18 of 33
sp3 := meatAxe dA6d16
--R
--R   Fingerprint element in generated algebra is non-singular
--R   Fingerprint element in generated algebra is singular

--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R   Fingerprint element in generated algebra is non-singular
--R   Fingerprint element in generated algebra is singular
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R   Fingerprint element in generated algebra is singular
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     We know that all the cyclic submodules generated by all
--R       non-trivial element of the singular matrix under view are
--R       not proper, hence Norton's irreducibility test can be done:
--R     The generated cyclic submodule was not proper
--R     Representation is irreducible, but we don't know
--R       whether it is absolutely irreducible
--R
--R   (18)
--R   [
--R      +0  0  0  0  0  0  0  0  1  0  1  0  0  0  0  0+
--R      |                                              |
--R      |0  0  0  0  0  0  0  0  0  1  1  1  0  0  0  0|
--R      |                                              |
--R      |0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0|
--R      |                                              |
--R      |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
--R      |                                              |
--R      |1  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R      |                                              |
--R      |0  1  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R      |                                              |
--R      |1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R      |                                              |
--R      |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R     [|                                              |,
--R      |0  0  0  0  1  0  1  0  0  0  0  0  0  0  0  0|
--R      |                                              |
--R      |0  0  0  0  0  1  1  1  0  0  0  0  0  0  0  0|
--R      |                                              |
--R      |0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0|

--R      |                                              |
--R      |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
--R      |                                              |
--R      |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  0|
--R      |                                              |
--R      |0  0  0  0  0  0  0  0  0  0  0  0  0  1  1  1|
--R      |                                              |
--R      |0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0|
--R      |                                              |
--R      +0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0+
--R      +0  0  0  0  0  1  1  0  0  1  1  0  0  1  1  0+
--R      |                                              |
--R      |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
--R      |                                              |
--R      |0  0  0  0  1  1  1  0  1  1  1  0  1  1  1  0|
--R      |                                              |
--R      |0  0  0  0  0  1  1  1  0  1  1  1  0  1  1  1|
--R      |                                              |
--R      |0  1  1  0  0  1  1  0  0  1  1  0  0  1  1  0|
--R      |                                              |
--R      |0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1|
--R      |                                              |
--R      |1  1  1  0  1  1  1  0  1  1  1  0  1  1  1  0|
--R      |                                              |
--R      |0  1  1  1  0  1  1  1  0  1  1  1  0  1  1  1|
--R      |                                              |]
--R      |0  1  1  0  0  0  0  0  0  1  1  0  0  1  1  0|
--R      |                                              |
--R      |0  1  0  1  0  0  0  0  0  1  0  1  0  1  0  1|
--R      |                                              |
--R      |1  1  1  0  0  0  0  0  1  1  1  0  1  1  1  0|
--R      |                                              |
--R      |0  1  1  1  0  0  0  0  0  1  1  1  0  1  1  1|
--R      |                                              |
--R      |0  1  1  0  0  1  1  0  0  0  0  0  0  1  1  0|
--R      |                                              |
--R      |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
--R      |                                              |
--R      |1  1  1  0  1  1  1  0  0  0  0  0  1  1  1  0|
--R      |                                              |
--R      +0  1  1  1  0  1  1  1  0  0  0  0  0  1  1  1+
--R     ]
--R                                      Type: List(List(Matrix(PrimeField(2))))
--E 18

———-

The representation is irreducible, but may be not absolutely irreducible.

— * —

--S 19 of 33

isAbsolutelyIrreducible? dA6d16
--R
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel

--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   We have not found a one-dimensional kernel so far,
--R     as we do a random search you could try again
--R
--R   (19)  false
--R                                                                Type: Boolean
--E 19

———-

So let’s try the same over the field with 4 elements:

— * —

--S 20 of 33
gf4 := FiniteField(2,2)
--R
--R
--R   (20)  FiniteField(2,2)
--R                                                                 Type: Domain
--E 20
--S 21 of 33
dA6d16gf4 : List Matrix gf4 := dA6d16
--R
--R
--R   (21)
--R    +0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0+
--R    |                                              |
--R    |0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
--R   [|                                              |,
--R    |1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|

--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0|
--R    |                                              |
--R    +0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0+
--R    +0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0+
--R    |                                              |
--R    |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
--R    |                                              |
--R    |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
--R    |                                              |
--R    |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
--R    |                                              |
--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1|
--R    |                                              |
--R    |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
--R    |                                              |]
--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
--R    |                                              |
--R    |0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0|
--R    |                                              |
--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1|
--R    |                                              |
--R    +0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1+
--R                                         Type: List(Matrix(FiniteField(2,2)))
--E 21
--S 22 of 33
sp4 := meatAxe dA6d16gf4
--R
--R   Fingerprint element in generated algebra is non-singular
--R   Fingerprint element in generated algebra is singular
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper

--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R   Fingerprint element in generated algebra is non-singular
--R   Fingerprint element in generated algebra is singular
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R   Fingerprint element in generated algebra is singular
--R     The generated cyclic submodule was not proper
--R     The generated cyclic submodule was not proper
--R     A proper cyclic submodule is found.
--R     Transition matrix computed
--R     The inverse of the transition matrix computed
--R     Now transform the matrices
--R
--R   (22)
--R   [
--R      +  %A    %A + 1    0       %A      1     %A + 1    0     0 +
--R      |                                                          |
--R      |  0       0       %A    %A + 1    %A      %A      0     0 |
--R      |                                                          |
--R      |  %A    %A + 1    %A      1     %A + 1    0       0     0 |
--R      |                                                          |
--R      |  %A    %A + 1    %A      1       %A      0       0     0 |
--R     [|                                                          |,
--R      |%A + 1    1       1       1       0       0     %A + 1  %A|
--R      |                                                          |
--R      |  0       0     %A + 1    1       0       0       %A    0 |
--R      |                                                          |
--R      |  1       0       1       1       0       0       0     0 |
--R      |                                                          |
--R      +  1       1       0       0       0       0       0     0 +
--R      +  1       0       %A      0       1       1       %A    %A + 1+
--R      |                                                              |
--R      |  1     %A + 1    0       0       0     %A + 1    1     %A + 1|
--R      |                                                              |
--R      |  %A      1     %A + 1  %A + 1  %A + 1    1       %A      0   |
--R      |                                                              |
--R      |%A + 1  %A + 1    0       0       1     %A + 1    1       1   |
--R      |                                                              |]
--R      |  1       0     %A + 1    0       1       1       %A      %A  |
--R      |                                                              |
--R      |  0       0     %A + 1  %A + 1  %A + 1    1       1       %A  |
--R      |                                                              |

--R      |  0       0       1       0       0       1       0       1   |
--R      |                                                              |
--R      +  0       %A      0       %A      1     %A + 1  %A + 1    %A  +
--R     ,
--R
--R      +0     1       1     %A + 1  0  0  0  0+
--R      |                                      |
--R      |1     1     %A + 1    0     0  0  0  0|
--R      |                                      |
--R      |%A    0       0       0     0  0  0  0|
--R      |                                      |
--R      |1     %A      0       0     0  0  0  0|
--R     [|                                      |,
--R      |%A  %A + 1    1       1     1  0  1  1|
--R      |                                      |
--R      |0     0       %A      1     0  1  0  1|
--R      |                                      |
--R      |%A    1       0       1     1  1  0  0|
--R      |                                      |
--R      +1     %A    %A + 1    %A    0  1  0  0+
--R      +%A + 1    1       %A      0       0     %A + 1    0       1   +
--R      |                                                              |
--R      |  0       %A      1       1       1       0     %A + 1    %A  |
--R      |                                                              |
--R      |  0     %A + 1    0     %A + 1  %A + 1    1     %A + 1    %A  |
--R      |                                                              |
--R      |  1     %A + 1    1     %A + 1    0       0     %A + 1    1   |
--R      |                                                              |]
--R      |  0       %A      0     %A + 1  %A + 1    0       0     %A + 1|
--R      |                                                              |
--R      |%A + 1    0     %A + 1    %A      0     %A + 1    0     %A + 1|
--R      |                                                              |
--R      |  0       1       0       1     %A + 1    0     %A + 1  %A + 1|
--R      |                                                              |
--R      +  %A      %A      %A      1       %A      %A      1     %A + 1+
--R     ]
--R                                   Type: List(List(Matrix(FiniteField(2,2))))
--E 22

———-

Now we find two 8-dimensional ones, dA6d8a and dA6d8b.

— * —

--S 23 of 33
dA6d8a : List Matrix gf4  := sp4.1
--R
--R
--R   (23)
--R    +  %A    %A + 1    0       %A      1     %A + 1    0     0 +

--R    |                                                          |
--R    |  0       0       %A    %A + 1    %A      %A      0     0 |
--R    |                                                          |
--R    |  %A    %A + 1    %A      1     %A + 1    0       0     0 |
--R    |                                                          |
--R    |  %A    %A + 1    %A      1       %A      0       0     0 |
--R   [|                                                          |,
--R    |%A + 1    1       1       1       0       0     %A + 1  %A|
--R    |                                                          |
--R    |  0       0     %A + 1    1       0       0       %A    0 |
--R    |                                                          |
--R    |  1       0       1       1       0       0       0     0 |
--R    |                                                          |
--R    +  1       1       0       0       0       0       0     0 +
--R    +  1       0       %A      0       1       1       %A    %A + 1+
--R    |                                                              |
--R    |  1     %A + 1    0       0       0     %A + 1    1     %A + 1|
--R    |                                                              |
--R    |  %A      1     %A + 1  %A + 1  %A + 1    1       %A      0   |
--R    |                                                              |
--R    |%A + 1  %A + 1    0       0       1     %A + 1    1       1   |
--R    |                                                              |]
--R    |  1       0     %A + 1    0       1       1       %A      %A  |
--R    |                                                              |
--R    |  0       0     %A + 1  %A + 1  %A + 1    1       1       %A  |
--R    |                                                              |
--R    |  0       0       1       0       0       1       0       1   |
--R    |                                                              |
--R    +  0       %A      0       %A      1     %A + 1  %A + 1    %A  +
--R                                         Type: List(Matrix(FiniteField(2,2)))
--E 23
--S 24 of 33
dA6d8b : List Matrix gf4  := sp4.2
--R
--R
--R   (24)
--R    +0     1       1     %A + 1  0  0  0  0+
--R    |                                      |
--R    |1     1     %A + 1    0     0  0  0  0|
--R    |                                      |
--R    |%A    0       0       0     0  0  0  0|
--R    |                                      |
--R    |1     %A      0       0     0  0  0  0|
--R   [|                                      |,
--R    |%A  %A + 1    1       1     1  0  1  1|
--R    |                                      |
--R    |0     0       %A      1     0  1  0  1|
--R    |                                      |
--R    |%A    1       0       1     1  1  0  0|

--R    |                                      |
--R    +1     %A    %A + 1    %A    0  1  0  0+
--R    +%A + 1    1       %A      0       0     %A + 1    0       1   +
--R    |                                                              |
--R    |  0       %A      1       1       1       0     %A + 1    %A  |
--R    |                                                              |
--R    |  0     %A + 1    0     %A + 1  %A + 1    1     %A + 1    %A  |
--R    |                                                              |
--R    |  1     %A + 1    1     %A + 1    0       0     %A + 1    1   |
--R    |                                                              |]
--R    |  0       %A      0     %A + 1  %A + 1    0       0     %A + 1|
--R    |                                                              |
--R    |%A + 1    0     %A + 1    %A      0     %A + 1    0     %A + 1|
--R    |                                                              |
--R    |  0       1       0       1     %A + 1    0     %A + 1  %A + 1|
--R    |                                                              |
--R    +  %A      %A      %A      1       %A      %A      1     %A + 1+
--R                                         Type: List(Matrix(FiniteField(2,2)))
--E 24

———-

Both are absolutely irreducible...

— * —

--S 25 of 33 random generation, failure ok.
isAbsolutelyIrreducible? dA6d8a
--R
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra has
--R     one-dimensional kernel
--R     We know that all the cyclic submodules generated by all
--R       non-trivial element of the singular matrix under view are

--R       not proper, hence Norton's irreducibility test can be done:
--R     The generated cyclic submodule was not proper
--R     Representation is absolutely irreducible
--R
--R   (25)  true
--R                                                                Type: Boolean
--E 25
--S 26 of 33 random generation, failure ok.
isAbsolutelyIrreducible? dA6d8b
--R
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra has
--R     one-dimensional kernel
--R     We know that all the cyclic submodules generated by all
--R       non-trivial element of the singular matrix under view are
--R       not proper, hence Norton's irreducibility test can be done:
--R     The generated cyclic submodule was not proper
--R     Representation is absolutely irreducible
--R
--R   (26)  true
--R                                                                Type: Boolean
--E 26

———-

And they are not equivalent...

— * —

--S 27 of 33 random generation, failure ok.
areEquivalent? ( dA6d8a, dA6d8b )
--R
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra does
--R     not have a one-dimensional kernel
--R   Random element in generated algebra has

--R     one-dimensional kernel
--R     There is no isomorphism, as the only possible one
--R       fails to do the necessary base change
--R
--R   Representations are not equivalent.
--R
--R   (27)  [0]
--R                                               Type: Matrix(FiniteField(2,2))
--E 27

———-

So we have found five absolutely irreducible representations of A6 in characteristic 2.

The theory tells us that there are no more irreducible ones. Here again are all absolutely irreducible 2-modular representations of A6

— * —

--S 28 of 33
dA6d1
--R
--R
--R   (28)  [[1],[1]]
--R                                            Type: List(Matrix(PrimeField(2)))
--E 28
--S 29 of 33
dA6d4a
--R
--R
--R          +0  1  0  0+ +0  1  1  1+
--R          |          | |          |
--R          |0  0  1  0| |1  1  0  1|
--R   (29)  [|          |,|          |]
--R          |1  0  0  0| |1  1  1  0|
--R          |          | |          |
--R          +0  0  0  1+ +1  1  1  1+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 29
--S 30 of 33
dA6d4b
--R
--R
--R          +1  0  1  1+ +0  0  1  0+
--R          |          | |          |
--R          |0  1  0  1| |1  1  1  1|
--R   (30)  [|          |,|          |]
--R          |1  1  0  0| |1  0  1  1|
--R          |          | |          |
--R          +0  1  0  0+ +0  1  0  1+

--R                                            Type: List(Matrix(PrimeField(2)))
--E 30
--S 31 of 33
dA6d8a
--R
--R
--R   (31)
--R    +  %A    %A + 1    0       %A      1     %A + 1    0     0 +
--R    |                                                          |
--R    |  0       0       %A    %A + 1    %A      %A      0     0 |
--R    |                                                          |
--R    |  %A    %A + 1    %A      1     %A + 1    0       0     0 |
--R    |                                                          |
--R    |  %A    %A + 1    %A      1       %A      0       0     0 |
--R   [|                                                          |,
--R    |%A + 1    1       1       1       0       0     %A + 1  %A|
--R    |                                                          |
--R    |  0       0     %A + 1    1       0       0       %A    0 |
--R    |                                                          |
--R    |  1       0       1       1       0       0       0     0 |
--R    |                                                          |
--R    +  1       1       0       0       0       0       0     0 +
--R    +  1       0       %A      0       1       1       %A    %A + 1+
--R    |                                                              |
--R    |  1     %A + 1    0       0       0     %A + 1    1     %A + 1|
--R    |                                                              |
--R    |  %A      1     %A + 1  %A + 1  %A + 1    1       %A      0   |
--R    |                                                              |
--R    |%A + 1  %A + 1    0       0       1     %A + 1    1       1   |
--R    |                                                              |]
--R    |  1       0     %A + 1    0       1       1       %A      %A  |
--R    |                                                              |
--R    |  0       0     %A + 1  %A + 1  %A + 1    1       1       %A  |
--R    |                                                              |
--R    |  0       0       1       0       0       1       0       1   |
--R    |                                                              |
--R    +  0       %A      0       %A      1     %A + 1  %A + 1    %A  +
--R                                         Type: List(Matrix(FiniteField(2,2)))
--E 31
--S 32 of 33
dA6d8b
--R
--R
--R   (32)
--R    +0     1       1     %A + 1  0  0  0  0+
--R    |                                      |
--R    |1     1     %A + 1    0     0  0  0  0|
--R    |                                      |

--R    |%A    0       0       0     0  0  0  0|
--R    |                                      |
--R    |1     %A      0       0     0  0  0  0|
--R   [|                                      |,
--R    |%A  %A + 1    1       1     1  0  1  1|
--R    |                                      |
--R    |0     0       %A      1     0  1  0  1|
--R    |                                      |
--R    |%A    1       0       1     1  1  0  0|
--R    |                                      |
--R    +1     %A    %A + 1    %A    0  1  0  0+
--R    +%A + 1    1       %A      0       0     %A + 1    0       1   +
--R    |                                                              |
--R    |  0       %A      1       1       1       0     %A + 1    %A  |
--R    |                                                              |
--R    |  0     %A + 1    0     %A + 1  %A + 1    1     %A + 1    %A  |
--R    |                                                              |
--R    |  1     %A + 1    1     %A + 1    0       0     %A + 1    1   |
--R    |                                                              |]
--R    |  0       %A      0     %A + 1  %A + 1    0       0     %A + 1|
--R    |                                                              |
--R    |%A + 1    0     %A + 1    %A      0     %A + 1    0     %A + 1|
--R    |                                                              |
--R    |  0       1       0       1     %A + 1    0     %A + 1  %A + 1|
--R    |                                                              |
--R    +  %A      %A      %A      1       %A      %A      1     %A + 1+
--R                                         Type: List(Matrix(FiniteField(2,2)))
--E 32

———-

And here again is the irreducible, but not absolutely irreducible representations of A6 over PrimeField 2

— * —

--S 33 of 33
dA6d16
--R
--R
--R   (33)
--R    +0  0  0  0  1  0  1  1  0  0  0  0  0  0  0  0+
--R    |                                              |
--R    |0  0  0  0  0  1  0  1  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  1  1  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  1  0  1  1  0  0  0  0|
--R    |                                              |

--R    |0  0  0  0  0  0  0  0  0  1  0  1  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  1  1  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0|
--R   [|                                              |,
--R    |1  0  1  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  1  0  1  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |1  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  0  1  1|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  1|
--R    |                                              |
--R    |0  0  0  0  0  0  0  0  0  0  0  0  1  1  0  0|
--R    |                                              |
--R    +0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0+
--R    +0  0  0  0  0  0  1  0  0  0  1  0  0  0  1  0+
--R    |                                              |
--R    |0  0  0  0  1  1  1  1  1  1  1  1  1  1  1  1|
--R    |                                              |
--R    |0  0  0  0  1  0  1  1  1  0  1  1  1  0  1  1|
--R    |                                              |
--R    |0  0  0  0  0  1  0  1  0  1  0  1  0  1  0  1|
--R    |                                              |
--R    |0  0  1  0  0  0  1  0  0  0  0  0  0  0  1  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  0  0  0  0  1  1  1  1|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  0  0  0  0  1  0  1  1|
--R    |                                              |
--R    |0  1  0  1  0  1  0  1  0  0  0  0  0  1  0  1|
--R    |                                              |]
--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  0  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  1  1  1  1  0  0  0  0|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  1  0  1  1  0  0  0  0|
--R    |                                              |
--R    |0  1  0  1  0  1  0  1  0  1  0  1  0  0  0  0|
--R    |                                              |
--R    |0  0  1  0  0  0  1  0  0  0  1  0  0  0  1  0|
--R    |                                              |
--R    |1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1|
--R    |                                              |
--R    |1  0  1  1  1  0  1  1  1  0  1  1  1  0  1  1|

--R    |                                              |
--R    +0  1  0  1  0  1  0  1  0  1  0  1  0  1  0  1+
--R                                            Type: List(Matrix(PrimeField(2)))
--E 33
)spool
)lisp (bye)

———-

References^{[1] nothing}

27