help links subscribe changes refresh edit

	SandBox
SandBoxRemember ←	↑	→ SandBoxRootOfUnity

SandBoxRepresentationTheory

Edit detail for SandBoxRepresentationTheory revision 1 of 2

	1 2
Editor: Bill Page Time: 2007/09/19 16:19:42 GMT-7
Note:

changed:
-
From: input/repa6.input
\begin{axiom}
)cls

-- 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)
 
genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]
 
-- pRA6 is the permutation representation over the Integers...
 
pRA6 := permutationRepresentation (genA6, 6)
 
-- ... and pRA6m2 is the permutation representation over PrimeField 2:
 
pRA6m2 : List Matrix PrimeField 2 := pRA6
 
-- Now try to split pRA6m2:
 
sp0 := meatAxe pRA6m2
 
-- We have found the trivial module as a factormodule
-- and a 5-dimensional submodule.
 
dA6d1 := sp0.2
 
-- Try to split again...
 
sp1 := meatAxe sp0.1
 
-- ... and find a 4-dimensional submodule, say dA6d4a, and the
-- trivial one again.
 
dA6d4a := sp1.2
 
-- Now we want to test, whether dA6d4a is absolutely irreducible...
 
isAbsolutelyIrreducible? dA6d4a
 
-- ...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]
lambda := [2,2,1,1]
dimIrrRepSym lambda
 
-- now create the restriction to A6:
 
d2211  := irrRepSymNat(lambda, genA6)
 
-- ... and d2211m2 is the representation over PrimeField 2:
 
d2211m2 : List Matrix PrimeField 2 := d2211
 
-- and split it:
 
sp2 := meatAxe d2211m2
 
-- A 5 and a 4-dimensional one.
 
-- we take the 4-dimensional one, say dA6d4b:
 
dA6d4b := sp2.1
 
-- This is absolutely irreducible, too ...
 
isAbsolutelyIrreducible? dA6d4b
 
-- ... and dA6d4a and dA6d4b are not equivalent:
 
areEquivalent? ( dA6d4a , dA6d4b )
 
-- So the third irreducible representation is found.
 
-- Now construct a new representation with the help of the tensorproduct
 
dA6d16 := tensorProduct ( dA6d4a , dA6d4b )
 
-- and try to split it...
 
sp3 := meatAxe dA6d16
 
-- The representation is irreducible, but may be not
-- absolutely irreducible.
 
isAbsolutelyIrreducible? dA6d16
 
-- So let's try the same over the field with 4 elements:
 
gf4 := FiniteField(2,2)
 
dA6d16gf4 : List Matrix gf4 := dA6d16
sp4 := meatAxe dA6d16gf4
 
-- Now we find two 8-dimensional ones, dA6d8a and dA6d8b.
 
dA6d8a : List Matrix gf4  := sp4.1
dA6d8b : List Matrix gf4  := sp4.2
 
-- Both are absolutely irreducible...
 
isAbsolutelyIrreducible? dA6d8a
isAbsolutelyIrreducible? dA6d8b
 
-- and they are not equivalent...
 
areEquivalent? ( dA6d8a, dA6d8b )
 
-- 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
 
dA6d1
dA6d4a
dA6d4b
dA6d8a
dA6d8b
 
-- And here again is the irreducible, but not absolutely irreducible
-- representations of A6 over PrimeField 2
 
dA6d16
\end{axiom}

From: input/repa6.input

axiom

)cls

   Your argument list is not valid.

-- 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)

genA6 : List PERM INT := [cycle [1,2,3], cycle [2,3,4,5,6]]

$\label{eq1}\left[{\left(1 \ 2 \ 3 \right)}, \:{\left(2 \ 3 \ 4 \ 5 \ 6 \right)}\right]$

(1)

Type: List(Permutation(Integer))

axiom

-- pRA6 is the permutation representation over the Integers...

pRA6 := permutationRepresentation (genA6, 6)

$\label{eq2}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccccc} 0 & 0 & 1 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 1$

(2)

Type: List(Matrix(Integer))

axiom

-- ... and pRA6m2 is the permutation representation over PrimeField 2:

pRA6m2 : List Matrix PrimeField 2 := pRA6

$\label{eq3}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccccc} 0 & 0 & 1 & 0 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 & 0 \ 0 & 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 0 & 1$

(3)

Type: List(Matrix(PrimeField(2)))

axiom

-- Now try to split pRA6m2:

sp0 := meatAxe pRA6m2

   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

$\label{eq4}\begin{array}{@{}l} \displaystyle \left[{ \begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{ccccc} 0 & 0 & 1 & 0 & 0 \ 1 & 0 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 0 & 1$

(4)

Type: List(List(Matrix(PrimeField(2))))

axiom

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

dA6d1 := sp0.2

$\label{eq5}\left[{\left[ \begin{array}{c} 1$

(5)

Type: List(Matrix(PrimeField(2)))

axiom

-- Try to split again...

sp1 := meatAxe sp0.1

   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 all
       non-trivial element of the singular matrix under view 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 split:

$\label{eq6}\begin{array}{@{}l} \displaystyle \left[{\left[{\left[ \begin{array}{c} 1$

(6)

Type: List(List(Matrix(PrimeField(2))))

axiom

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

dA6d4a := sp1.2

$\label{eq7}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1$

(7)

Type: List(Matrix(PrimeField(2)))

axiom

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

isAbsolutelyIrreducible? dA6d4a

   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 all
       non-trivial element of the singular matrix under view are
       not proper, hence Norton's irreducibility test can be done:
     The generated cyclic submodule was not proper
     Representation is absolutely irreducible

$\label{eq8} \mbox{\rm true}$

(8)

Type: Boolean

axiom

-- ...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]
lambda := [2,2,1,1]

$\label{eq9}\left[ 2, \: 2, \: 1, \: 1 \right]$

(9)

Type: List(PositiveInteger)

axiom

dimIrrRepSym lambda

   There are no library operations named dimIrrRepSym
      Use HyperDoc Browse or issue
                            )what op dimIrrRepSym
      to learn if there is any operation containing " dimIrrRepSym " in
      its name.

   Cannot find a definition or applicable library operation named
      dimIrrRepSym with argument type(s)
                            List(PositiveInteger)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- now create the restriction to A6:

d2211  := irrRepSymNat(lambda, genA6)

   There are no library operations named irrRepSymNat
      Use HyperDoc Browse or issue
                            )what op irrRepSymNat
      to learn if there is any operation containing " irrRepSymNat " in
      its name.

   Cannot find a definition or applicable library operation named
      irrRepSymNat with argument type(s)
                            List(PositiveInteger)
                         List(Permutation(Integer))

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- ... and d2211m2 is the representation over PrimeField 2:

d2211m2 : List Matrix PrimeField 2 := d2211

   Cannot convert right-hand side of assignment
   d2211

      to an object of the type List(Matrix(PrimeField(2))) of the
      left-hand side.

-- and split it:

sp2 := meatAxe d2211m2

   d2211m2 is declared as being in List(Matrix(PrimeField(2))) but has
      not been given a value.

-- A 5 and a 4-dimensional one.

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

dA6d4b := sp2.1

   There are no library operations named sp2
      Use HyperDoc Browse or issue
                                )what op sp2
      to learn if there is any operation containing " sp2 " in its
      name.

   Cannot find a definition or applicable library operation named sp2
      with argument type(s)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- This is absolutely irreducible, too ...

isAbsolutelyIrreducible? dA6d4b

   There are 1 exposed and 0 unexposed library operations named
      isAbsolutelyIrreducible? having 1 argument(s) but none was
      determined to be applicable. Use HyperDoc Browse, or issue
                    )display op isAbsolutelyIrreducible?
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named
      isAbsolutelyIrreducible? with argument type(s)
                              Variable(dA6d4b)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- ... and dA6d4a and dA6d4b are not equivalent:

areEquivalent? ( dA6d4a , dA6d4b )

   There are 1 exposed and 0 unexposed library operations named
      areEquivalent? having 2 argument(s) but none was determined to be
      applicable. Use HyperDoc Browse, or issue
                         )display op areEquivalent?
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named
      areEquivalent? with argument type(s)
                         List(Matrix(PrimeField(2)))
                              Variable(dA6d4b)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- So the third irreducible representation is found.

-- Now construct a new representation with the help of the tensorproduct

dA6d16 := tensorProduct ( dA6d4a , dA6d4b )

   There are 2 exposed and 0 unexposed library operations named
      tensorProduct having 2 argument(s) but none was determined to be
      applicable. Use HyperDoc Browse, or issue
                          )display op tensorProduct
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named
      tensorProduct with argument type(s)
                         List(Matrix(PrimeField(2)))
                              Variable(dA6d4b)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- and try to split it...

sp3 := meatAxe dA6d16

   There are 1 exposed and 0 unexposed library operations named meatAxe
      having 1 argument(s) but none was determined to be applicable.
      Use HyperDoc Browse, or issue
                             )display op meatAxe
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named
      meatAxe with argument type(s)
                              Variable(dA6d16)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

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

isAbsolutelyIrreducible? dA6d16

   There are 1 exposed and 0 unexposed library operations named
      isAbsolutelyIrreducible? having 1 argument(s) but none was
      determined to be applicable. Use HyperDoc Browse, or issue
                    )display op isAbsolutelyIrreducible?
      to learn more about the available operations. Perhaps
      package-calling the operation or using coercions on the arguments
      will allow you to apply the operation.

   Cannot find a definition or applicable library operation named
      isAbsolutelyIrreducible? with argument type(s)
                              Variable(dA6d16)

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

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

gf4 := FiniteField(2,2)

$\label{eq10}\hbox{\axiomType{FiniteField}\ } (2, 2)$

(10)

Type: Domain

axiom

dA6d16gf4 : List Matrix gf4 := dA6d16

   Cannot convert right-hand side of assignment
   dA6d16

      to an object of the type List(Matrix(FiniteField(2,2))) of the
      left-hand side.
sp4 := meatAxe dA6d16gf4

   dA6d16gf4 is declared as being in List(Matrix(FiniteField(2,2))) but
      has not been given a value.

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

dA6d8a : List Matrix gf4  := sp4.1

   There are no library operations named sp4
      Use HyperDoc Browse or issue
                                )what op sp4
      to learn if there is any operation containing " sp4 " in its
      name.

   Cannot find a definition or applicable library operation named sp4
      with argument type(s)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.
dA6d8b : List Matrix gf4  := sp4.2

   There are no library operations named sp4
      Use HyperDoc Browse or issue
                                )what op sp4
      to learn if there is any operation containing " sp4 " in its
      name.

   Cannot find a definition or applicable library operation named sp4
      with argument type(s)
                               PositiveInteger

      Perhaps you should use "@" to indicate the required return type,
      or "$" to specify which version of the function you need.

-- Both are absolutely irreducible...

isAbsolutelyIrreducible? dA6d8a

   dA6d8a is declared as being in List(Matrix(FiniteField(2,2))) but
      has not been given a value.
isAbsolutelyIrreducible? dA6d8b

   dA6d8b is declared as being in List(Matrix(FiniteField(2,2))) but
      has not been given a value.

-- and they are not equivalent...

areEquivalent? ( dA6d8a, dA6d8b )

   dA6d8a is declared as being in List(Matrix(FiniteField(2,2))) but
      has not been given a value.

-- 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

dA6d1

$\label{eq11}\left[{\left[ \begin{array}{c} 1$

(11)

Type: List(Matrix(PrimeField(2)))

axiom

dA6d4a

$\label{eq12}\begin{array}{@{}l} \displaystyle \left[{\left[ \begin{array}{cccc} 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 1 & 0 & 0 & 0 \ 0 & 0 & 0 & 1$

(12)

Type: List(Matrix(PrimeField(2)))

axiom

dA6d4b

$\label{eq13}dA 6 d 4 b$

(13)

Type: Variable(dA6d4b)

axiom

dA6d8a

   dA6d8a is declared as being in List(Matrix(FiniteField(2,2))) but
      has not been given a value.
dA6d8b

   dA6d8b is declared as being in List(Matrix(FiniteField(2,2))) but
      has not been given a value.

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

dA6d16

$\label{eq14}dA 6 d 16$

(14)

Type: Variable(dA6d16)