GAP (ref) - Chapter 69: The MeatAxe

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69 The MeatAxe
69.1 MeatAxe Modules
69.1-1 GModuleByMats
69.2 Module Constructions
69.2-1 NaturalGModule
69.2-2 PermutationGModule
69.2-3 TensorProductGModule
69.2-4 WedgeGModule
69.3 Selecting a Different MeatAxe
69.3-1 MTX
69.4 Accessing a Module
69.4-1 MTX.Generators
69.4-2 MTX.Dimension
69.4-3 MTX.Field
69.5 Irreducibility Tests
69.5-1 MTX.IsIrreducible
69.5-2 MTX.IsAbsolutelyIrreducible
69.5-3 MTX.DegreeSplittingField
69.6 Decomposition of modules
69.6-1 MTX.IsIndecomposable
69.6-2 MTX.Indecomposition
69.6-3 MTX.HomogeneousComponents
69.7 Finding Submodules
69.7-1 MTX.SubmoduleGModule
69.7-2 MTX.ProperSubmoduleBasis
69.7-3 MTX.BasesSubmodules
69.7-4 MTX.BasesMinimalSubmodules
69.7-5 MTX.BasesMaximalSubmodules
69.7-6 MTX.BasisRadical
69.7-7 MTX.BasisSocle
69.7-8 MTX.BasesMinimalSupermodules
69.7-9 MTX.BasesCompositionSeries
69.7-10 MTX.CompositionFactors
69.7-11 MTX.CollectedFactors
69.8 Induced Actions
69.8-1 MTX.NormedBasisAndBaseChange
69.8-2 MTX.InducedActionSubmodule
69.8-3 MTX.InducedActionFactorModule
69.8-4 MTX.InducedActionSubMatrix
69.8-5 MTX.InducedAction
69.9 Module Homomorphisms
69.9-1 MTX.BasisModuleHomomorphisms
69.9-2 MTX.BasisModuleEndomorphisms
69.9-3 MTX.IsomorphismModules
69.9-4 MTX.ModuleAutomorphisms
69.10 Module Homomorphisms for irreducible modules
69.10-1 MTX.IsEquivalent
69.10-2 MTX.IsomorphismIrred
69.10-3 MTX.Homomorphism
69.10-4 MTX.Homomorphisms
69.10-5 MTX.Distinguish
69.11 MeatAxe Functionality for Invariant Forms
69.11-1 MTX.InvariantBilinearForm
69.11-2 MTX.InvariantSesquilinearForm
69.11-3 MTX.InvariantQuadraticForm
69.11-4 MTX.BasisInOrbit
69.11-5 MTX.OrthogonalSign
69.12 The Smash MeatAxe
69.12-1 SMTX.RandomIrreducibleSubGModule
69.12-2 SMTX.GoodElementGModule
69.12-3 SMTX.SortHomGModule
69.12-4 SMTX.MinimalSubGModules
69.12-5 SMTX.Setter
69.12-6 SMTX.Getter
69.12-7 SMTX.IrreducibilityTest
69.12-8 SMTX.AbsoluteIrreducibilityTest
69.12-9 SMTX.MinimalSubGModule
69.12-10 SMTX.MatrixSum
69.12-11 SMTX.CompleteBasis
69.13 Smash MeatAxe Flags
69.13-1 SMTX.Subbasis
69.13-2 SMTX.AlgEl
69.13-3 SMTX.AlgElMat
69.13-4 SMTX.AlgElCharPol
69.13-5 SMTX.AlgElCharPolFac
69.13-6 SMTX.AlgElNullspaceVec
69.13-7 SMTX.AlgElNullspaceDimension
69.13-8 SMTX.CentMat
69.13-9 SMTX.CentMatMinPoly

### 69 The MeatAxe

The MeatAxe [Par84] is a tool for the examination of submodules of a group algebra. It is a basic tool for the examination of group actions on finite-dimensional modules.

GAP uses the improved MeatAxe of Derek Holt and Sarah Rees, and also incorporates further improvements of Ivanyos and Lux.

Please note that, consistently with the convention for group actions, the action of the GAP MeatAxe is always that of matrices on row vectors by multiplication on the right. If you want to investigate left modules you will have to transpose the matrices.

#### 69.1 MeatAxe Modules

##### 69.1-1 GModuleByMats

`‣ GModuleByMats`( gens, field ) | ( function )

`‣ GModuleByMats`( emptygens, dim, field ) | ( function )

creates a MeatAxe module over field from a list of invertible matrices gens which reflect a group's action. If the list of generators is empty, the dimension must be given as second argument.

MeatAxe routines are on a level with Gaussian elimination. Therefore they do not deal with GAP modules but essentially with lists of matrices. For the MeatAxe, a module is a record with components

`generators`

  A list of matrices which represent a group operation on a finite dimensional row vector space.

`dimension`

  The dimension of the vector space (this is the common length of the row vectors (see `DimensionOfVectors` (61.9-6))).

`field`

  The field over which the vector space is defined.

Once a module has been created its entries may not be changed. A MeatAxe may create a new component NameOfMeatAxe in which it can store private information. By a MeatAxe submodule or factor module we denote actually the induced action on the submodule, respectively factor module. Therefore the submodules or factor modules are again MeatAxe modules. The arrangement of `generators` is guaranteed to be the same for the induced modules, but to obtain the complete relation to the original module, the bases used are needed as well.

#### 69.2 Module Constructions

##### 69.2-1 NaturalGModule

`‣ NaturalGModule`( group[, field] ) | ( function )

creates a MeatAxe module over field from the generators of the matrix group group. If field is not provided then the value returned by `DefaultFieldOfMatrixGroup` (44.2-2) is used instead.

##### 69.2-2 PermutationGModule

`‣ PermutationGModule`( G, F ) | ( function )

Called with a permutation group G and a field F (F may be infinite), `PermutationGModule` returns the natural permutation module M over F for the group of permutation matrices that acts on the canonical basis of M in the same way as G acts on the points up to its largest moved point (see `LargestMovedPoint` (42.3-2)).

##### 69.2-3 TensorProductGModule

`‣ TensorProductGModule`( m1, m2 ) | ( function )

`TensorProductGModule` calculates the tensor product of the modules m1 and m2. They are assumed to be modules over the same algebra so, in particular, they should have the same number of generators.

##### 69.2-4 WedgeGModule

`‣ WedgeGModule`( module ) | ( function )

`WedgeGModule` calculates the wedge product of a G-module. That is the action on antisymmetric tensors.

#### 69.3 Selecting a Different MeatAxe

##### 69.3-1 MTX

`‣ MTX` | ( global variable )

All MeatAxe routines are accessed via the global variable `MTX`, which is a record whose components hold the various functions. It is possible to have several implementations of a MeatAxe available. Each MeatAxe represents its routines in an own global variable and assigning `MTX` to this variable selects the corresponding MeatAxe.

#### 69.4 Accessing a Module

Even though a MeatAxe module is a record, its components should never be accessed outside of MeatAxe functions. Instead the following operations should be used:

##### 69.4-1 MTX.Generators

`‣ MTX.Generators`( module ) | ( function )

returns a list of matrix generators of module.

##### 69.4-2 MTX.Dimension

`‣ MTX.Dimension`( module ) | ( function )

returns the dimension in which the matrices act.

##### 69.4-3 MTX.Field

`‣ MTX.Field`( module ) | ( function )

returns the field over which module is defined.

#### 69.5 Irreducibility Tests

##### 69.5-1 MTX.IsIrreducible

`‣ MTX.IsIrreducible`( module ) | ( function )

tests whether the module module is irreducible (i.e. contains no proper submodules.)

##### 69.5-2 MTX.IsAbsolutelyIrreducible

`‣ MTX.IsAbsolutelyIrreducible`( module ) | ( function )

A module is absolutely irreducible if it remains irreducible over the algebraic closure of the field. (Formally: If the tensor product L ⊗_K M is irreducible where M is the module defined over K and L is the algebraic closure of K.)

##### 69.5-3 MTX.DegreeSplittingField

`‣ MTX.DegreeSplittingField`( module ) | ( function )

returns the degree of the splitting field as extension of the prime field.

#### 69.6 Decomposition of modules

A module is decomposable if it can be written as the direct sum of two proper submodules (and indecomposable if not). Obviously every finite dimensional module is a direct sum of its indecomposable parts. The homogeneous components of a module are the direct sums of isomorphic indecomposable components. They are uniquely determined.

##### 69.6-1 MTX.IsIndecomposable

`‣ MTX.IsIndecomposable`( module ) | ( function )

returns whether module is indecomposable.

##### 69.6-2 MTX.Indecomposition

`‣ MTX.Indecomposition`( module ) | ( function )

returns a decomposition of module as a direct sum of indecomposable modules. It returns a list, each entry is a list of form [B,ind] where B is a list of basis vectors for the indecomposable component and ind the induced module action on this component. (Such a decomposition is not unique.)

##### 69.6-3 MTX.HomogeneousComponents

`‣ MTX.HomogeneousComponents`( module ) | ( function )

computes the homogeneous components of module given as sums of indecomposable components. The function returns a list, each entry of which is a record corresponding to one isomorphism type of indecomposable components. The record has the following components.

`indices`

  the index numbers of the indecomposable components, as given by `MTX.Indecomposition` (69.6-2), that are in the homogeneous component,

`component`

  one of the indecomposable components,

`images`

  a list of the remaining indecomposable components, each given as a record with the components `component` (the component itself) and `isomorphism` (an isomorphism from the defining component to this one).

#### 69.7 Finding Submodules

##### 69.7-1 MTX.SubmoduleGModule

`‣ MTX.SubmoduleGModule`( module, subspace ) | ( function )

`‣ MTX.SubGModule`( module, subspace ) | ( function )

subspace should be a subspace of (or a vector in) the underlying vector space of module i.e. the full row space of the same dimension and over the same field as module. A normalized basis of the submodule of module generated by subspace is returned.

##### 69.7-2 MTX.ProperSubmoduleBasis

`‣ MTX.ProperSubmoduleBasis`( module ) | ( function )

returns the basis of a proper submodule of module and `fail` if no proper submodule exists.

##### 69.7-3 MTX.BasesSubmodules

`‣ MTX.BasesSubmodules`( module ) | ( function )

returns a list containing a basis for every submodule.

##### 69.7-4 MTX.BasesMinimalSubmodules

`‣ MTX.BasesMinimalSubmodules`( module ) | ( function )

returns a list of bases of all minimal submodules.

##### 69.7-5 MTX.BasesMaximalSubmodules

`‣ MTX.BasesMaximalSubmodules`( module ) | ( function )

returns a list of bases of all maximal submodules.

##### 69.7-6 MTX.BasisRadical

`‣ MTX.BasisRadical`( module ) | ( function )

returns a basis of the radical of module.

##### 69.7-7 MTX.BasisSocle

`‣ MTX.BasisSocle`( module ) | ( function )

returns a basis of the socle of module.

##### 69.7-8 MTX.BasesMinimalSupermodules

`‣ MTX.BasesMinimalSupermodules`( module, sub ) | ( function )

returns a list of bases of all minimal supermodules of the submodule given by the basis sub.

##### 69.7-9 MTX.BasesCompositionSeries

`‣ MTX.BasesCompositionSeries`( module ) | ( function )

returns a list of bases of submodules in a composition series in ascending order.

##### 69.7-10 MTX.CompositionFactors

`‣ MTX.CompositionFactors`( module ) | ( function )

returns a list of composition factors of module in ascending order.

##### 69.7-11 MTX.CollectedFactors

`‣ MTX.CollectedFactors`( module ) | ( function )

returns a list giving all irreducible composition factors with their frequencies.

#### 69.8 Induced Actions

##### 69.8-1 MTX.NormedBasisAndBaseChange

`‣ MTX.NormedBasisAndBaseChange`( sub ) | ( function )

returns a list `[bas, change ]` where bas is a normed basis (i.e. in echelon form with pivots normed to 1) for sub and change is the base change from bas to sub (the basis vectors of bas expressed in coefficients for sub).

##### 69.8-2 MTX.InducedActionSubmodule

`‣ MTX.InducedActionSubmodule`( module, sub ) | ( function )

`‣ MTX.InducedActionSubmoduleNB`( module, sub ) | ( function )

creates a new module corresponding to the action of module on the non-trivial submodule sub. In the `NB` version the basis sub must be normed. (That is it must be in echelon form with pivots normed to 1, see `MTX.NormedBasisAndBaseChange` (69.8-1).)

##### 69.8-3 MTX.InducedActionFactorModule

`‣ MTX.InducedActionFactorModule`( module, sub[, compl] ) | ( function )

creates a new module corresponding to the action of module on the factor of the proper submodule sub. If compl is given, it has to be a basis of a (vector space-)complement of sub. The action then will correspond to compl.

The basis sub has to be given in normed form. (That is it must be in echelon form with pivots normed to 1, see `MTX.NormedBasisAndBaseChange` (69.8-1))

##### 69.8-4 MTX.InducedActionSubMatrix

`‣ MTX.InducedActionSubMatrix`( mat, sub ) | ( function )

`‣ MTX.InducedActionSubMatrixNB`( mat, sub ) | ( function )

`‣ MTX.InducedActionFactorMatrix`( mat, sub[, compl] ) | ( function )

work the same way as the above functions for modules, but take as input only a single matrix.

##### 69.8-5 MTX.InducedAction

`‣ MTX.InducedAction`( module, sub[, type] ) | ( function )

Computes induced actions on submodules or factor modules and also returns the corresponding bases. The action taken is binary encoded in type: `1` stands for subspace action, `2` for factor action, and `4` for action of the full module on a subspace adapted basis. The routine returns the computed results in a list in sequence (sub,quot,both,basis) where basis is a basis for the whole space, extending sub. (Actions which are not computed are omitted, so the returned list may be shorter.) If no type is given, it is assumed to be `7`. The basis given in sub must be normed!

All these routines return `fail` if sub is not a proper subspace.

#### 69.9 Module Homomorphisms

##### 69.9-1 MTX.BasisModuleHomomorphisms

`‣ MTX.BasisModuleHomomorphisms`( module1, module2 ) | ( function )

returns a basis of all module homomorphisms from module1 to module2. Homomorphisms are by matrices, whose rows give the images of the standard basis vectors of module1 in the standard basis of module2.

##### 69.9-2 MTX.BasisModuleEndomorphisms

`‣ MTX.BasisModuleEndomorphisms`( module ) | ( function )

returns a basis of all module homomorphisms from module to module.

##### 69.9-3 MTX.IsomorphismModules

`‣ MTX.IsomorphismModules`( module1, module2 ) | ( function )

If module1 and module2 are isomorphic modules, this function returns an isomorphism from module1 to module2 in form of a matrix. It returns `fail` if the modules are not isomorphic.

##### 69.9-4 MTX.ModuleAutomorphisms

`‣ MTX.ModuleAutomorphisms`( module ) | ( function )

returns the module automorphisms of module (the set of all isomorphisms from module to itself) as a matrix group.

#### 69.10 Module Homomorphisms for irreducible modules

The following are lower-level functions that provide homomorphism functionality for irreducible modules. Generic code should use the functions in Section 69.9 instead.

##### 69.10-1 MTX.IsEquivalent

`‣ MTX.IsEquivalent`( module1, module2 ) | ( function )

tests two irreducible modules for equivalence.

##### 69.10-2 MTX.IsomorphismIrred

`‣ MTX.IsomorphismIrred`( module1, module2 ) | ( function )

returns an isomorphism from module1 to module2 (if one exists), and `fail` otherwise. It requires that one of the modules is known to be irreducible. It implicitly assumes that the same group is acting, otherwise the results are unpredictable. The isomorphism is given by a matrix M, whose rows give the images of the standard basis vectors of module1 in the standard basis of module2. That is, conjugation of the generators of module2 with M yields the generators of module1.

##### 69.10-3 MTX.Homomorphism

`‣ MTX.Homomorphism`( module1, module2, mat ) | ( function )

mat should be a dim1 × dim2 matrix defining a homomorphism from module1 to module2. This function verifies that mat really does define a module homomorphism, and then returns the corresponding homomorphism between the underlying row spaces of the modules. This can be used for computing kernels, images and pre-images.

##### 69.10-4 MTX.Homomorphisms

`‣ MTX.Homomorphisms`( module1, module2 ) | ( function )

returns a basis of the space of all homomorphisms from the irreducible module module1 to module2.

##### 69.10-5 MTX.Distinguish

`‣ MTX.Distinguish`( cf, nr ) | ( function )

Let cf be the output of `MTX.CollectedFactors` (69.7-11). This routine tries to find a group algebra element that has nullity zero on all composition factors except number nr.

#### 69.11 MeatAxe Functionality for Invariant Forms

The functions in this section can only be applied to an absolutely irreducible MeatAxe module.

##### 69.11-1 MTX.InvariantBilinearForm

`‣ MTX.InvariantBilinearForm`( module ) | ( function )

returns an invariant bilinear form, which may be symmetric or anti-symmetric, of module, or `fail` if no such form exists.

##### 69.11-2 MTX.InvariantSesquilinearForm

`‣ MTX.InvariantSesquilinearForm`( module ) | ( function )

returns an invariant hermitian (= self-adjoint) sesquilinear form of module, which must be defined over a finite field whose order is a square, or `fail` if no such form exists.

##### 69.11-3 MTX.InvariantQuadraticForm

`‣ MTX.InvariantQuadraticForm`( module ) | ( function )

returns either the matrix of an invariant quadratic form of the absolutely irreducible module module, or `fail`.

If the characteristic of module is odd then `fail` is returned if there is no nonzero invariant bilinear form, otherwise a matrix of the bilinear form divided by 2 is returned; note that this matrix may be antisymmetric and thus describe the zero quadratic form. If the characteristic of module is 2 then `fail` is returned if module does not admit a nonzero quadratic form, otherwise a lower triangular matrix describing the form is returned.

An error is signalled if module is not absolutely irreducible.

    gap> g:= SO(-1, 4, 2);;
    gap> m:= NaturalGModule( g );;
    gap> Display( MTX.InvariantQuadraticForm( m ) );
     . . . .
     1 . . .
     . . 1 .
     . . 1 1
    gap> g:= Sp(4, 2);;
    gap> m:= NaturalGModule( g );;
    gap> MTX.InvariantQuadraticForm( m );
    fail
    gap> g:= Sp(4, 3);;
    gap> m:= NaturalGModule( g );;
    gap> q:= MTX.InvariantQuadraticForm( m );;
    gap> q = - TransposedMat( q );  # antisymmetric inv. bilinear form
    true

##### 69.11-4 MTX.BasisInOrbit

`‣ MTX.BasisInOrbit`( module ) | ( function )

returns a basis of the underlying vector space of module which is contained in an orbit of the action of the generators of module on that space. This is used by `MTX.InvariantQuadraticForm` (69.11-3) in characteristic 2.

##### 69.11-5 MTX.OrthogonalSign

`‣ MTX.OrthogonalSign`( module ) | ( function )

Let module be an absolutely irreducible G-module. If module does not fix a nondegenerate quadratic form see `MTX.InvariantQuadraticForm` (69.11-3) then `fail` is returned. Otherwise the sign ϵ ∈ { -1, 0, 1 } is returned such that G embeds into the general orthogonal group GO^ϵ(d, q) w.r.t. the invariant quadratic form, see `GeneralOrthogonalGroup` (50.2-6). That is, `0` is returned if module has odd dimension, and `1` or `-1` is returned if the orthogonal group has plus or minus type, respectively.

An error is signalled if module is not absolutely irreducible.

The `SMTX` implementation uses an algorithm due to Jon Thackray.

    gap> mats:= GeneratorsOfGroup( GO(1,4,2) );;
    gap> MTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );
    1
    gap> mats:= GeneratorsOfGroup( GO(-1,4,2) );;
    gap> MTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );
    -1
    gap> mats:= GeneratorsOfGroup( GO(5,3) );;
    gap> MTX.OrthogonalSign( GModuleByMats( mats, GF(3) ) );
    0
    gap> mats:= GeneratorsOfGroup( SP(4,2) );;
    gap> MTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );
    fail

#### 69.12 The Smash MeatAxe

The standard MeatAxe provided in the GAP library is based on the MeatAxe in the GAP 3 package Smash, originally written by Derek Holt and Sarah Rees [HR94]. It is accessible via the variable `SMTX` to which `MTX` (69.3-1) is assigned by default. For the sake of completeness the remaining sections document more technical functions of this MeatAxe.

##### 69.12-1 SMTX.RandomIrreducibleSubGModule

`‣ SMTX.RandomIrreducibleSubGModule`( module ) | ( function )

returns the module action on a random irreducible submodule.

##### 69.12-2 SMTX.GoodElementGModule

`‣ SMTX.GoodElementGModule`( module ) | ( function )

finds an element with minimal possible nullspace dimension if module is known to be irreducible.

##### 69.12-3 SMTX.SortHomGModule

`‣ SMTX.SortHomGModule`( module1, module2, homs ) | ( function )

Function to sort the output of `Homomorphisms`.

##### 69.12-4 SMTX.MinimalSubGModules

`‣ SMTX.MinimalSubGModules`( module1, module2[, max] ) | ( function )

returns (at most max) bases of submodules of module2 which are isomorphic to the irreducible module module1.

##### 69.12-5 SMTX.Setter

`‣ SMTX.Setter`( string ) | ( function )

returns a setter function for the component `smashMeataxe.(string)`.

##### 69.12-6 SMTX.Getter

`‣ SMTX.Getter`( string ) | ( function )

returns a getter function for the component `smashMeataxe.(string)`.

##### 69.12-7 SMTX.IrreducibilityTest

`‣ SMTX.IrreducibilityTest`( module ) | ( function )

Tests for irreducibility and sets a subbasis if reducible. It neither sets an irreducibility flag, nor tests it. Thus the routine also can simply be called to obtain a random submodule.

##### 69.12-8 SMTX.AbsoluteIrreducibilityTest

`‣ SMTX.AbsoluteIrreducibilityTest`( module ) | ( function )

Tests for absolute irreducibility and sets splitting field degree. It neither sets an absolute irreducibility flag, nor tests it.

##### 69.12-9 SMTX.MinimalSubGModule

`‣ SMTX.MinimalSubGModule`( module, cf, nr ) | ( function )

returns the basis of a minimal submodule of module containing the indicated composition factor. It assumes `Distinguish` has been called already.

##### 69.12-10 SMTX.MatrixSum

`‣ SMTX.MatrixSum`( matrices1, matrices2 ) | ( function )

creates the direct sum of two matrix lists.

##### 69.12-11 SMTX.CompleteBasis

`‣ SMTX.CompleteBasis`( module, pbasis ) | ( function )

extends the partial basis pbasis to a basis of the full space by action of module. It returns whether it succeeded.

#### 69.13 Smash MeatAxe Flags

The following getter routines access internal flags. For each routine, the appropriate setter's name is prefixed with `Set`.

##### 69.13-1 SMTX.Subbasis

`‣ SMTX.Subbasis`( module ) | ( function )

Basis of a submodule.

##### 69.13-2 SMTX.AlgEl

`‣ SMTX.AlgEl`( module ) | ( function )

list `[newgens,coefflist]` giving an algebra element used for chopping.

##### 69.13-3 SMTX.AlgElMat

`‣ SMTX.AlgElMat`( module ) | ( function )

matrix of `SMTX.AlgEl` (69.13-2).

##### 69.13-4 SMTX.AlgElCharPol

`‣ SMTX.AlgElCharPol`( module ) | ( function )

minimal polynomial of `SMTX.AlgEl` (69.13-2).

##### 69.13-5 SMTX.AlgElCharPolFac

`‣ SMTX.AlgElCharPolFac`( module ) | ( function )

uses factor of `SMTX.AlgEl` (69.13-2).

##### 69.13-6 SMTX.AlgElNullspaceVec

`‣ SMTX.AlgElNullspaceVec`( module ) | ( function )

nullspace of the matrix evaluated under this factor.

##### 69.13-7 SMTX.AlgElNullspaceDimension

`‣ SMTX.AlgElNullspaceDimension`( module ) | ( function )

dimension of the nullspace.

##### 69.13-8 SMTX.CentMat

`‣ SMTX.CentMat`( module ) | ( function )

matrix centralising all generators which is computed as a byproduct of `SMTX.AbsoluteIrreducibilityTest` (69.12-8).

##### 69.13-9 SMTX.CentMatMinPoly

`‣ SMTX.CentMatMinPoly`( module ) | ( function )

minimal polynomial of `SMTX.CentMat` (69.13-8).

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