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# 67 The MeatAxe

### Sections

1. MeatAxe Modules
2. Module Constructions
3. Selecting a Different MeatAxe
4. Accessing a Module
5. Irreducibility Tests
6. Finding Submodules
7. Induced Actions
8. Module Homomorphisms
9. Invariant Forms
10. The Smash MeatAxe
11. Smash MeatAxe Flags

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.

## 67.1 MeatAxe Modules

- `GModuleByMats(` gens`,` field`)`
- `GModuleByMats(` emptygens`,` dim`,` field`)`

  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)).

  `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.

  ## 67.2 Module Constructions

- `TensorProductGModule (`m1`,`m2`) F`

  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.

- `WedgeGModule (`module`)`

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

  ## 67.3 Selecting a Different MeatAxe

  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.

  ## 67.4 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:

- `MTX.Generators(` module`)`

  returns a list of matrix generators of module.

- `MTX.Dimension(` module`)`

  returns the dimension in which the matrices act.

- `MTX.Field(` module`)`

  returns the field over which module is defined.

  ## 67.5 Irreducibility Tests

- `MTX.IsIrreducible(` module`) AST`

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

- `MTX.IsAbsolutelyIrreducible(` module`) AST`

  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.)

- `MTX.DegreeSplittingField(` module`)`

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

  ## 67.6 Finding Submodules

- `MTX.SubmoduleGModule(` module`,` subspace`) F`
- `MTX.SubGModule(` module`,` subspace`) F`

  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.

- `MTX.ProperSubmoduleBasis(` module`) F`

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

- `MTX.BasesSubmodules(` module`) F`

  returns a list containing a basis for every submodule.

- `MTX.BasesMinimalSubmodules(` module`) F`

  returns a list of bases of all minimal submodules.

- `MTX.BasesMaximalSubmodules(` module`) F`

  returns a list of bases of all maximal submodules.

- `MTX.BasisRadical(` module`) F`

  returns a basis of the radical of module.

- `MTX.BasisSocle(` module`) F`

  returns a basis of the socle of module.

- `MTX.BasesMinimalSupermodules(` module`,` sub`) F`

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

- `MTX.BasesCompositionSeries(` module`) F`

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

- `MTX.CompositionFactors(` module`) F`

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

- `MTX.CollectedFactors(` module`) F`

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

  ## 67.7 Induced Actions

- `MTX.NormedBasisAndBaseChange(`sub`)`

  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)

- `MTX.InducedActionSubmodule(` module`,` sub`) F`
- `MTX.InducedActionSubmoduleNB(` module`,` sub`) F`

  creates a new module corresponding to the action of module on 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`)

- `MTX.InducedActionFactorModule(` module`,` sub`[,` compl`] ) F`

  creates a new module corresponding to the action of module on the factor of 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`)

- `MTX.InducedActionMatrix(`mat`,`sub`)`
- `MTX.InducedActionMatrixNB(`mat`,`sub`)`
- `MTX.InducedActionFactorMatrix(` mat`,` sub`[,` compl`] ) F`

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

- `MTX.InducedAction(` module`,` sub`[,` type`] ) F`

  Computes induced actions on submodules or factormodules 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.

  ## 67.8 Module Homomorphisms

- `MTX.IsEquivalent(` module1`,` module2`) F`

  tests two irreducible modules for equivalence.

- `MTX.Isomorphism(` module1`,` module2`) F`

  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 module2 in the standard basis of module1. That is, conjugation of the generators of module2 with M yields the generators of module1.

- `MTX.Homomorphism(` module1`,` module2`,` mat`) F`

  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.

- `MTX.Homomorphisms(` module1`,` module2`) F`

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

- `MTX.Distinguish(` cf`,` nr`) F`

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

  ## 67.9 Invariant Forms

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

- `MTX.InvariantBilinearForm(`module`) F`

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

- `MTX.InvariantSesquilinearForm(`module`) F`

  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.

- `MTX.InvariantQuadraticForm(`module`) F`

  returns an invariant quadratic form of module, or `fail` if no such form exists. If the characteristic of the field over which module is defined is not 2, then the invariant bilinear form (if any) divided by two will be returned. In characteristic 2, the form returned will be lower triangular.

- `MTX.BasisInOrbit(`module`) F`

  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` in characteristic 2.

- `MTX.OrthogonalSign(`module`) F`

  for an even dimensional module, returns 1 or -1, according as `MTX.InvariantQuadraticForm(`module`)` is of + or - type. For an odd dimensional module, returns 0. For a module with no invariant quadratic form, returns `fail`. This calculation uses an algorithm due to Jon Thackray.

  ## 67.10 The Smash MeatAxe

  The standard MeatAxe provided in the GAP library is 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` is assigned by default. For the sake of completeness the remaining sections document more technical functions of this MeatAxe.

- `SMTX.RandomIrreducibleSubGModule(` module`) F`

  returns the module action on a random irreducible submodule.

- `SMTX.GoodElementGModule(` module`) F`

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

- `SMTX.SortHomGModule(` module1`,` module2`,` homs`) F`

  Function to sort the output of `Homomorphisms`.

- `SMTX.MinimalSubGModules(` module1`,` module2`[,` max`] )`

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

- `SMTX.Setter(` string`)`

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

- `SMTX.Getter(` string`)`

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

- `SMTX.IrreducibilityTest(` module`)`

  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.

- `SMTX.AbsoluteIrreducibilityTest(` module`)`

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

- `SMTX.MinimalSubGModule(` module`,` cf`,` nr`)`

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

- `SMTX.MatrixSum(` matrices1`,` matrices2`)`

  creates the direct sum of two matrix lists.

- `SMTX.CompleteBasis(` module`,` pbasis`)`

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

  ## 67.11 Smash MeatAxe Flags

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

- `SMTX.Subbasis`

  Basis of a submodule.

- `SMTX.AlgEl`

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

- `SMTX.AlgElMat`

  matrix of `SMTX.AlgEl`.

- `SMTX.AlgElCharPol`

  minimal polynomial of `SMTX.AlgEl`.

- `SMTX.AlgElCharPolFac`

  uses factor of `SMTX.AlgEl`.

- `SMTX.AlgElNullspaceVec`

  nullspace of the matrix evaluated under this factor.

- `SMTX.AlgElNullspaceDimension`

  dimension of the nullspace.

- `SMTX.CentMat`

- `SMTX.CentMatMinPoly`

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  GAP 4 manual
  March 2004