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.
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
dimension
field
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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 2006