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.
‣ 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
generatorsA list of matrices which represent a group operation on a finite dimensional row vector space.
dimensionThe dimension of the vector space (this is the common length of the row vectors (see DimensionOfVectors (61.9-6))).
fieldThe 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.
‣ 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)).
‣ 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.
‣ WedgeGModule( module ) | ( function ) |
WedgeGModule calculates the wedge product of a G-module. That is the action on antisymmetric tensors.
‣ 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.
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 ) | ( function ) |
returns a list of matrix generators of module.
‣ MTX.Dimension( module ) | ( function ) |
returns the dimension in which the matrices act.
‣ MTX.Field( module ) | ( function ) |
returns the field over which module is defined.
‣ MTX.IsIrreducible( module ) | ( function ) |
tests whether the module module is irreducible (i.e. contains no proper submodules.)
‣ 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 \(L \otimes_K M\) is irreducible where \(M\) is the module defined over \(K\) and \(L\) is the algebraic closure of \(K\).)
‣ MTX.DegreeSplittingField( module ) | ( function ) |
returns the degree of the splitting field as extension of the prime field.
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.
‣ MTX.IsIndecomposable( module ) | ( function ) |
returns whether module is indecomposable.
‣ 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.)
‣ 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.
indicesthe index numbers of the indecomposable components, as given by MTX.Indecomposition (69.6-2), that are in the homogeneous component,
componentone of the indecomposable components,
imagesa 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).
‣ 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.
‣ MTX.ProperSubmoduleBasis( module ) | ( function ) |
returns the basis of a proper submodule of module and fail if no proper submodule exists.
‣ MTX.BasesSubmodules( module ) | ( function ) |
returns a list containing a basis for every submodule.
‣ MTX.BasesMinimalSubmodules( module ) | ( function ) |
returns a list of bases of all minimal submodules.
‣ MTX.BasesMaximalSubmodules( module ) | ( function ) |
returns a list of bases of all maximal submodules.
‣ MTX.BasisRadical( module ) | ( function ) |
returns a basis of the radical of module.
‣ MTX.BasisSocle( module ) | ( function ) |
returns a basis of the socle of module.
‣ MTX.BasesMinimalSupermodules( module, sub ) | ( function ) |
returns a list of bases of all minimal supermodules of the submodule given by the basis sub.
‣ MTX.BasesCompositionSeries( module ) | ( function ) |
returns a list of bases of submodules in a composition series in ascending order.
‣ MTX.CompositionFactors( module ) | ( function ) |
returns a list of composition factors of module in ascending order.
‣ MTX.CollectedFactors( module ) | ( function ) |
returns a list giving all irreducible composition factors with their frequencies.
‣ 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).
‣ 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).)
‣ 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))
‣ 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.
‣ 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.
‣ 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.
‣ MTX.BasisModuleEndomorphisms( module ) | ( function ) |
returns a basis of all module homomorphisms from module to module.
‣ 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.
‣ MTX.ModuleAutomorphisms( module ) | ( function ) |
returns the module automorphisms of module (the set of all isomorphisms from module to itself) as a matrix group.
The following are lower-level functions that provide homomorphism functionality for irreducible modules. Generic code should use the functions in Section 69.9 instead.
‣ MTX.IsEquivalent( module1, module2 ) | ( function ) |
tests two irreducible modules for equivalence.
‣ 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.
‣ MTX.Homomorphism( module1, module2, mat ) | ( function ) |
mat should be a dim1 \(\times\) 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 ) | ( function ) |
returns a basis of the space of all homomorphisms from the irreducible module module1 to module2.
‣ 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.
The functions in this section can only be applied to an absolutely irreducible MeatAxe module.
‣ 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.
‣ 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.
‣ 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:= GModuleByMats( GeneratorsOfGroup( g ), GF(2) );; gap> Display( MTX.InvariantQuadraticForm( m ) ); . . . . 1 . . . . . 1 . . . 1 1 gap> g:= SP(4, 2);; gap> m:= GModuleByMats( GeneratorsOfGroup( g ), GF(2) );; gap> MTX.InvariantQuadraticForm( m ); fail gap> g:= SP(4, 3);; gap> m:= GModuleByMats( GeneratorsOfGroup( g ), GF(3) );; gap> q:= MTX.InvariantQuadraticForm( m );; gap> q = - TransposedMat( q ); # antisymmetric inv. bilinear form true
‣ 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.
‣ 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 \(\epsilon \in \{ -1, 0, 1 \}\) is returned such that \(G\) embeds into the general orthogonal group \(GO^{\epsilon}(d, q)\) 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
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.
‣ SMTX.RandomIrreducibleSubGModule( module ) | ( function ) |
returns the module action on a random irreducible submodule.
‣ SMTX.GoodElementGModule( module ) | ( function ) |
finds an element with minimal possible nullspace dimension if module is known to be irreducible.
‣ SMTX.SortHomGModule( module1, module2, homs ) | ( function ) |
Function to sort the output of Homomorphisms.
‣ SMTX.MinimalSubGModules( module1, module2[, max] ) | ( function ) |
returns (at most max) bases of submodules of module2 which are isomorphic to the irreducible module module1.
‣ SMTX.Setter( string ) | ( function ) |
returns a setter function for the component smashMeataxe.(string).
‣ SMTX.Getter( string ) | ( function ) |
returns a getter function for the component smashMeataxe.(string).
‣ 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.
‣ SMTX.AbsoluteIrreducibilityTest( module ) | ( function ) |
Tests for absolute irreducibility and sets splitting field degree. It neither sets an absolute irreducibility flag, nor tests it.
‣ 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.
‣ SMTX.MatrixSum( matrices1, matrices2 ) | ( function ) |
creates the direct sum of two matrix lists.
‣ 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.
The following getter routines access internal flags. For each routine, the appropriate setter's name is prefixed with Set.
‣ SMTX.Subbasis( module ) | ( function ) |
Basis of a submodule.
‣ SMTX.AlgEl( module ) | ( function ) |
list [newgens,coefflist] giving an algebra element used for chopping.
‣ SMTX.AlgElMat( module ) | ( function ) |
matrix of SMTX.AlgEl (69.13-2).
‣ SMTX.AlgElCharPol( module ) | ( function ) |
minimal polynomial of SMTX.AlgEl (69.13-2).
‣ SMTX.AlgElCharPolFac( module ) | ( function ) |
uses factor of SMTX.AlgEl (69.13-2).
‣ SMTX.AlgElNullspaceVec( module ) | ( function ) |
nullspace of the matrix evaluated under this factor.
‣ SMTX.AlgElNullspaceDimension( module ) | ( function ) |
dimension of the nullspace.
‣ SMTX.CentMat( module ) | ( function ) |
matrix centralising all generators which is computed as a byproduct of SMTX.AbsoluteIrreducibilityTest (69.12-8).
‣ SMTX.CentMatMinPoly( module ) | ( function ) |
minimal polynomial of SMTX.CentMat (69.13-8).
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