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 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)). 69.2-2 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. 69.2-3 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.  Example  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  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.  Example  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).