  
  [1X69 [33X[0;0YThe MeatAxe[133X[101X
  
  [33X[0;0YThe  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.[133X
  
  [33X[0;0Y[5XGAP[105X  uses  the  improved  MeatAxe  of  Derek  Holt  and Sarah Rees, and also
  incorporates further improvements of Ivanyos and Lux.[133X
  
  [33X[0;0YPlease  note  that,  consistently with the convention for group actions, the
  action  of  the  [5XGAP[105X  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.[133X
  
  
  [1X69.1 [33X[0;0YMeatAxe Modules[133X[101X
  
  
  [1X69.1-1 [33X[0;0YGModuleByMats[133X[101X
  
  [33X[1;0Y[29X[2XGModuleByMats[102X( [3Xgens[103X, [3Xfield[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGModuleByMats[102X( [3Xemptygens[103X, [3Xdim[103X, [3Xfield[103X ) [32X function[133X
  
  [33X[0;0Ycreates  a MeatAxe module over [3Xfield[103X from a list of invertible matrices [3Xgens[103X
  which  reflect  a  group's  action.  If the list of generators is empty, the
  dimension must be given as second argument.[133X
  
  [33X[0;0YMeatAxe routines are on a level with Gaussian elimination. Therefore they do
  not  deal  with  [5XGAP[105X modules but essentially with lists of matrices. For the
  MeatAxe, a module is a record with components[133X
  
  [8X[10Xgenerators[110X[8X[108X
        [33X[0;6YA  list  of  matrices  which  represent  a group operation on a finite
        dimensional row vector space.[133X
  
  [8X[10Xdimension[110X[8X[108X
        [33X[0;6YThe  dimension  of  the vector space (this is the common length of the
        row vectors (see [2XDimensionOfVectors[102X ([14X61.9-6[114X))).[133X
  
  [8X[10Xfield[110X[8X[108X
        [33X[0;6YThe field over which the vector space is defined.[133X
  
  [33X[0;0YOnce a module has been created its entries may not be changed. A MeatAxe may
  create  a  new  component  [3XNameOfMeatAxe[103X  in  which  it  can  store  private
  information.  By a MeatAxe [21Xsubmodule[121X or [21Xfactor module[121X we denote actually the
  [13Xinduced  action[113X  on the submodule, respectively factor module. Therefore the
  submodules  or  factor modules are again MeatAxe modules. The arrangement of
  [10Xgenerators[110X  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.[133X
  
  
  [1X69.2 [33X[0;0YModule Constructions[133X[101X
  
  
  [1X69.2-1 [33X[0;0YNaturalGModule[133X[101X
  
  [33X[1;0Y[29X[2XNaturalGModule[102X( [3Xgroup[103X[, [3Xfield[103X] ) [32X function[133X
  
  [33X[0;0Ycreates  a MeatAxe module over [3Xfield[103X from the generators of the matrix group
  [3Xgroup[103X.   If   [3Xfield[103X   is   not   provided   then   the   value  returned  by
  [2XDefaultFieldOfMatrixGroup[102X ([14X44.2-2[114X) is used instead.[133X
  
  [1X69.2-2 PermutationGModule[101X
  
  [33X[1;0Y[29X[2XPermutationGModule[102X( [3XG[103X, [3XF[103X ) [32X function[133X
  
  [33X[0;0YCalled  with  a  permutation  group  [3XG[103X  and  a  field [3XF[103X ([3XF[103X may be infinite),
  [2XPermutationGModule[102X  returns  the natural permutation module [22XM[122X over [3XF[103X for the
  group  of  permutation matrices that acts on the canonical basis of [22XM[122X in the
  same   way  as  [3XG[103X  acts  on  the  points  up  to  its  largest  moved  point
  (see [2XLargestMovedPoint[102X ([14X42.3-2[114X)).[133X
  
  [1X69.2-3 TrivialGModule[101X
  
  [33X[1;0Y[29X[2XTrivialGModule[102X( [3XG[103X, [3XF[103X ) [32X function[133X
  
  [33X[0;0YCalled  with  a  group  [3XG[103X  and a field [3XF[103X ([3XF[103X may be infinite), [2XTrivialGModule[102X
  returns the trivial module over [3XF[103X.[133X
  
  [1X69.2-4 TensorProductGModule[101X
  
  [33X[1;0Y[29X[2XTensorProductGModule[102X( [3Xm1[103X, [3Xm2[103X ) [32X function[133X
  
  [33X[0;0Y[2XTensorProductGModule[102X calculates the tensor product of the modules [3Xm1[103X and [3Xm2[103X.
  They are assumed to be modules over the same algebra so, in particular, they
  should have the same number of generators.[133X
  
  [1X69.2-5 DirectSumGModule[101X
  
  [33X[1;0Y[29X[2XDirectSumGModule[102X( [3Xm1[103X, [3Xm2[103X ) [32X function[133X
  
  [33X[0;0Y[2XDirectSumGModule[102X  calculates  the  direct sum of the modules [3Xm1[103X and [3Xm2[103X. They
  are  assumed  to  be  modules  over the same algebra so, in particular, they
  should have the same number of generators.[133X
  
  [1X69.2-6 WedgeGModule[101X
  
  [33X[1;0Y[29X[2XWedgeGModule[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Y[2XWedgeGModule[102X  calculates the wedge product of a [3XG[103X-module. That is the action
  on antisymmetric tensors.[133X
  
  
  [1X69.3 [33X[0;0YSelecting a Different MeatAxe[133X[101X
  
  [1X69.3-1 MTX[101X
  
  [33X[1;0Y[29X[2XMTX[102X [32X global variable[133X
  
  [33X[0;0YAll  MeatAxe  routines  are accessed via the global variable [2XMTX[102X, 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 [2XMTX[102X to this variable
  selects the corresponding MeatAxe.[133X
  
  
  [1X69.4 [33X[0;0YAccessing a Module[133X[101X
  
  [33X[0;0YEven  though  a  MeatAxe  module is a record, its components should never be
  accessed  outside  of  MeatAxe  functions.  Instead the following operations
  should be used:[133X
  
  [1X69.4-1 MTX.Generators[101X
  
  [33X[1;0Y[29X[2XMTX.Generators[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a list of matrix generators of [3Xmodule[103X.[133X
  
  [1X69.4-2 MTX.Dimension[101X
  
  [33X[1;0Y[29X[2XMTX.Dimension[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns the dimension in which the matrices act.[133X
  
  [1X69.4-3 MTX.Field[101X
  
  [33X[1;0Y[29X[2XMTX.Field[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns the field over which [3Xmodule[103X is defined.[133X
  
  
  [1X69.5 [33X[0;0YIrreducibility Tests[133X[101X
  
  [1X69.5-1 MTX.IsIrreducible[101X
  
  [33X[1;0Y[29X[2XMTX.IsIrreducible[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ytests  whether  the  module  [3Xmodule[103X  is irreducible (i.e. contains no proper
  submodules.)[133X
  
  [1X69.5-2 MTX.IsAbsolutelyIrreducible[101X
  
  [33X[1;0Y[29X[2XMTX.IsAbsolutelyIrreducible[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0YA  module  is  absolutely  irreducible  if  it  remains irreducible over the
  algebraic  closure of the field. (Formally: If the tensor product [22XL ⊗_K M[122X is
  irreducible  where  [22XM[122X  is  the  module defined over [22XK[122X and [22XL[122X is the algebraic
  closure of [22XK[122X.)[133X
  
  [1X69.5-3 MTX.DegreeSplittingField[101X
  
  [33X[1;0Y[29X[2XMTX.DegreeSplittingField[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns the degree of the splitting field as extension of the prime field.[133X
  
  
  [1X69.6 [33X[0;0YDecomposition of modules[133X[101X
  
  [33X[0;0YA  module  is  [13Xdecomposable[113X  if  it  can be written as the direct sum of two
  proper  submodules  (and  [13Xindecomposable[113X  if  not).  Obviously  every finite
  dimensional  module  is  a  direct  sum  of  its  indecomposable  parts. The
  [13Xhomogeneous  components[113X  of  a  module  are  the  direct  sums of isomorphic
  indecomposable components. They are uniquely determined.[133X
  
  [1X69.6-1 MTX.IsIndecomposable[101X
  
  [33X[1;0Y[29X[2XMTX.IsIndecomposable[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns whether [3Xmodule[103X is indecomposable.[133X
  
  [1X69.6-2 MTX.Indecomposition[101X
  
  [33X[1;0Y[29X[2XMTX.Indecomposition[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a decomposition of [3Xmodule[103X as a direct sum of indecomposable modules.
  It returns a list, each entry is a list of form [[3XB[103X,[3Xind[103X] where [3XB[103X is a list of
  basis  vectors  for  the indecomposable component and [3Xind[103X the induced module
  action on this component. (Such a decomposition is not unique.)[133X
  
  [1X69.6-3 MTX.HomogeneousComponents[101X
  
  [33X[1;0Y[29X[2XMTX.HomogeneousComponents[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ycomputes   the   homogeneous   components   of   [3Xmodule[103X  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.[133X
  
  [8X[10Xindices[110X[8X[108X
        [33X[0;6Ythe  index  numbers  of  the  indecomposable  components,  as given by
        [2XMTX.Indecomposition[102X ([14X69.6-2[114X), that are in the homogeneous component,[133X
  
  [8X[10Xcomponent[110X[8X[108X
        [33X[0;6Yone of the indecomposable components,[133X
  
  [8X[10Ximages[110X[8X[108X
        [33X[0;6Ya  list  of  the  remaining indecomposable components, each given as a
        record  with  the  components  [10Xcomponent[110X  (the  component  itself) and
        [10Xisomorphism[110X (an isomorphism from the defining component to this one).[133X
  
  
  [1X69.7 [33X[0;0YFinding Submodules[133X[101X
  
  [1X69.7-1 MTX.SubmoduleGModule[101X
  
  [33X[1;0Y[29X[2XMTX.SubmoduleGModule[102X( [3Xmodule[103X, [3Xsubspace[103X ) [32X function[133X
  [33X[1;0Y[29X[2XMTX.SubGModule[102X( [3Xmodule[103X, [3Xsubspace[103X ) [32X function[133X
  
  [33X[0;0Y[3Xsubspace[103X  should  be  a  subspace  of (or a vector in) the underlying vector
  space  of  [3Xmodule[103X i.e. the full row space of the same dimension and over the
  same  field  as  [3Xmodule[103X.  A  normalized  basis  of  the  submodule of [3Xmodule[103X
  generated by [3Xsubspace[103X is returned.[133X
  
  [1X69.7-2 MTX.ProperSubmoduleBasis[101X
  
  [33X[1;0Y[29X[2XMTX.ProperSubmoduleBasis[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  basis  of  a  proper submodule of [3Xmodule[103X and [9Xfail[109X if no proper
  submodule exists.[133X
  
  [1X69.7-3 MTX.BasesSubmodules[101X
  
  [33X[1;0Y[29X[2XMTX.BasesSubmodules[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a list containing a basis for every submodule.[133X
  
  [1X69.7-4 MTX.BasesMinimalSubmodules[101X
  
  [33X[1;0Y[29X[2XMTX.BasesMinimalSubmodules[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a list of bases of all minimal submodules.[133X
  
  [1X69.7-5 MTX.BasesMaximalSubmodules[101X
  
  [33X[1;0Y[29X[2XMTX.BasesMaximalSubmodules[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a list of bases of all maximal submodules.[133X
  
  [1X69.7-6 MTX.BasisRadical[101X
  
  [33X[1;0Y[29X[2XMTX.BasisRadical[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a basis of the radical of [3Xmodule[103X.[133X
  
  [1X69.7-7 MTX.BasisSocle[101X
  
  [33X[1;0Y[29X[2XMTX.BasisSocle[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a basis of the socle of [3Xmodule[103X.[133X
  
  [1X69.7-8 MTX.BasesMinimalSupermodules[101X
  
  [33X[1;0Y[29X[2XMTX.BasesMinimalSupermodules[102X( [3Xmodule[103X, [3Xsub[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list of bases of all minimal supermodules of the submodule given
  by the basis [3Xsub[103X.[133X
  
  [1X69.7-9 MTX.BasesCompositionSeries[101X
  
  [33X[1;0Y[29X[2XMTX.BasesCompositionSeries[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list of bases of submodules in a composition series in ascending
  order.[133X
  
  [1X69.7-10 MTX.CompositionFactors[101X
  
  [33X[1;0Y[29X[2XMTX.CompositionFactors[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a list of composition factors of [3Xmodule[103X in ascending order.[133X
  
  [1X69.7-11 MTX.CollectedFactors[101X
  
  [33X[1;0Y[29X[2XMTX.CollectedFactors[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list  giving  all  irreducible  composition  factors  with their
  frequencies.[133X
  
  
  [1X69.8 [33X[0;0YInduced Actions[133X[101X
  
  [1X69.8-1 MTX.NormedBasisAndBaseChange[101X
  
  [33X[1;0Y[29X[2XMTX.NormedBasisAndBaseChange[102X( [3Xsub[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  list [10X[[3Xbas[103X[10X, [3Xchange[103X[10X ][110X where [3Xbas[103X is a normed basis (i.e. in echelon
  form with pivots normed to 1) for [3Xsub[103X and [3Xchange[103X is the base change from [3Xbas[103X
  to [3Xsub[103X (the basis vectors of [3Xbas[103X expressed in coefficients for [3Xsub[103X).[133X
  
  [1X69.8-2 MTX.InducedActionSubmodule[101X
  
  [33X[1;0Y[29X[2XMTX.InducedActionSubmodule[102X( [3Xmodule[103X, [3Xsub[103X ) [32X function[133X
  [33X[1;0Y[29X[2XMTX.InducedActionSubmoduleNB[102X( [3Xmodule[103X, [3Xsub[103X ) [32X function[133X
  
  [33X[0;0Ycreates  a  new  module  corresponding  to  the  action  of  [3Xmodule[103X  on  the
  non-trivial  submodule  [3Xsub[103X. In the [10XNB[110X version the basis [3Xsub[103X must be normed.
  (That   is   it   must   be  in  echelon  form  with  pivots  normed  to  1,
  see [2XMTX.NormedBasisAndBaseChange[102X ([14X69.8-1[114X).)[133X
  
  [1X69.8-3 MTX.InducedActionFactorModule[101X
  
  [33X[1;0Y[29X[2XMTX.InducedActionFactorModule[102X( [3Xmodule[103X, [3Xsub[103X[, [3Xcompl[103X] ) [32X function[133X
  
  [33X[0;0Ycreates  a new module corresponding to the action of [3Xmodule[103X on the factor of
  the  proper  submodule  [3Xsub[103X.  If  [3Xcompl[103X  is given, it has to be a basis of a
  (vector space-)complement of [3Xsub[103X. The action then will correspond to [3Xcompl[103X.[133X
  
  [33X[0;0YThe basis [3Xsub[103X has to be given in normed form. (That is it must be in echelon
  form with pivots normed to 1, see [2XMTX.NormedBasisAndBaseChange[102X ([14X69.8-1[114X))[133X
  
  [1X69.8-4 MTX.InducedActionSubMatrix[101X
  
  [33X[1;0Y[29X[2XMTX.InducedActionSubMatrix[102X( [3Xmat[103X, [3Xsub[103X ) [32X function[133X
  [33X[1;0Y[29X[2XMTX.InducedActionSubMatrixNB[102X( [3Xmat[103X, [3Xsub[103X ) [32X function[133X
  [33X[1;0Y[29X[2XMTX.InducedActionFactorMatrix[102X( [3Xmat[103X, [3Xsub[103X[, [3Xcompl[103X] ) [32X function[133X
  
  [33X[0;0Ywork the same way as the above functions for modules, but take as input only
  a single matrix.[133X
  
  [1X69.8-5 MTX.InducedAction[101X
  
  [33X[1;0Y[29X[2XMTX.InducedAction[102X( [3Xmodule[103X, [3Xsub[103X[, [3Xtype[103X] ) [32X function[133X
  
  [33X[0;0YComputes  induced  actions  on submodules or factor modules and also returns
  the  corresponding  bases.  The  action  taken  is binary encoded in [3Xtype[103X: [10X1[110X
  stands  for  subspace  action,  [10X2[110X for factor action, and [10X4[110X for action of the
  full  module  on  a subspace adapted basis. The routine returns the computed
  results  in  a list in sequence ([3Xsub[103X,[3Xquot[103X,[3Xboth[103X,[3Xbasis[103X) where [3Xbasis[103X is a basis
  for  the  whole  space,  extending  [3Xsub[103X. (Actions which are not computed are
  omitted,  so  the  returned list may be shorter.) If no [3Xtype[103X is given, it is
  assumed to be [10X7[110X. The basis given in [3Xsub[103X must be normed![133X
  
  [33X[0;0YAll these routines return [9Xfail[109X if [3Xsub[103X is not a proper subspace.[133X
  
  
  [1X69.9 [33X[0;0YModule Homomorphisms[133X[101X
  
  [1X69.9-1 MTX.BasisModuleHomomorphisms[101X
  
  [33X[1;0Y[29X[2XMTX.BasisModuleHomomorphisms[102X( [3Xmodule1[103X, [3Xmodule2[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  basis  of  all  module  homomorphisms  from  [3Xmodule1[103X to [3Xmodule2[103X.
  Homomorphisms  are  by  matrices, whose rows give the images of the standard
  basis vectors of [3Xmodule1[103X in the standard basis of [3Xmodule2[103X.[133X
  
  [1X69.9-2 MTX.BasisModuleEndomorphisms[101X
  
  [33X[1;0Y[29X[2XMTX.BasisModuleEndomorphisms[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns a basis of all module homomorphisms from [3Xmodule[103X to [3Xmodule[103X.[133X
  
  [1X69.9-3 MTX.IsomorphismModules[101X
  
  [33X[1;0Y[29X[2XMTX.IsomorphismModules[102X( [3Xmodule1[103X, [3Xmodule2[103X ) [32X function[133X
  
  [33X[0;0YIf  [3Xmodule1[103X  and  [3Xmodule2[103X  are  isomorphic modules, this function returns an
  isomorphism  from [3Xmodule1[103X to [3Xmodule2[103X in form of a matrix. It returns [9Xfail[109X if
  the modules are not isomorphic.[133X
  
  [1X69.9-4 MTX.ModuleAutomorphisms[101X
  
  [33X[1;0Y[29X[2XMTX.ModuleAutomorphisms[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns the module automorphisms of [3Xmodule[103X (the set of all isomorphisms from
  [3Xmodule[103X to itself) as a matrix group.[133X
  
  
  [1X69.10 [33X[0;0YModule Homomorphisms for irreducible modules[133X[101X
  
  [33X[0;0YThe   following   are   lower-level   functions  that  provide  homomorphism
  functionality for irreducible modules. Generic code should use the functions
  in Section [14X69.9[114X instead.[133X
  
  [1X69.10-1 MTX.IsEquivalent[101X
  
  [33X[1;0Y[29X[2XMTX.IsEquivalent[102X( [3Xmodule1[103X, [3Xmodule2[103X ) [32X function[133X
  
  [33X[0;0Ytests two irreducible modules for equivalence.[133X
  
  [1X69.10-2 MTX.IsomorphismIrred[101X
  
  [33X[1;0Y[29X[2XMTX.IsomorphismIrred[102X( [3Xmodule1[103X, [3Xmodule2[103X ) [32X function[133X
  
  [33X[0;0Yreturns  an  isomorphism  from  [3Xmodule1[103X to [3Xmodule2[103X (if one exists), and [9Xfail[109X
  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 [22XM[122X, whose rows give
  the images of the standard basis vectors of [3Xmodule1[103X in the standard basis of
  [3Xmodule2[103X. That is, conjugation of the generators of [3Xmodule2[103X with [22XM[122X yields the
  generators of [3Xmodule1[103X.[133X
  
  [1X69.10-3 MTX.Homomorphism[101X
  
  [33X[1;0Y[29X[2XMTX.Homomorphism[102X( [3Xmodule1[103X, [3Xmodule2[103X, [3Xmat[103X ) [32X function[133X
  
  [33X[0;0Y[3Xmat[103X  should  be a [3Xdim1[103X [22X×[122X [3Xdim2[103X matrix defining a homomorphism from [3Xmodule1[103X to
  [3Xmodule2[103X.  This  function  verifies  that  [3Xmat[103X  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.[133X
  
  [1X69.10-4 MTX.Homomorphisms[101X
  
  [33X[1;0Y[29X[2XMTX.Homomorphisms[102X( [3Xmodule1[103X, [3Xmodule2[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  basis  of  the  space  of all homomorphisms from the irreducible
  module [3Xmodule1[103X to [3Xmodule2[103X.[133X
  
  [1X69.10-5 MTX.Distinguish[101X
  
  [33X[1;0Y[29X[2XMTX.Distinguish[102X( [3Xcf[103X, [3Xnr[103X ) [32X function[133X
  
  [33X[0;0YLet  [3Xcf[103X  be the output of [2XMTX.CollectedFactors[102X ([14X69.7-11[114X). This routine tries
  to  find  a  group  algebra element that has nullity zero on all composition
  factors except number [3Xnr[103X.[133X
  
  
  [1X69.11 [33X[0;0YMeatAxe Functionality for Invariant Forms[133X[101X
  
  [33X[0;0YThe  functions  in  this  section  can  only  be  applied  to  an absolutely
  irreducible MeatAxe module.[133X
  
  [1X69.11-1 MTX.InvariantBilinearForm[101X
  
  [33X[1;0Y[29X[2XMTX.InvariantBilinearForm[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns   an   invariant   bilinear   form,   which   may  be  symmetric  or
  anti-symmetric, of [3Xmodule[103X, or [9Xfail[109X if no such form exists.[133X
  
  [1X69.11-2 MTX.InvariantSesquilinearForm[101X
  
  [33X[1;0Y[29X[2XMTX.InvariantSesquilinearForm[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns an invariant hermitian (= self-adjoint) sesquilinear form of [3Xmodule[103X,
  which  must  be defined over a finite field whose order is a square, or [9Xfail[109X
  if no such form exists.[133X
  
  [1X69.11-3 MTX.InvariantQuadraticForm[101X
  
  [33X[1;0Y[29X[2XMTX.InvariantQuadraticForm[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns  either  the matrix of an invariant quadratic form of the absolutely
  irreducible module [3Xmodule[103X, or [9Xfail[109X.[133X
  
  [33X[0;0YIf  the characteristic of [3Xmodule[103X is odd then [9Xfail[109X is returned if there is no
  nonzero  invariant  bilinear  form,  otherwise a matrix of the bilinear form
  divided  by  [22X2[122X  is  returned; note that this matrix may be antisymmetric and
  thus  describe the zero quadratic form. If the characteristic of [3Xmodule[103X is [22X2[122X
  then  [9Xfail[109X  is  returned  if [3Xmodule[103X does not admit a nonzero quadratic form,
  otherwise a lower triangular matrix describing the form is returned.[133X
  
  [33X[0;0YAn error is signalled if [3Xmodule[103X is not absolutely irreducible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= SO(-1, 4, 2);;[127X[104X
    [4X[25Xgap>[125X [27Xm:= NaturalGModule( g );;[127X[104X
    [4X[25Xgap>[125X [27XDisplay( MTX.InvariantQuadraticForm( m ) );[127X[104X
    [4X[28X . . . .[128X[104X
    [4X[28X 1 . . .[128X[104X
    [4X[28X . . 1 .[128X[104X
    [4X[28X . . 1 1[128X[104X
    [4X[25Xgap>[125X [27Xg:= Sp(4, 2);;[127X[104X
    [4X[25Xgap>[125X [27Xm:= NaturalGModule( g );;[127X[104X
    [4X[25Xgap>[125X [27XMTX.InvariantQuadraticForm( m );[127X[104X
    [4X[28Xfail[128X[104X
    [4X[25Xgap>[125X [27Xg:= Sp(4, 3);;[127X[104X
    [4X[25Xgap>[125X [27Xm:= NaturalGModule( g );;[127X[104X
    [4X[25Xgap>[125X [27Xq:= MTX.InvariantQuadraticForm( m );;[127X[104X
    [4X[25Xgap>[125X [27Xq = - TransposedMat( q );  # antisymmetric inv. bilinear form[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X69.11-4 MTX.BasisInOrbit[101X
  
  [33X[1;0Y[29X[2XMTX.BasisInOrbit[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a basis of the underlying vector space of [3Xmodule[103X which is contained
  in an orbit of the action of the generators of module on that space. This is
  used by [2XMTX.InvariantQuadraticForm[102X ([14X69.11-3[114X) in characteristic 2.[133X
  
  [1X69.11-5 MTX.OrthogonalSign[101X
  
  [33X[1;0Y[29X[2XMTX.OrthogonalSign[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0YLet  [3Xmodule[103X  be an absolutely irreducible [22XG[122X-module. If [3Xmodule[103X does not fix a
  nondegenerate  quadratic  form see [2XMTX.InvariantQuadraticForm[102X ([14X69.11-3[114X) then
  [9Xfail[109X  is returned. Otherwise the sign [22Xϵ ∈ { -1, 0, 1 }[122X is returned such that
  [22XG[122X  embeds  into the general orthogonal group [22XGO^ϵ(d, q)[122X w.r.t. the invariant
  quadratic  form, see [2XGeneralOrthogonalGroup[102X ([14X50.2-6[114X). That is, [10X0[110X is returned
  if [3Xmodule[103X has odd dimension, and [10X1[110X or [10X-1[110X is returned if the orthogonal group
  has plus or minus type, respectively.[133X
  
  [33X[0;0YAn error is signalled if [3Xmodule[103X is not absolutely irreducible.[133X
  
  [33X[0;0YThe [10XSMTX[110X implementation uses an algorithm due to Jon Thackray.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= GeneratorsOfGroup( GO(1,4,2) );;[127X[104X
    [4X[25Xgap>[125X [27XMTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27Xmats:= GeneratorsOfGroup( GO(-1,4,2) );;[127X[104X
    [4X[25Xgap>[125X [27XMTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );[127X[104X
    [4X[28X-1[128X[104X
    [4X[25Xgap>[125X [27Xmats:= GeneratorsOfGroup( GO(5,3) );;[127X[104X
    [4X[25Xgap>[125X [27XMTX.OrthogonalSign( GModuleByMats( mats, GF(3) ) );[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xmats:= GeneratorsOfGroup( SP(4,2) );;[127X[104X
    [4X[25Xgap>[125X [27XMTX.OrthogonalSign( GModuleByMats( mats, GF(2) ) );[127X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  
  [1X69.12 [33X[0;0YThe Smash MeatAxe[133X[101X
  
  [33X[0;0YThe  standard MeatAxe provided in the [5XGAP[105X library is based on the MeatAxe in
  the  [5XGAP[105X 3  package  [5XSmash[105X,  originally written by Derek Holt and Sarah Rees
  [HR94].  It  is  accessible  via  the variable [10XSMTX[110X to which [2XMTX[102X ([14X69.3-1[114X) is
  assigned  by  default.  For  the sake of completeness the remaining sections
  document more technical functions of this MeatAxe.[133X
  
  [1X69.12-1 SMTX.RandomIrreducibleSubGModule[101X
  
  [33X[1;0Y[29X[2XSMTX.RandomIrreducibleSubGModule[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yreturns the module action on a random irreducible submodule.[133X
  
  [1X69.12-2 SMTX.GoodElementGModule[101X
  
  [33X[1;0Y[29X[2XSMTX.GoodElementGModule[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yfinds  an  element  with  minimal  possible nullspace dimension if [3Xmodule[103X is
  known to be irreducible.[133X
  
  [1X69.12-3 SMTX.SortHomGModule[101X
  
  [33X[1;0Y[29X[2XSMTX.SortHomGModule[102X( [3Xmodule1[103X, [3Xmodule2[103X, [3Xhoms[103X ) [32X function[133X
  
  [33X[0;0YFunction to sort the output of [10XHomomorphisms[110X.[133X
  
  [1X69.12-4 SMTX.MinimalSubGModules[101X
  
  [33X[1;0Y[29X[2XSMTX.MinimalSubGModules[102X( [3Xmodule1[103X, [3Xmodule2[103X[, [3Xmax[103X] ) [32X function[133X
  
  [33X[0;0Yreturns (at most [3Xmax[103X) bases of submodules of [3Xmodule2[103X which are isomorphic to
  the irreducible module [3Xmodule1[103X.[133X
  
  [1X69.12-5 SMTX.Setter[101X
  
  [33X[1;0Y[29X[2XSMTX.Setter[102X( [3Xstring[103X ) [32X function[133X
  
  [33X[0;0Yreturns a setter function for the component [10XsmashMeataxe.(string)[110X.[133X
  
  [1X69.12-6 SMTX.Getter[101X
  
  [33X[1;0Y[29X[2XSMTX.Getter[102X( [3Xstring[103X ) [32X function[133X
  
  [33X[0;0Yreturns a getter function for the component [10XsmashMeataxe.(string)[110X.[133X
  
  [1X69.12-7 SMTX.IrreducibilityTest[101X
  
  [33X[1;0Y[29X[2XSMTX.IrreducibilityTest[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0YTests  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.[133X
  
  [1X69.12-8 SMTX.AbsoluteIrreducibilityTest[101X
  
  [33X[1;0Y[29X[2XSMTX.AbsoluteIrreducibilityTest[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0YTests  for  absolute  irreducibility  and  sets  splitting  field degree. It
  neither sets an absolute irreducibility flag, nor tests it.[133X
  
  [1X69.12-9 SMTX.MinimalSubGModule[101X
  
  [33X[1;0Y[29X[2XSMTX.MinimalSubGModule[102X( [3Xmodule[103X, [3Xcf[103X, [3Xnr[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the basis of a minimal submodule of [3Xmodule[103X containing the indicated
  composition factor. It assumes [10XDistinguish[110X has been called already.[133X
  
  [1X69.12-10 SMTX.MatrixSum[101X
  
  [33X[1;0Y[29X[2XSMTX.MatrixSum[102X( [3Xmatrices1[103X, [3Xmatrices2[103X ) [32X function[133X
  
  [33X[0;0Ycreates the direct sum of two matrix lists.[133X
  
  [1X69.12-11 SMTX.CompleteBasis[101X
  
  [33X[1;0Y[29X[2XSMTX.CompleteBasis[102X( [3Xmodule[103X, [3Xpbasis[103X ) [32X function[133X
  
  [33X[0;0Yextends  the  partial basis [3Xpbasis[103X to a basis of the full space by action of
  [3Xmodule[103X. It returns whether it succeeded.[133X
  
  
  [1X69.13 [33X[0;0YSmash MeatAxe Flags[133X[101X
  
  [33X[0;0YThe  following  getter routines access internal flags. For each routine, the
  appropriate setter's name is prefixed with [10XSet[110X.[133X
  
  [1X69.13-1 SMTX.Subbasis[101X
  
  [33X[1;0Y[29X[2XSMTX.Subbasis[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0YBasis of a submodule.[133X
  
  [1X69.13-2 SMTX.AlgEl[101X
  
  [33X[1;0Y[29X[2XSMTX.AlgEl[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ylist [10X[newgens,coefflist][110X giving an algebra element used for chopping.[133X
  
  [1X69.13-3 SMTX.AlgElMat[101X
  
  [33X[1;0Y[29X[2XSMTX.AlgElMat[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ymatrix of [2XSMTX.AlgEl[102X ([14X69.13-2[114X).[133X
  
  [1X69.13-4 SMTX.AlgElCharPol[101X
  
  [33X[1;0Y[29X[2XSMTX.AlgElCharPol[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yminimal polynomial of [2XSMTX.AlgEl[102X ([14X69.13-2[114X).[133X
  
  [1X69.13-5 SMTX.AlgElCharPolFac[101X
  
  [33X[1;0Y[29X[2XSMTX.AlgElCharPolFac[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yuses factor of [2XSMTX.AlgEl[102X ([14X69.13-2[114X).[133X
  
  [1X69.13-6 SMTX.AlgElNullspaceVec[101X
  
  [33X[1;0Y[29X[2XSMTX.AlgElNullspaceVec[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ynullspace of the matrix evaluated under this factor.[133X
  
  [1X69.13-7 SMTX.AlgElNullspaceDimension[101X
  
  [33X[1;0Y[29X[2XSMTX.AlgElNullspaceDimension[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ydimension of the nullspace.[133X
  
  [1X69.13-8 SMTX.CentMat[101X
  
  [33X[1;0Y[29X[2XSMTX.CentMat[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Ymatrix  centralising  all  generators  which  is  computed as a byproduct of
  [2XSMTX.AbsoluteIrreducibilityTest[102X ([14X69.12-8[114X).[133X
  
  [1X69.13-9 SMTX.CentMatMinPoly[101X
  
  [33X[1;0Y[29X[2XSMTX.CentMatMinPoly[102X( [3Xmodule[103X ) [32X function[133X
  
  [33X[0;0Yminimal polynomial of [2XSMTX.CentMat[102X ([14X69.13-8[114X).[133X
  
