Impressum meataxe.xml   Sprache: XML

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<!-- %W  meataxe.tex                 GAP documentation        Alexander Hulpke -->
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<!-- %Y  Copyright 1997,  Lehrstuhl D für Mathematik,  RWTH Aachen,   Germany -->
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<!-- %%  This file contains a description of the MeatAxe functions. -->
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<Chapter Label="The MeatAxe">
<Heading>The MeatAxe</Heading>

The MeatAxe <Cite Key="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.
<P/>
&GAP; uses the improved MeatAxe of Derek Holt and Sarah Rees, and
also incorporates further improvements of Ivanyos and Lux.
<P/>
Please note that, consistently with Therefore the submodules or factor modu are again  modules. The
on row vectors by multiplication on the right. If you want to investigate
left modules youC>generators</C is guaranteed to be the same for the induced


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<Section Label="MeatAxe Modules">
<Heading>MeatAxe</>

<ManSection java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
<Heading>ModuleByMats</eading
<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
 Label="for generators and a field"/>
<Func Name="GModuleByMats" Arg='emptygens, dim, field'
 Label="for empty list, the dimension, and a fieldFunc ="NaturalGModule"Arg=group[ '="  

<Description>
creates a MeatAxe module over <A>field</A> from a list of invertible group <A>group</A>. If <A>field</A> is not provided then the value returned by
<A>gens<A    'action the of  empty,
 java.lang.StringIndexOutOfBoundsException: Range [22, 13) out of bounds for length 47
<P/>
MeatAxe routines are on a level with Gaussian java.lang.StringIndexOutOfBoundsException: Range [0, 57) out of bounds for length 0
 &;    of. 
MeatAxe, a module is a record with components
<P/>
<t
<Mark></>/java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 30
<
   A list of matrices which represent a point ( (ee&<"java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 104
   Funcjava.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 40
</
<Description
<java.lang.StringIndexOutOfBoundsException: Range [11, 6) out of bounds for length 77
   Thedimensionof the vector  t    of
   the row vectors (see <Ref Attr="DimensionOfVectors"/>)).
</>
<Mark><C</escription>
<Item>
   The field over which the vector space is defined.
</Item>
</List>
<P/>
Once a module has been created its entriesManSection>
create a new component <A>NameOfMeatAxe</A> in which it can store private
module/> or<>actor />denote
actually the <E>java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
Therefore the submodules of the modules <A>m1</A> and <A>m2</A>.
arrangement of <C>generators</C>They are assumed the same algebra so,in particular,
,butto complete  original 
bases used are needed as well.
</Description>
<</>

<Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->Description
<Section  Func="WedgeGModulejava.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 77
<Heading>Module Constructions</Heading>

<ManSection>
<Heading>NaturalGModule</Heading>
<Func Name="NaturalGModule" Arg='group[, field]' Label="for matrix group and a field"/>
<Description>
creates a MeatAxe module over <A>field</A> from the generators of the matrix
group <A>group</A>. That is the action on antisymmetric tensors.
<Ref Attr="DefaultFieldOfMatrixGroup"/> is used instead./>
<>
</ManSection>

<ManSection>
<Func java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0

<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
Called with a java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<Ref Func"PermutationGModule/> returns the natural permutation module
<M>M</M> over <A>F</A>
for the group of permutation matrices that acts on<>Selecting a Different MeatAxe/>
<in thesame way as A></>on   up   
moved point (see <Ref is possibleto haveseveral  of MeatAxe available.
</Description>
</ManSection>

<ManSection>
<Func Each MeatAxerepresents routines  anown java.lang.StringIndexOutOfBoundsException: Range [67, 62) out of bounds for length 76

<Description>
 witha  A>G<>andafield <>/>(<><A>be)
<Ref 
/>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<ManSection>
<Func Name=<=AccessingModule>

<Description>
<Ref Func="TensorProductGModule"/> calculates the<> Module<
/A> <>2/>
They are assumed to      following
they  should have be :
</Description>
<< Name"Generators"=java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 42

<ManSection>
<Func Name="WedgeGModule" Arg/

<>
<Ref Func="WedgeGModuleManSection>
<FuncName"MTX.Dimension" Arg=module'/
</>
</ManSection>returns the dimension in which matricesact

</ection>


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<Section Label="Selecting a Different MeatAxe">
<Heading>Selecting aFunc Name="TX.ield" Arg='module'/>

<ManSection>
<Varjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Description>/>
All MeatAxeManSection>
which is a java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
It is possibleLabel"java.lang.StringIndexOutOfBoundsException: Range [38, 30) out of bounds for length 38
Each java.lang.StringIndexOutOfBoundsException: Range [24, 13) out of bounds for length 78
<Ref Var="MTX"/> to.java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
</java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 12
</ManSection>

</Section


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<ection Label="Accessing a Module">
<Heading>Accessing a Module</Heading>

Even though a MeatAxe module java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
accessed  of Instead the following operations
should be used:

<ManSection>
<Func Name="MTX.Generators" Arg='module'/>

<Description>
returns a list of matrix generators of <A>module</A>.
<ption>
</ManSection>

<algebraic c of<M>K</>.java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
< Name"TX.imension" Arg='module'/>

<Description>
returns the dimension in which the matrices java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.Field" Arg='module'/>

<Description>
returns the field over which <A>module</Ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</Description
</ManSection>

<//Descript>


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<Section
<>IrreducibilityTests</Heading

<java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 0
<Func Name="MTX.IsIrreducible"< Label="Decomposition  modules"java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 42

<Description>
java.lang.StringIndexOutOfBoundsException: Range [13, 5) out of bounds for length 78
)
</Description>
</ManSection>

<ManSection>
< =MTXjava.lang.StringIndexOutOfBoundsException: Range [41, 39) out of bounds for length 55

<java.lang.StringIndexOutOfBoundsException: Range [25, 14) out of bounds for length 56
A =java.lang.StringIndexOutOfBoundsException: Range [34, 32) out of bounds for length 48
Description
(  java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 68
<M> module ></> M</>isthe
algebraic closure of <M>K</M>.</anSection>
</Description>
</ManSection>

<ManSection>
<Func Name="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 4

<Description>
returns the degree of the 
</Description decompositionof Amodule<A>as a 
</ManSection>

</Section>


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<Section Label="Decomposition of modules">
<Heading>Decomposition of modules</Heading>

A module is <E>decomposable</E> if it can be written as the direct sum of two
proper submodules (and <E>indecomposablethe inducedmodule action onthis component. (Such a decomposition is not
dimensional module is a direct sum of its indecomposable parts.
The <E>/Description
ecomponents.Theyare uniquely determined.
<P/>
<>
<Func Name=<>

<Description
returns whether>
</computes the homogeneous components of <A>module</A> given as sums of
</ManSection>
<P/>
<ManSection>
<Func Name="MTX  record  isomorphism type of indecomposable

<java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 4
returns a  numbers of  components,
modulesIndecomposition"/>java.lang.StringIndexOutOfBoundsException: Index 48 out of bounds for length 48
<A>B</A> is a that are in thehomogeneous component,
theinducedmoduleactionon thiscomponent  a decomposition is java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
unique.)
</escription
</<Ccomponent<>(component )
<P/  isomorphism/>(an isomorphism from definingcomponent to thisone)
<ManSection>
<Func

<Description>
computes the homogeneous components of <A>moduleDescriptionjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
is a record corresponding to oneSectionLabel=FindingSubmodules>
ManSection
The Func NameMTXS" Arg='odule,subspace'/>
<Fun =MSubGModule" =module,subspace'>
<Descr
<Mark><<A>subsp/beo a)java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 78
<the sa /. basisofjava.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 71
  the index java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 14
  =".java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 48
     component
<>
<<component</<Mark
<ManS>
  one of the indecomposable components,
</Item>
<Mark><C>images</C></Mark>
<Item>
  a list of the remaining indecomposable components,
  each given as a record with the components
  <C><ManSection>
  <Func Name="TX.asesSubmodules" Arg='module'/>
</Item>
</List>
<<Description>
</returns list containing a basis for every submodule.

</Section>


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<Func Name="MTX.BasesMinimalSubmodules" Argjava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
<ManSection

<ManSection>java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
<Func/>
FuncName"TXSubGModule" Arg='module, subspace'/>

<Description>
<A>subspace</A> should be a subspace of (or a vector <A>subspace</A> should be a subspace of (or a vector in
spaceof<>module<A . thefullspaceof thesame dimensionand over
the same field as <A
<A><escription>
</>
</ManSection/Description>

<ManSection>
<Func ManSection


returns the basis of a proper 
 .
</Descriptionreturnsbasis of the socle  <>module<A>
</ManSection>

<ManSection>
<Func Name="MTX.BasesSubmodules" Arg='module'/>

<Description>
returns a list containing a basis for every submodule.
</Description
java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13

<
<Func Name". 'odule/

</Descriptio
java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<Description
</ManSection>

<ManSection>
<Name="MTX." Arg='odule'>

<Description>
returns a list of bases of all maximal submodules.
</Description>
</ManSection>

<ManSection
<Func Name="MTX<ManSectionjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

<Description>
returns a basis of the radical of <A>module</A>.
</Description>
</ManSection>

<anSection>
<Func Name="MTX.BasisSocle" Arg='module'/>

<Description>
returnsiption>
</Description>
</>

<ManSection>
<java.lang.StringIndexOutOfBoundsException: Range [10, 5) out of bounds for length 61

<Description>
returns <unc Name="MTX.CollectedFactors"Arg=odule/
the basis <<>
<java.lang.StringIndexOutOfBoundsException: Range [8, 7) out of bounds for length 68
</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

<ManSection>
<Func Name="MTX.BasesCompositionSeriesSection Label="Induced Actions">

<Description>
returns a list of bases of submodules in a java.lang.StringIndexOutOfBoundsException: Index 47 out of bounds for length 34
java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.CompositionFactors" Arg='module'/>

<>
returns a returns a list <C>[<A>bas</A>, <A>change</A> ]</C> where <A>bas</A> is normed basis (.e. echelon with pivotsnormedto1)for <>ub/>
</>
</ManSection>

<>
<Func Description

<Description>
returns aManSection>
frequenciesjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
<Description
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<SectionLabel"nduced java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 34

<ManSection<Ref".java.lang.StringIndexOutOfBoundsException: Range [53, 48) out of bounds for length 53
<"NormedBasisAndBaseChange=sub'/>

<Description>
returns a list <C>[<A>bas</A>, <A>change</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
normedbasis(.e.  echelonformwith  normedto1  <A>sub</A>
and <A>change</A> is the base changefactor oftheproper  A></A> If <complA>isgiven,ithas tobe a   a
t   of<><A expressedincoefficients for<sub/>.
<Description>
</ManSection>

<ManSection>
<Name"MTX."Argsub'>
<Name=MTXInducedActionSubmoduleNB='module,sub'java.lang.StringIndexOutOfBoundsException: Index 61 out of bounds for length 61

<Description>
java.lang.StringIndexOutOfBoundsException: Range [13, 7) out of bounds for length 68
the non-trivial submodule
<A>ub</>
In/anSection
(That ManSection
&bsp;<Ref )
</Description>
</ManSection>

<Func ="Arg=mat,sub'>
<Func Name="MTX.InducedActionFactorModule" Arg='module, sub[, compl]'/>

<Description>
creates a new module corresponding to the action of <A>module</
factorjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 0
(A>. Theaction   correspond to
<A>compl</A>.
<Pcorresponding bases.Theaction  isbinary encoded Atype
 A/ has to  given innormed.(That is  must be in
echelon form factor ,
see <Ref Func="and <C>4</C> for action<C>4<C> for  of the full module on a subspace adapted basis.
</Description>
</ManSection>

<>
<Func Name="MTX.(<A>sub</A>,<A>quot</A>both/><A>basis</A>)
<Func Name="MTX.InducedActionSubMatrixNB" Arg='mat, sub'/>
<uncName="TX.nducedActionFactorMatrix" Arg='mat, sub[, compl]'/>

<Description>
work the same way as the above functions for modules, but extending <A>sub</A>. (Actions which are not computed are omit so java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
a single matrix.
</Description>
</ManSection>

<ManSection>
<Func NameP>

<Description>
Computes induced actions on submodules or factor modules and also returns the
corresponding>
<C
<> forfactor ,
and <Cjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
The routine returns the computed results in a list in sequence
(A><A>A></A>/>,basis/>java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
where <A>java.lang.StringIndexOutOfBoundsException: Range [0, 14) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [30, 9) out of bounds for length 73
returned list may be shorter.)
If no <A>java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [21, 3) out of bounds for length 45
<P/>
All A>module2.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->A/A>java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<Section 
<Headingjava.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12

<ManSection>
<Func Name="MTX.BasisModuleHomomorphisms" Arg='module1, module2'/>

<java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
returns returns a basisofall module homomorphismsfrom<>/>to <A>java.lang.StringIndexOutOfBoundsException: Range [76, 75) out of bounds for length 80
<A>module2</A>.
Homomorphisms are by matrices, whose rows give the images of the
standard basis vectors of <A>module1</A> in the standard basis of
<A>module2</A>.
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.BasisModuleEndomorphisms" Arg='module'/>

<Description>
returns a  ofall module homomorphisms from <A>module</A> to <A>module</A>.
</Description>
</ManSection>

<ManSection>
< Name"TX.IsomorphismModules"Arg=,module2'>

<Description>
If <A>module1</A> and <A>module2</A> are isomorphic modules,
this function java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 12
informmatrix.
It returns <K>fail</K> if the modules are not isomorphic.
</Description>
</ManSection>

<Description>
<Name""Arg='/

<Description>
returns the module automorphisms of <A>module</A> (the set of all isomorphisms
from <A>module</A> to itselftfunction  isomorphismfrom <>/A <>/>
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Module Homomorphisms for irreducible modules">
<Heading>Module Homomorphisms form a .

The following are lower-level functions that provide homomorphism <>/>themodulesnotisomorphicjava.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 57
functionality for irreducible modules. Generic code should use the functions
in Section <Ref Sect="Module Homomorphisms"/> instead.

<ManSection
<Func Name="MTX.IsEquivalent" java.lang.StringIndexOutOfBoundsException: Range [0, 33) out of bounds for length 13

<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
tests two irreducible modules for equivalence.
</Description>
</ManSection>


<ManSection>
FuncName"IsomorphismIrred" Arg='module1, module2'/>

<Description>
returns an isomorphism from <A></escription
and <K>fail</K> otherwise. It requires that one of the modules is known to be
irreducible./Section
the results are unpredictable.
The isomorphism is <java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77
the  basis vectors of <A>module1</A> in the standard basis of
<A>module2</A>.
That is, <Heading>ModuleHomomorphismsfor irreducible modules</Heading>
the generators of <A>module1</A>.
</The   -    
</   .     


<java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
<Func Func Name"TX.IsEquivalent" =module1,module2'>

<
<A>mat<<Description>
defining a <>
java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
really does  Name="MTX.IsomorphismIrred=module1,'>
Description>
modules. This can be used for computing kernels, images and pre-images.
<>
<>


<ManSection>
<Func Name="MTX.Homomorphisms" Arg='module1, module2'/>

<Description>
java.lang.StringIndexOutOfBoundsException: Range [18, 7) out of bounds for length 77
Amodule1/>A<>
</>
</the generators of <A>module1</A>.


<ManSection>
Func ""',nr/

<Description>
Let <A><Func Nam"Homomorphism =mjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 59
This A>mat</A> should be a <A>dim1</A> <M>\times</M> <A>defining a homomorphism from <A>module1</A> to <A>module2</A>.
java.lang.StringIndexOutOfBoundsException: Range [26, 2) out of bounds for length 51
</Description  underlyingspaces the
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="meataxe:java.lang.StringIndexOutOfBoundsException: Index 32 out of bounds for length 0
<>MeatAxeFunctionality Invariant/>

The A>module1.
MeatAxe module.

<ManSection>
Func Name="MTX.InvariantBilinearForm" Arg='module'/>

<Description>
returns an java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 12
of>fail</K if no no suchform exists.
</Description>
</ManSection>


<ManSection>
<Func Name="MTX.InvariantSesquilinearForm" Arg='module'/>

<Description>
returns an invariant hermitian (= self-adjoint) sesquilinear form of
<A>module</>,
which routine tries  findgroupalgebraelementthat  nullityzero
or <K>fail< factorsexcept  <>r/>java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 51
</ManSection>
</ManSection>


<#Include Label="MTX.InvariantQuadraticForm">


<ManSection
<Func Name="MTX.BasisInOrbit" Arg='moduleSection Label="eataxe:Invariant Forms"

<<
The functions this  can   to absolutelyirreducible
in an orbit of  module.
This is used by <Ref Func="MTXFunc "MTXInvariantBilinearFormjava.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 53
</>
</>


<#Include Label="MTX.OrthogonalSign">

/>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --><!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label=eturns  hermitian ( self-adjoint) sesquilinearform of
<Heading>TheA><A>

The standard> ifno  form exists.
basedDescription>
originally>
It
is assigned by default#ncludeLabel"TX.InvariantQuadraticForm">
For the 
functions=MTXBasisInOrbit Arg='odule'>

<ManSection>
<Func Name="returns a basis of the underlying vector space of <A>module</A> which is contained

<Description>
returns the module action on a random irreducible This is used by <Ref Func="MTX.InvariantQuadraticForm"/> in characteristic 2.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.GoodElementGModule" Arg='module'/>

<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
findsan element  minimal nullspacedimension <module/>
is known to be irreducible.
<Description>
</ManSection>


<anSection
<Func Name="SMTX.java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0

<Description>
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 52
</Description>
</ManSection


<ManSection>
<Func Name="SMTX.MinimalSubGModules" Arg='module1, module2[, max]'/>

<Description>
returns(at most  <A>max</A> bases of submodules of <>module2</> which are theremaining sections document more technical
isomorphic to the irreducible module  <A>module1</A>.
<
</ManSection>


<ManSection>
<Func =SMTX."Arg='tring'/

<java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
returns a setter function for the component <C>smashMeataxe.(string)</C>.
</Description>
</ManSection>


<ManSection>
<uncName="MTX.Getter"Arg=string'/

<Description
returns a java.lang.StringIndexOutOfBoundsException: Range [0, 16) out of bounds for length 0
<Name="SMTX.GoodElementGModule" Arg='module'/>
</ManSection>


<ManSection>
<Func Name="SMTX.IrreducibilityTestfinds   with  possible nullspacedimension if <></A>

<Description>
Tests for irreducibility and sets a subbasis if reducible. It/>
an irreducibility flag, java.lang.StringIndexOutOfBoundsException: Range [0, 27) out of bounds for length 0
calledto obtainarandomsubmodule
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 13

<>
FuncName=SMTXMinimalSubGModules Arg',module2[ max]'/>
neither sets an absolute irreducibility flag, nor tests it.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.MinimalSubGModule" Arg='module, cf, nr'/>

<Description>
returns the java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for length 0
indicated factor.  assumes <></> has  called
already.
>
</ManSection


<ManSection<
< =SMTX =m,'

<Description>
  directsum  two  lists.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.CompleteBasis" Arg='module, pbasis'/>

<Description>
extends the partialDescription>
by action Tests for irreducibilityand  subbasis  reducible.It neither sets
<Description>
</

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Smash
h MeatAxe Flags<Tests for   and  splitting degree.It

The following getter routines access internal flags. For each routine, the
appropriate<>

<<ManSection
<=".Subbasis"'>

<Description>
Basis of a submodule.
</indicated factor.Itassumes <></> hasbeen called
</ManSection


<ManSection>
<Func Name=<ManSectionjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13

<Description>
list <C>[<escription>
</>
</>


<ManSection>
<Func Name=ManSection>

<escription
matrix of <Ref Func="SMTX.AlgEl"/>.
</Description>
</ManSection>


<ManSection> action  A><A.Itreturnswhetherjava.lang.StringIndexOutOfBoundsException: Range [50, 49) out of bounds for length 60
<Func Name="SMTX.java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0

<Description>
minimal polynomial of Ref ="SMTX.lgEl">
</Description>
</anSection


<ManSection>
<Funcjava.lang.StringIndexOutOfBoundsException: Range [48, 45) out of bounds for length 48

<>
uses factor of <Ref Func"SMTX.AlgEl">.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.AlgElNullspaceVec" Arg='module'/>

<Description>
nullspace of the matrix evaluated under this factor.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 0

<
dimension 
</ManSection>
</ManSection>


<ManSection>
<Func Name="SMTX.CentMat" Arg='module'/>

<Description>
matrix centralising all generators which is computed as
a byproduct of <Ref Func="SMTX.AbsoluteIrreducibilityTest"/>.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.CentMatMinPoly" Arg='module'/>

<Description>
minimal polynomial of <Ref Func="list <C>[newgens,coefflist]</C> giving an algebra element used for chopping.
</Description>
</ManSection>

</Section>
</Chapter>