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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 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.


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

<ManSection>
<Heading>GModuleByMats</Heading>
<Func Name="GModuleByMats" Arg='gens, field'
 Label="for generators and a field"/>
<Func Name="GModuleByMats" Arg='emptygens, dim, field'
 Label="for empty list, the dimension, and a field"/>

<Description>
creates a MeatAxe module over <A>field</A> from a list of invertible matrices
<A>gens</A> which reflect a group's action. If the list of generators is empty,
the dimension must be given as second argument.
<P/>
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
<P/>
<List>
<Mark><C>generators</C></Mark>
<Item>
   A list of matrices which represent a group operation on a
   finite dimensional row vector space.
</Item>
<Mark><C>dimension</C></Mark>
<Item>
   The dimension of the vector space (this is the common length of
   the row vectors (see <Ref Attr="DimensionOfVectors"/>)).
</Item>
<Mark><C>field</C></Mark>
<Item>
   The field over which the vector space is defined.
</Item>
</List>
<P/>
Once a module has been created its entries may not be changed. A MeatAxe may
create a new component <A>NameOfMeatAxe</A> in which it can store private
information. By a
java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 80
lesMeatAxe java.lang.StringIndexOutOfBoundsException: Index 73 out of bounds for length 73
C/> java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 77
Heading Modules/eading>
bases used are needed as well.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Module Constructions">
<Heading>Module Constructions</>/>

<ManSection>
<Heading>java.lang.StringIndexOutOfBoundsException: Range [30, 25) out of bounds for length 54
<Name"Arg=,field] Label="ormatrixgroup and a field"/>
<Description>
creates a MeatAxe module over <A>field</A> from java.lang.StringIndexOutOfBoundsException: Range [0, 51) out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [75, 5) out of bounds for length 78
<Ref Attr="DefaultFieldOfMatrixGroup"/> is used instead.
</>/>whichreflecta group' .If thelist of generatorsis ,
<the dimension must be given as second argument.

<ManSection>
<java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 4

<Description>
Called with a permutation group <A>G</A> and a field <A>F</A> (<not dealwith &AP; modulesbut essentiallywith listsof matrices Forthe
<<Lis>
<Mark><Cgenerators</C><Mark>
for the group of permutation matrices that acts on the canonical basis of
<M>M</Item>
moved pointsee <ef Attr="argestMovedPoint" Label="for a list or collection of permutations"/>).
</Description>
</ManSection>

<ManSection>
< Name="TrivialGModule" Arg='G, F'/>

<Description>
Called with a group <A>G</A> and a field <A>F</A> (<A>F</A> may be    The dimension  vectorspace (his isthe common length of
<Item
</java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 14
</

<a   
<Func Name="TensorProductGModule<Q or Q>actor module<Q we denote

<Description>
<Ref Func="TensorProductGModule"/> calculates the tensor product
java.lang.StringIndexOutOfBoundsException: Range [27, 2) out of bounds for length 39
are to be modules oversame ,,
they  should have the same number modules,  obtain the  relation to the original module,the
</Description
</ManSection/Section>

<ManSection
<Func

<Description>
<Ref"/> calculates the wedge product of a <A>G</A>-module.
java.lang.StringIndexOutOfBoundsException: Range [22, 4) out of bounds for length 44
</Description
</<Description>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section=PermutationGModule"/ java.lang.StringIndexOutOfBoundsException: Range [41, 40) out of bounds for length 71
Heading<Heading>

<ManSection>
<Var <M>M</M>   way as<>G/A>acts  thepointsup toits largest

<Description>
All MeatAxe routines are accessed via the global variable <Ref Var="MTX"/>,
which is a record whose components hold the various functions.
It possible to  several implementations a MeatAxe java.lang.StringIndexOutOfBoundsException: Range [70, 69) out of bounds for length 70
Each MeatAxe  itsroutines inan ownglobal variable and assigning
<Ref Var="MTX"/> to this variable selects the corresponding MeatAxe.
</Description>
</lledwith agroup <<A>and  field <><A (<>F</>may  infinite,

<Description>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<ection Label"Accessing a "java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 36
<eading>Accessinga Module<Heading>

Even though a of the modules <A>m1<and <>2<A.
accessedoutsideofMeatAxe functions.Instead the following operations
should be used

<ManSection>
<unc Name=MTX. Arg=module'/>

<Description>
returns a list of matrix java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</Description>
</Description

<ManSection>
<= =/

<Description
theinwhichthe  .
<//ection>
</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<ManSection>
<Name"Fjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 37

<Description>
returns the field over which <
<Description
</>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section =Irreducibility Tests">
<Heading>Irreducibility Tests</Heading>

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

<Description>
tests whether the module <A>module</A> is irreducible (i.e. contains no proper
submodules)
</Description>
</ManSection>

<ManSection>
<Func <

<Description>
A module<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 36
algebraic closure ofthe field.
(Formally: If the tensor product <M>L accessedoutside MeatAxe functions. java.lang.StringIndexOutOfBoundsException: Index 59 out of bounds for length 15
where </Descri>
losure K/)
</Description>
</Func=M.java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 41

<java.lang.StringIndexOutOfBoundsException: Range [12, 13) out of bounds for length 12
<Func java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Description>
returns the degree of the splitting field />
</ion
</java.lang.StringIndexOutOfBoundsException: Index 3 out of bounds for length 0

</Heading Tests/Heading>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<SectionLabel"of modules"
<Heading>Decomposition of modules</Heading>

A module is <E>
proper tests whether the module <A>module</A> is irreducible (i.esubmodules.java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 12
FuncName=".IsAbsolutelyIrreducible" Arg='module'/>
The <E>homogeneous components</E> of a module are the direct sums of isomorphic
indecomposable components. They are uniquely determined.
<P/>
<ManSection>
<Func  Name"MTX.IsIndecomposable" Arg='module'/>

<>
Formally:If the tensor product <M>L \otimes_K M</M> is irreducible
<where <></M isthe  defined over<MK/M>and<>L/ is the
<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
<P/>
<ManSection>
<Func Name="MTX.Indecomposition" java.lang.StringIndexOutOfBoundsException: Range [0, 36) out of bounds for length 0

<Description>
returns a decomposition <>module/A  direct sum of indecomposable
modules. Section
<A>B
   java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 73
unique.)
<>
</indecomposabl    uniquely determined.
<P/>ManSection
<anSection
<Func >

<Description
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 69
indecomposable components. java.lang.StringIndexOutOfBoundsException: Index 28 out of bounds for length 13
is a record correspondingtoone isomorphism of
components.
The
<P/>
<List>
<Mark><C>indices</C></Mark>
<Item>
  the index theindecomposable components,
  as given by <Ref Func="MTX.,
   java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for length 40
</Item>
<Mark><C>component</C></Mark>
<Item>
  one of the indecomposable components,
</Item>
<Mark><C>images</C></Mark>
<Item>
  a list of the remaining indecomposable componentsthe induced   on  .(Suchdecomposition isnot
  each given as a record withunique./>
  <C><C (the componentitself and
  <C><C (an isomorphism fromthe  component to  one)
<>
</java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0
</>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
< Label" "
<Heading>Finding Submodules</Heading>

<>
<=".ubmoduleGModule Arg=m 
<c Name"TX." Arg= /

<iption>
<ce<A> should be a subspace of (ra vector in the underlying vector
space of <A>module</A> i.e. the full row space of the same dimension and over
theme field  as <A>module</>.Anormalized basis of the submodule of
<A>module</A> generated by <A>subspace</A> is returned.
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.as given by <Ref Func"TXIndecomposition"/>,

<Description>
returns  thatare in the homogeneouscomponent,
submodule exists/Item
</<Mark>C></>/>
</ectionjava.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 52

<"Bjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 47

>
 ajava.lang.StringIndexOutOfBoundsException: Range [15, 14) out of bounds for length 54
</java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 10
</ManSection>

<ManSection>
<java.lang.StringIndexOutOfBoundsException: Range [10, 5) out of bounds for length 54

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

<>
<Func 

<Description>
returns a list of bases of all maximal submodules.
</escription
</Func =M.java.lang.StringIndexOutOfBoundsException: Range [40, 39) out of bounds for length 52

<ManSection>
<Func Name="MTX.BasisRadical  A/>i.. full row  of the  dimension and 

<escription
Description
</Description
</ManSection

<>
<Func Name="MTX<Func Name="MTX.BasisSocle"

<submoduleexists.
returns a the socleof <>module/>
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.BasesMinimalSupermodules" Arg='module, sub'/>

<Description>
returns a list of ManSection>
the ="TX.asesMinimalSubmodules" Arg=''/
<n>
</ManSection>

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

<Description>
returns Func MTXBasesMaximalSubmodulesArg'/
orderjava.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13
<>
/>

<ManSection>
<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0

<<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12

iption
</ManSection

<Func Name="MTX.BasesMinimalSupermodules"
< =''>

<Description basis <>sub/A>.
returns a list giving all irreducible composition factors with their
frequencies.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 33
<Heading>Induced Actions</Heading>

<ManSection>
<java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13

<Description
java.lang.StringIndexOutOfBoundsException: Range [14, 7) out of bounds for length 72
normed(e.inechelonform pivots  to 1)for<>ub<A>
and <A>change</A> is the base change from <A>bas</A> to Description
(the basis vectors ManSection
</>
</ManSection

<>
<Func frequencies.
<Func Name="MTX.InducedActionSubmoduleNB" Arg='module/Description>

<Description>
creates java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
the non ="Actions">
<A>sub</A>.
In the <C>NBHeading>Induced Actions</Heading>
(That is it must be in echelon 
see  Func="TXNormedBasisAndBaseChange"/>.)
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.InducedActionFactorModule" Arg='module, sub[, compl]<Func Name=MTX." Arg=java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53

<Description>
creates a new module corresponding to the action of <A>normed  (.in echelon  with pivots to 1)for 
factor of  propersubmodule<sub/A> If A></  given  has to  basisof a
(vector space-(hebasisvectors Abas/>expressed  coefficients for A><A)
<A><Descriptionjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<PFunc =".nducedActionSubmodule Arg='module, sub'/
Func Name".InducedActionSubmoduleNB" Arg'module,/>
echelon form with pivots normed to 1,
see <creates a new module corresponding to the action of <A>module</A> on
</DescriptionA>ub</>
</>

<>
<Funcsee&;<Ref Func="MTX.NormedBasisAndBaseChange"/>.java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 53
<Name=MTX.InducedActionSubMatrixNB = 'java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 58
<Func Name="MTX.InducedActionFactorMatrix" Arg='matjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Description>
work the same way as the above functions for modules, but take as input only
a single matrix.
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.InducedAction" Arg='module, sub[, type]'/>

<Description>
Computes induced actions on (vector space-)complement of <A>sub< actionthenwill 
corresponding bases  takenis   in<></A>:
<C>1</C> stands The basis <>sub</> to begiven in  form isit 
action
and C<actionjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 71
The ManSection
(,<A<A,java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 49
where <A>basis</A> is a basis< ".java.lang.StringIndexOutOfBoundsException: Range [52, 41) out of bounds for length 68
ted,the
returned list may be shorter.)
If no <A>type</Ajava.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 16
The basis ManSection
</
All these java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
</Description>
</ManSection

</C>2</C  action


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label<sub/><quot/A><A>both</>,A></)
<Heading>Module Homomorphisms</Heading>

<ManSection>
<Func Name="MTX.extending <A>sub</A>. (Actions which are not computed are omitted, so the

<Description>
returns a basis of The basis given in <A>sub</A> must be normed!
<</A>java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
Homomorphisms
standard
<>module2<.
</Description>
</ManSection>

<ManSection>
<Func Name="MTX.BasisModuleEndomorphismsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<Description>
returns basis  all module  from Amodule<A to <A>module</A>.
</basis java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 80
</Func=" =module1 /

<ManSection>
<Func Name="MTX.IsomorphismModules" Arg='module1  of a .

<
If <A<unc Name=MTX.ModuleAutomorphisms Arg='odule'>
his returnsan isomorphism <>module1</A> to Amodule2<Ajava.lang.StringIndexOutOfBoundsException: Index 74 out of bounds for length 74
inform of matrixjava.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
Itreturns K>ail<K>if the modules are not isomorphic.
</Description>
</ManSection>

<ManSection>
<Func Name="MTX<escription>

<Description>
returns java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
from <A<Func Name=MTX.java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 58
/>
</ManSection>

<>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<standardjava.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 69
<  irreduciblejava.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 63

 followingare lower-evel functionsthat providehomomorphism
functionalityforirreduciblemodules.Genericcodeshoulduse the functions
in Section

<ManSection>
<="IsEquivalentArg=, /

<Description>
Description>
<Description
<ManSection


<ManSection>
<Func" Arg=module1,module2'>

<
returns <Description
and </ManSection
irreducible. It java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
the results are unpredictable.
The isomorphism is given by
the standard returns a basis of the space of all homomorphisms from the irreducible module
<A<A>module1<A to <A>module2/A>
That/Description
java.lang.StringIndexOutOfBoundsException: Range [13, 3) out of bounds for length 33
</java.lang.StringIndexOutOfBoundsException: Range [0, 13) out of bounds for length 0
</FuncName=MTX.Distinguish Arg=cf '>


<ManSection>
<e="TX.Homomorphism"Arg='odule1, module2, mat'/>

<Description>
<java.lang.StringIndexOutOfBoundsException: Range [9, 2) out of bounds for length 67
java.lang.StringIndexOutOfBoundsException: Range [57, 8) out of bounds for length 62
This function verifies that <A>mat</A>
really does define on all composition factors except number <A>nr</A>.
corresponding homomorphism between the  row of java.lang.StringIndexOutOfBoundsException: Range [67, 68) out of bounds for length 67
modules. This can
</Description>
</ManSection>


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

<Description>
returns a basis of the space of all homomorphismsHeading>MeatAxe  for Forms<Heading
<</A> to <A>module2</A>java.lang.StringIndexOutOfBoundsException: Index 33 out of bounds for length 33
</Description>java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 53
</ManSection>


<ManSection>
<Func Name="MTX.of <A>module</A>, or <K>  java.lang.StringIndexOutOfBoundsException: Range [49, 48) out of bounds for length 56

<java.lang.StringIndexOutOfBoundsException: Range [10, 7) out of bounds for length 68
LetmoduleA,
This  tofind a   element thathas 
nall composition  exceptnumberA><A.
</Description>
</java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 13

</Sectionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->>
<SectionLabel" >
<Heading>MeatAxe Functionality for

The in  sectioncanonlybe applied to anabsolutely irreducible
MeatAxejava.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15

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

<Description>
returns Description
of ManSection
</Description>
<Section



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

<Description>
eturns an invarianthermitian (-adjoint) sesquilinear formof
<A>module/>
which
K  nosuchform  exists
</Description
</ManSection


<# =MInvariantQuadraticForm


<ManSection>
e"TX."Arg='odule/>

<Description>
java.lang.StringIndexOutOfBoundsException: Range [15, 7) out of bounds for length 82
in an orbit of the action of the java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Range [55, 4) out of bounds for length 77
</Description>
</ManSection>


<#Include Label="java.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds for length 0

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> element with possible nullspace  ifA><A
<Section Label="/java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
<>

The standard MeatAxe provided in the 
based on the MeatAxe in the Function to sort the output of <C>Homomorphisms</C>.
originally />
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
is max>A/ 
For the sake of completeness java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 75
functions of this MeatAxe

<ManSection>
<Func Name="Name"MTXSetter '>

<Description>
returns java.lang.StringIndexOutOfBoundsException: Range [26, 25) out of bounds for length 73
</ S. '>
</ManSection>


<ManSection>
<Func java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 51

<Description>
anelementminimal    if<module
is known to be irreducible.
</java.lang.StringIndexOutOfBoundsException: Index 13 out of bounds for length 13
<ManSection


<ManSection>
<Func Name="called obtain a  submodule.

<Description>
Function
</Description>
</ManSection>


<ManSection>Description
< Name=."=module1 module2[ 

<Description>
returns (at most <A>max</A>) bases of submodules of  composition It<C>istinguishC has beencalled
isomorphic to the irreducible module  <A>module1</A>.
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX.Setter" Arg='string'/>

<Description>
returns a setter function for the component <C>smashMeataxe.(string)</C>.
</Description>
</</Description


<>
<Func Name

<escription>
returnsFunc Name".MatrixSum"Arg'atrices1,matrices2'/>
</Description>
</ManSectioncreates the  sum oftwomatrixlists.


<java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 0
<Func Name=

<>
for andsetsa ifreducible. neithersets
an irreducibility flag, <Description
ManSection>
</Description>
</ManSection>


<ManSection>
<Func Name="SMTX<java.lang.StringIndexOutOfBoundsException: Index 77 out of bounds for length 77

<Description>
Tests  absoluteirreducibilitysets splitting fielddegree.It
neither setsjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</Description>
</anSection


>
<Func Func NameSMTX.Subbasis" Arg='odule/>

<Description>
returns the basis of a
 compositionfactor  assumes CDistinguish/ has  called
</>
</java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 0
<ManSection>


<ManSection>
<Func Name="SMTX.MatrixSum" Arg='matrices1, matrices2'/>

<escription
Description
</ManSection>
</


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

<</Descriptiojava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 14
extends
yof<>module/>  returns  it succeeded.
</Description>
</ManSection>

</Section>


<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <FuncSMTXA/>
<Section Label="Smash MeatAxe Flags/anSection>
<Heading>Smash MeatAxe

The following getter  Name="SMTX.AlgElCharPolFac" Arg='module'/>
Description ="

<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
<Funcjava.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

<Description>
Basis
</<Description
</ManSection>


<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
<Func Name="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 13

<Description>
java.lang.StringIndexOutOfBoundsException: Range [38, 4) out of bounds for length 76
</Description>
</ManSection>


<ManSection>
<Func Name="

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


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

<Description>
minimal polynomial of <Ref Func="SMTX.AlgEl"/>.
</Description>
</ManSection>


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

<Description>
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.AlgElNullspaceDimension" Arg='module'/>

<Description>
dimension of the nullspace.
</Description>
</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="SMTX.CentMat"/>.
</Description>
</ManSection>

</Section>
</Chapter>