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Quelle mindeg.gd
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#############################################################################
##
#W mindeg.gd GAP 4 package AtlasRep Thomas Breuer
##
## This file contains declarations for dealing with information about
## permutation and matrix representations of minimal degree
## for selected groups.
##
#############################################################################
##
## <#GAPDoc Label="subsect:minimality-criteria">
## <Subsection Label="subsect:minimality-criteria">
## <Heading>Criteria Used to Compute Minimality Information</Heading>
##
## The information about the minimal degree of a faithful <E>matrix
## representation</E> of <M>G</M> in a given characteristic or over a given
## field in positive characteristic is derived from the relevant (ordinary
## or modular) character table of <M>G</M>,
## except in a few cases where this table itself is not known but enough
## information about the degrees is available in <Cite Key="HL89"/> and
## <Cite Key="Jan05"/>.
## <P/>
## The following criteria are used for deriving the minimal degree of a
## faithful <E>permutation representation</E> of <M>G</M>
## from the information in the &GAP; libraries of character tables and of
## tables of marks.
## <P/>
## <List>
## <Item>
## If the name of <M>G</M> has the form <C>"A</C><M>n</M><C>"</C> or
## <C>"A</C><M>n</M><C>.2"</C>
## (denoting alternating and symmetric groups, respectively)
## then the minimal degree is <M>n</M>, except if <M>n</M> is smaller than
## <M>3</M> or <M>2</M>, respectively.
## </Item>
## <Item>
## If the name of <M>G</M> has the form <C>"L2(</C><M>q</M><C>)"</C>
## (denoting projective special linear groups in dimension two)
## then the minimal degree is <M>q + 1</M>,
## except if <M>q \in \{ 2, 3, 5, 7, 9, 11 \}</M>,
## see <Cite Key="Hup67" Where="Satz II.8.28"/>.
## </Item>
## <Item>
## If the largest maximal subgroup of <M>G</M> is core-free
## then the index of this subgroup is the minimal degree.
## (This is used when the two character tables in question and the class
## fusion are available in &GAP;'s Character Table Library
## <Cite Key="CTblLib"/>;
## this happens for many character tables of simple groups.)
## </Item>
## <Item>
## If <M>G</M> has a unique minimal normal subgroup then each minimal
## faithful permutation representation is transitive.
## (Note that the core of each point stabilizer is either trivial or
## contains the unique minimal normal subgroup.)
## <P/>
## In this case, the minimal degree can be computed directly from the
## information in the table of marks of <M>G</M> if this is available in
## &GAP;'s Library of Tables of Marks <Cite Key="TomLib"/>.
## <P/>
## Suppose that the largest maximal subgroup of <M>G</M> is not core-free
## but simple and normal in <M>G</M>, and that the other maximal subgroups
## of <M>G</M> are core-free.
## In this case, we take the minimum of the indices of the core-free
## maximal subgroups and of the product of index and minimal degree of
## the normal maximal subgroup.
## (This suffices since no core-free subgroup of the whole group can
## contain a nontrivial normal subgroup of a normal maximal subgroup.)
## <P/>
## Let <M>N</M> be the unique minimal normal subgroup of <M>G</M>,
## and assume that <M>G/N</M> is simple and has minimal degree <M>n</M>,
## say.
## If there is a subgroup <M>U</M> of index <M>n \cdot |N|</M> in <M>G</M>
## that intersects <M>N</M> trivially
## then the minimal degree of <M>G</M> is <M>n \cdot |N|</M>.
## (This is used for the case that <M>N</M> is central in <M>G</M>
## and <M>N \times U</M> occurs as a subgroup of <M>G</M>.)
## </Item>
## <Item>
## If we know a subgroup of <M>G</M> whose minimal degree is <M>n</M>,
## say, and if we know either (a class fusion from) a core-free subgroup
## of index <M>n</M> in <M>G</M> or a faithful permutation representation
## of degree <M>n</M> for <M>G</M>
## then <M>n</M> is the minimal degree for <M>G</M>.
## (This happens often for tables of almost simple groups.)
## </Item>
## </List>
## </Subsection>
## <#/GAPDoc>
##
#############################################################################
##
#F MinimalPermutationRepresentationInfo( <grpname> )
##
DeclareGlobalFunction( "MinimalPermutationRepresentationInfo" );
#############################################################################
##
#F MinimalRepresentationInfo( <grpname>, NrMovedPoints[, <mode>] )
#F MinimalRepresentationInfo( <grpname>, Characteristic, <p>[, <mode>] )
#F MinimalRepresentationInfo( <grpname>, Size, <q>[, <mode>] )
##
## <#GAPDoc Label="MinimalRepresentationInfo">
## <ManSection>
## <Func Name="MinimalRepresentationInfo" Arg='grpname, conditions'/>
##
## <Returns>
## a record with the components <C>value</C> and <C>source</C>,
## or <K>fail</K>
## </Returns>
## <Description>
## Let <A>grpname</A> be the &GAP; name of a group <M>G</M>, say.
## If the information described by <A>conditions</A> about minimal
## representations of this group can be computed or is stored
## then <Ref Func="MinimalRepresentationInfo"/> returns a record with the
## components <C>value</C> and <C>source</C>,
## otherwise <K>fail</K> is returned.
## <P/>
## The following values for <A>conditions</A> are supported.
## <P/>
## <List>
## <Item>
## If <A>conditions</A> is <Ref Attr="NrMovedPoints" BookName="ref"/>
## then <C>value</C>, if known, is the degree of a minimal faithful
## (not necessarily transitive) permutation representation for <M>G</M>.
## </Item>
## <Item>
## If <A>conditions</A> consists of
## <Ref Attr="Characteristic" BookName="ref"/> and a prime integer
## <A>p</A> then <C>value</C>, if known, is the dimension of a minimal
## faithful (not necessarily irreducible) matrix representation
## in characteristic <A>p</A> for <M>G</M>.
## </Item>
## <Item>
## If <A>conditions</A> consists of <Ref Attr="Size" BookName="ref"/> and
## a prime power <A>q</A> then <C>value</C>, if known, is the dimension
## of a minimal faithful (not necessarily irreducible)
## matrix representation over the field of size <A>q</A> for <M>G</M>.
## </Item>
## </List>
## <P/>
## In all cases, the value of the component <C>source</C> is a list of
## strings that describe sources of the information,
## which can be the ordinary or modular character table of <M>G</M>
## (see <Cite Key="CCN85"/>, <Cite Key="JLPW95"/>, <Cite Key="HL89"/>),
## the table of marks of <M>G</M>, or <Cite Key="Jan05"/>.
## For an overview of minimal degrees of faithful matrix representations for
## sporadic simple groups and their covering groups, see also
## <P/>
## <URL>http://www.math.rwth-aachen.de/~MOC/mindeg/</URL>.
## <P/>
## Note that <Ref Func="MinimalRepresentationInfo"/> cannot provide any
## information about minimal representations over prescribed fields in
## characteristic zero.
## <P/>
## Information about groups that occur in the &AtlasRep; package
## is precomputed in <Ref Var="MinimalRepresentationInfoData"/>,
## so the packages <Package>CTblLib</Package> and <Package>TomLib</Package>
## are not needed when <Ref Func="MinimalRepresentationInfo"/> is called for
## these groups.
## (The only case that is not covered by this list is that one asks for the
## minimal degree of matrix representations over a prescribed field in
## characteristic coprime to the group order.)
## <P/>
## One of the following strings can be given as an additional last argument.
## <P/>
## <List>
## <Mark><C>"cache"</C></Mark>
## <Item>
## means that the function tries to compute (and then store) values that
## are not stored in <Ref Var="MinimalRepresentationInfoData"/>,
## but stored values are preferred; this is also the default.
## </Item>
## <Mark><C>"lookup"</C></Mark>
## <Item>
## means that stored values are returned but the function
## does not attempt to compute values that are not stored in
## <Ref Var="MinimalRepresentationInfoData"/>.
## </Item>
## <Mark><C>"recompute"</C></Mark>
## <Item>
## means that the function always tries to compute the
## desired value, and checks the result against stored values.
## </Item>
## </List>
## <P/>
## <Example><![CDATA[
## gap> MinimalRepresentationInfo( "A5", NrMovedPoints );
## rec(
## source := [ "computed (alternating group)",
## "computed (char. table)", "computed (subgroup tables)",
## "computed (subgroup tables, known repres.)",
## "computed (table of marks)" ], value := 5 )
## gap> MinimalRepresentationInfo( "A5", Characteristic, 2 );
## rec( source := [ "computed (char. table)" ], value := 2 )
## gap> MinimalRepresentationInfo( "A5", Size, 2 );
## rec( source := [ "computed (char. table)" ], value := 4 )
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "MinimalRepresentationInfo" );
#############################################################################
##
#V MinimalRepresentationInfoData
##
## <#GAPDoc Label="MinimalRepresentationInfoData">
## <ManSection>
## <Var Name="MinimalRepresentationInfoData"/>
##
## <Description>
## This is a record whose components are &GAP; names of groups for which
## information about minimal permutation and matrix representations were
## known in advance or have been computed in the current &GAP; session.
## The value for the group <M>G</M>, say,
## is a record with the following components.
## <P/>
## <List>
## <Mark><C>NrMovedPoints</C></Mark>
## <Item>
## a record with the components <C>value</C> (the degree of a smallest
## faithful permutation representation of <M>G</M>)
## and <C>source</C> (a string describing the source of this information).
## </Item>
## <Mark><C>Characteristic</C></Mark>
## <Item>
## a record whose components are at most <C>0</C> and strings
## corresponding to prime integers, each bound to a record with the
## components <C>value</C> (the degree of a smallest faithful matrix
## representation of <M>G</M> in this characteristic)
## and <C>source</C> (a string describing the source of this information).
## </Item>
## <Mark><C>CharacteristicAndSize</C></Mark>
## <Item>
## a record whose components are strings corresponding
## to prime integers <A>p</A>, each bound to a record with the components
## <C>sizes</C> (a list of powers <A>q</A> of <A>p</A>),
## <C>dimensions</C> (the corresponding list of minimal dimensions of
## faithful matrix representations of <M>G</M> over a field of size
## <A>q</A>),
## <C>sources</C> (the corresponding list of strings describing the source
## of this information), and
## <C>complete</C> (a record with the components <C>val</C>
## (<K>true</K> if the minimal dimension over <E>any</E> finite field in
## characteristic <A>p</A> can be derived from the values in the record,
## and <K>false</K> otherwise) and <C>source</C> (a string describing the
## source of this information)).
## </Item>
## </List>
## <P/>
## The values are set by <Ref Func="SetMinimalRepresentationInfo"/>.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
## We want to delay reading the data until they are actually accessed.
##
DeclareAutoreadableVariables( "atlasrep", "gap/mindeg.g",
[ "MinimalRepresentationInfoData" ] );
#############################################################################
##
#F SetMinimalRepresentationInfo( <grpname>, <op>, <value>, <source> )
##
## <#GAPDoc Label="SetMinimalRepresentationInfo">
## <ManSection>
## <Func Name="SetMinimalRepresentationInfo"
## Arg='grpname, op, value, source'/>
##
## <Returns>
## <K>true</K> if the values were successfully set,
## <K>false</K> if stored values contradict the given ones.
## </Returns>
## <Description>
## This function sets an entry in <Ref Var="MinimalRepresentationInfoData"/>
## for the group <M>G</M>, say, with &GAP; name <A>grpname</A>.
## <P/>
## Supported values for <A>op</A> are
## <P/>
## <List>
## <Item>
## <C>"NrMovedPoints"</C>
## (see <Ref Attr="NrMovedPoints" BookName="ref"/>),
## which means that <A>value</A> is the degree of minimal faithful
## (not necessarily transitive) permutation representations of <M>G</M>,
## </Item>
## <Item>
## a list of length two with first entry
## <C>"Characteristic"</C>
## (see <Ref Attr="Characteristic" BookName="ref"/>)
## and second entry <A>char</A> either zero or a prime integer,
## which means that <A>value</A> is the dimension of minimal faithful
## (not necessarily irreducible) matrix representations of <M>G</M>
## in characteristic <A>char</A>,
## </Item>
## <Item>
## a list of length two with first entry
## <C>"Size"</C>
## (see <Ref Attr="Size" BookName="ref"/>)
## and second entry a prime power <A>q</A>,
## which means that <A>value</A> is the dimension of minimal faithful
## (not necessarily irreducible) matrix representations of <M>G</M>
## over the field with <A>q</A> elements, and
## </Item>
## <Item>
## a list of length three with first entry
## <C>"Characteristic"</C>
## (see <Ref Attr="Characteristic" BookName="ref"/>),
## second entry a prime integer <A>p</A>,
## and third entry the string <C>"complete"</C>,
## which means that the information stored for characteristic <A>p</A>
## is complete in the sense that for any given power <M>q</M> of <A>p</A>,
## the minimal faithful degree over the field with <M>q</M> elements
## equals that for the largest stored field size of which <M>q</M> is a
## power.
## </Item>
## </List>
## <P/>
## In each case,
## <A>source</A> is a string describing the source of the data;
## <E>computed</E> values are detected from the prefix <C>"comp"</C>
## of <A>source</A>.
## <P/>
## If the intended value is already stored and differs from <A>value</A>
## then an error message is printed.
## <P/>
## <Example><![CDATA[
## gap> SetMinimalRepresentationInfo( "A5", "NrMovedPoints", 5,
## > "computed (alternating group)" );
## true
## gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 0 ], 3,
## > "computed (char. table)" );
## true
## gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 2 ], 2,
## > "computed (char. table)" );
## true
## gap> SetMinimalRepresentationInfo( "A5", [ "Size", 2 ], 4,
## > "computed (char. table)" );
## true
## gap> SetMinimalRepresentationInfo( "A5", [ "Size", 4 ], 2,
## > "computed (char. table)" );
## true
## gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 3 ], 3,
## > "computed (char. table)" );
## true
## ]]></Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareGlobalFunction( "SetMinimalRepresentationInfo" );
#############################################################################
##
#F ComputedMinimalRepresentationInfo()
##
## <ManSection>
## <Func Name="ComputedMinimalRepresentationInfo" Arg=''/>
##
## <Returns>
## a records in the format of
## <Ref Var="AtlasOfGroupRepresentationsInfoData"/>.
## </Returns>
## <Description>
## For the groups listed in the <C>GAPnames</C> component of
## <Ref Var="AtlasOfGroupRepresentationsInfoData"/>,
## <Ref Func="MinimalRepresentationInfo"/> is called with the last argument
## <C>"recompute"</C>, and with the relevant conditions.
## The completeness info is set for a characteristic if the Brauer table
## of the group proves that all relevant values are known.
## <P/>
## A record with the recomputed values is returned,
## the variable <Ref Var="MinimalRepresentationInfoData"/> itself is not
## changed.
## <P/>
## Information is printed about differences between the stored and the
## computed value.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "ComputedMinimalRepresentationInfo" );
#############################################################################
##
#F StringOfMinimalRepresentationInfoData( <record> )
##
## <ManSection>
## <Func Name="StringOfMinimalRepresentationInfoData" Arg='record'/>
##
## <Returns>
## a string that describes the contents of <A>record</A>.
## </Returns>
## <Description>
## Let <A>record</A> be a record in the format of
## <Ref Var="MinimalRepresentationInfoData"/>.
## This function returns a string that contains the assignments of values
## with <Ref Func="SetMinimalRepresentationInfo"/>,
## as given in the package's file <F>gap/mindeg.g</F>.
## </Description>
## </ManSection>
##
DeclareGlobalFunction( "StringOfMinimalRepresentationInfoData" );
#############################################################################
##
#E
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2026-03-28
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