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generalized CharacterTableIsoclinic
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Up to now, `CharacterTableIsoclinic` dealt only with character tables
of groups of the structure 2.G.2 or 4.G.2.
Now also the cases p.G.p, for odd primes p, are supported.
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@ThomasBreuer
ThomasBreuer committed Jan 31, 2019
1 parent c770b4a commit 2013da1
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192 changes: 164 additions & 28 deletions lib/ctbl.gd
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Expand Up @@ -3839,49 +3839,78 @@ DeclareOperation( "CharacterTableFactorGroup",

#############################################################################
##
#A CharacterTableIsoclinic( <tbl> )
#O CharacterTableIsoclinic( <tbl>, <arec> )
#O CharacterTableIsoclinic( <modtbl>, <ordiso> )
#O CharacterTableIsoclinic( <tbl>[, <classes>][, <centre>] )
#A SourceOfIsoclinicTable( <tbl> )
##
## <#GAPDoc Label="CharacterTableIsoclinic">
## <ManSection>
## <Oper Name="CharacterTableIsoclinic" Arg='tbl[, classes][, centre]'/>
## <Oper Name="CharacterTableIsoclinic" Arg='tbl[, arec]'/>
## <Oper Name="CharacterTableIsoclinic" Arg='tbl[, classes][, centre]'
## Label="for a character table and one or two lists"/>
## <Oper Name="CharacterTableIsoclinic" Arg='modtbl, ordiso'
## Label="for a Brauer table and an ordinary table"/>
## <Attr Name="SourceOfIsoclinicTable" Arg='tbl'/>
##
## <Description>
## If <A>tbl</A> is the (ordinary or modular) character table of a group
## with the structure <M>2.G.2</M> with a central subgroup <M>Z</M> of order
## <M>2</M> or <M>4</M> and a normal subgroup <M>N</M> of index <M>2</M>
## that contains <M>Z</M> then <Ref Oper="CharacterTableIsoclinic"/> returns
## the table of the isoclinic group in the sense of the
## Let <A>tbl</A> be the (ordinary or modular) character table of a group
## <M>H</M>, say, with the structure <M>p.G.p</M> for some prime <M>p</M>,
## that is, <M>H/Z</M> has a normal subgroup <M>N</M> of index <M>p</M>
## and a central subgroup <M>Z</M> of order <M>p</M> contained in <M>N</M>.
## <P/>
## Then <Ref Oper="CharacterTableIsoclinic"/> returns
## the table of an isoclinic group in the sense of the
## &ATLAS; of Finite Groups
## <Cite Key="CCN85" Where="Chapter 6, Section 7"/>.
## If <M>N</M> is not uniquely determined then the positions of the classes
## forming <M>N</M> must be entered as list <A>classes</A>.
## If <M>Z</M> is not unique inside <M>N</M> then the positions of the
## classes in <M>Z</M> must be entered as list <A>centre</A>;
## If <M>Z</M> has order <M>2</M> then <A>centre</A> can be also the
## position of the involution in <M>Z</M>.
## <P/>
## Note that also if <A>tbl</A> is a Brauer table then <A>classes</A> and
## <A>centre</A> denote class numbers w.r.t.&nbsp;the <E>ordinary</E>
## character table.
## If <M>p = 2</M> then also the case <M>H = 4.G.2</M> is supported,
## that is, <M>Z</M> has order four and <M>N</M> has index two in <M>H</M>.
## <P/>
## For an ordinary character table that has been constructed via
## <Ref Oper="CharacterTableIsoclinic"/>,
## the value of <Ref Attr="SourceOfIsoclinicTable"/> is the list of three
## arguments in the <Ref Oper="CharacterTableIsoclinic"/> call.
## The optional arguments are needed if <A>tbl</A> does not determine
## the class positions of <M>N</M> or <M>Z</M> uniquely,
## and in the case <M>p > 2</M> if one wants to specify a
## <Q>variant number</Q> for the result.
## <P/>
## Note that there is no default method for <E>computing</E> the value of
## <Ref Attr="SourceOfIsoclinicTable"/>.
## <List>
## <Item>
## In general, the values can be specified via a record <A>arec</A>.
## If <M>N</M> is not uniquely determined then the positions of the
## classes forming <M>N</M> must be entered as the value of the component
## <C>normalSubgroup</C>.
## If <M>Z</M> is not unique inside <M>N</M> then the class position of
## a generator of <M>Z</M> must be entered as the value of the
## component <C>centralElement</C>.
## </Item>
## <Item>
## If <M>p = 2</M> then one may specify the positions of the classes
## forming <M>N</M> via a list <A>classes</A>,
## and the positions of the classes in <M>Z</M> as a list <A>centre</A>;
## if <M>Z</M> has order <M>2</M> then <A>centre</A> can be also the
## position of the involution in <M>Z</M>.
## </Item>
## </List>
## <P/>
## Note that also if <A>tbl</A> is a Brauer table then <C>normalSubgroup</C>
## and <C>centralElement</C>, resp.&nbsp; <A>classes</A> and <A>centre</A>,
## denote class numbers w.r.t.&nbsp;the <E>ordinary</E> character table.
## <P/>
## If <M>p</M> is odd then the &ATLAS; construction describes <M>p</M>
## isoclinic variants that arise from <M>p.G.p</M>.
## (These groups need not be pairwise nonisomorphic.)
## Entering an integer <M>k \in \{ 1, 2, \ldots, k-1 \}</M> as the value of
## the component <C>k</C> of <A>arec</A> yields the <M>k</M>-th of the
## corresponding character tables; the default for <C>k</C> is <M>1</M>.
## <P/>
## <Example><![CDATA[
## gap> d8:= CharacterTable( "Dihedral", 8 );
## CharacterTable( "Dihedral(8)" )
## gap> nsg:= ClassPositionsOfNormalSubgroups( d8 );
## [ [ 1 ], [ 1, 3 ], [ 1 .. 3 ], [ 1, 3, 4 ], [ 1, 3 .. 5 ], [ 1 .. 5 ]
## ]
## gap> iso:= CharacterTableIsoclinic( d8, nsg[3] );;
## gap> Display( iso );
## gap> isod8:= CharacterTableIsoclinic( d8, nsg[3] );;
## gap> Display( isod8 );
## Isoclinic(Dihedral(8))
##
## 2 3 2 3 2 2
Expand All @@ -3894,16 +3923,126 @@ DeclareOperation( "CharacterTableFactorGroup",
## X.3 1 -1 1 1 -1
## X.4 1 -1 1 -1 1
## X.5 2 . -2 . .
## gap> SourceOfIsoclinicTable( iso );
## [ CharacterTable( "Dihedral(8)" ), [ 1, 2, 3 ], [ 3 ], 3 ]
## gap> t1:= CharacterTable( SmallGroup( 27, 3 ) );;
## gap> t2:= CharacterTable( SmallGroup( 27, 4 ) );;
## gap> nsg:= ClassPositionsOfNormalSubgroups( t1 );
## [ [ 1 ], [ 1, 4, 8 ], [ 1, 2, 4, 5, 8 ], [ 1, 3, 4, 7, 8 ],
## [ 1, 4, 6, 8, 11 ], [ 1, 4, 8, 9, 10 ], [ 1 .. 11 ] ]
## gap> iso1:= CharacterTableIsoclinic( t1, rec( k:= 1,
## > normalSubgroup:= nsg[3] ) );;
## gap> iso2:= CharacterTableIsoclinic( t1, rec( k:= 2,
## > normalSubgroup:= nsg[3] ) );;
## gap> TransformingPermutationsCharacterTables( iso1, t1 ) <> fail;
## false
## gap> TransformingPermutationsCharacterTables( iso1, t2 ) <> fail;
## true
## gap> TransformingPermutationsCharacterTables( iso2, t2 ) <> fail;
## true
## ]]></Example>
## <P/>
## For an ordinary character table that has been constructed via
## <Ref Oper="CharacterTableIsoclinic"/>,
## the value of <Ref Attr="SourceOfIsoclinicTable"/> encodes this
## construction, and is defined as follows.
## If <M>p = 2</M> then the value is the list with entries <A>tbl</A>,
## <A>classes</A>, the list of class positions of the nonidentity
## elements in <M>Z</M>, and the class position of a generator of <M>Z</M>.
## If <M>p</M> is an odd prime then the value is a record with the
## following components.
## <P/>
## <List>
## <Mark><C>table</C></Mark>
## <Item>
## the character table <A>tbl</A>,
## </Item>
## <Mark><C>p</C></Mark>
## <Item>
## the prime <M>p</M>,
## </Item>
## <Mark><C>k</C></Mark>
## <Item>
## the variant number <M>k</M>,
## </Item>
## <Mark><C>outerClasses</C></Mark>
## <Item>
## the list of length <M>p-1</M> that contains at position <M>i</M>
## the sorted list of class positions of the <M>i</M>-th coset of the
## normal subgroup <M>N</M>
## </Item>
## <Mark><C>centralElement</C></Mark>
## <Item>
## the class position of a generator of the central subgroup <M>Z</M>.
## </Item>
## </List>
## <P/>
## There is no default method for <E>computing</E> the value of
## <Ref Attr="SourceOfIsoclinicTable"/>.
## <P/>
## <Example><![CDATA[
## gap> SourceOfIsoclinicTable( isod8 );
## [ CharacterTable( "Dihedral(8)" ), [ 1 .. 3 ], [ 3 ], 3 ]
## gap> SourceOfIsoclinicTable( iso1 );
## rec( centralElement := 4, k := 1,
## outerClasses := [ [ 3, 6, 9 ], [ 7, 10, 11 ] ], p := 3,
## table := CharacterTable( <pc group of size 27 with 3 generators> ) )
## ]]></Example>
## <P/>
## If the arguments of <Ref Oper="CharacterTableIsoclinic"/> are
## a Brauer table <A>modtbl</A> and an ordinary table <A>ordiso</A>
## then the <Ref Attr="SourceOfIsoclinicTable"/> value of <A>ordiso</A>
## is assumed to be identical with the
## <Ref Attr="OrdinaryCharacterTable" Label="for a character table"/>
## value of <A>modtbl</A>,
## and the specified isoclinic table of <A>modtbl</A> is returned.
## This variant is useful if one has already constructed <A>ordiso</A>
## in advance.
## <P/>
## <Example><![CDATA[
## gap> g:= GL(2,3);;
## gap> t:= CharacterTable( g );;
## gap> iso:= CharacterTableIsoclinic( t );;
## gap> t3:= t mod 3;;
## gap> iso3:= CharacterTableIsoclinic( t3, iso );;
## gap> TransformingPermutationsCharacterTables( iso3,
## > CharacterTableIsoclinic( t3 ) ) <> fail;
## true
## ]]></Example>
## <P/>
## <E>Theoretical background:</E>
## Consider the central product <M>K</M> of <M>H</M> with a cyclic group
## <M>C</M> of order <M>p^2</M>.
## That is, <M>K = H C</M>, <M>C \leq Z(K)</M>, and the central subgroup
## <M>Z</M> of order <M>p</M> in <M>H</M> lies in <M>C</M>.
## There are <M>p+1</M> subgroups of <M>K</M> that contain
## the normal subgroup <M>N</M> of index <M>p</M> in <M>H</M>.
## One of them is the central product of <M>C</M> with <M>N</M>,
## the others are <M>H_0 = H</M> and its isoclinic variants
## <M>H_1, H_2, \ldots, H_{{p-1}}</M>.
## We fix <M>g \in H \setminus N</M> and a generator <M>z</M> of <M>C</M>,
## and get <M>H = N \cup N g \cup N g^2 \cup \cdots \cup N g^{{p-1}}</M>.
## Then <M>H_k</M>, <M>0 \leq k \leq p-1</M>, is given by
## <M>N \cup N gz^k \cup N (gz^k)^2 \cup \cdots \cup N (gz^k)^{{p-1}}</M>.
## The conjugacy classes of all <M>H_k</M> are in bijection via multiplying
## the elements with suitable powers of <M>z</M>,
## and the irreducible characters of all <M>H_k</M> extend to <M>K</M> and
## are in bijection via multiplying the character values with suitable
## <M>p^2</M>-th roots of unity.
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttributeSuppCT( "SourceOfIsoclinicTable", IsNearlyCharacterTable,
[] );

DeclareAttribute( "CharacterTableIsoclinic", IsNearlyCharacterTable );

DeclareOperation( "CharacterTableIsoclinic",
[ IsNearlyCharacterTable, IsRecord ] );
DeclareOperation( "CharacterTableIsoclinic",
[ IsNearlyCharacterTable, IsList and IsCyclotomicCollection ] );
DeclareOperation( "CharacterTableIsoclinic",
[ IsBrauerTable, IsOrdinaryTable and HasSourceOfIsoclinicTable ] );

DeclareOperation( "CharacterTableIsoclinic",
[ IsNearlyCharacterTable, IsPosInt ] );
DeclareOperation( "CharacterTableIsoclinic",
Expand All @@ -3912,9 +4051,6 @@ DeclareOperation( "CharacterTableIsoclinic",
[ IsNearlyCharacterTable, IsList and IsCyclotomicCollection,
IsList and IsCyclotomicCollection ] );

DeclareAttributeSuppCT( "SourceOfIsoclinicTable", IsNearlyCharacterTable,
[] );


#############################################################################
##
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