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##
#W dlnames.gd GAP 4 package CTblLib Michael Claßen-Houben
##
#Y Copyright (C) 2005, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
##
## This file contains declarations concerning Deligne-Lusztig names of
## unipotent characters of finite groups of Lie type.
##
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##
## <#GAPDoc Label="sec:unipot">
## Unipotent characters are defined for finite groups of Lie type.
## For most of these groups whose character table is in the &GAP; Character
## Table Library, the unipotent characters are known and parametrised by
## labels.
## This labeling is due to the work of P. Deligne and G. Lusztig,
## thus the label of a unipotent character is called its Deligne-Lusztig
## name (see <Cite Key="Cla05"/>).
## <#/GAPDoc>
##
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##
#A DeligneLusztigNames( <tbl> )
#A DeligneLusztigNames( <string> )
#A DeligneLusztigNames( <record> )
##
## <#GAPDoc Label="DeligneLusztigNames">
## <ManSection>
## <Attr Name="DeligneLusztigNames" Arg="obj"/>
##
## <Description>
## For a character table <A>obj</A>, <Ref Attr="DeligneLusztigNames"/>
## returns a list of Deligne-Lusztig names of the the unipotent characters
## of <A>obj</A>.
## If the <M>i</M>-th entry is bound then it is the name of the <M>i</M>-th
## irreducible character of <A>obj</A>, and this character is irreducible.
## If an irreducible character is not unipotent the accordant position is
## unbound.
## <P/>
## <Ref Attr="DeligneLusztigNames"/> called with a string <A>obj</A>,
## calls itself with the argument <C>CharacterTable( <A>obj</A> )</C>.
## <P/>
## When <Ref Attr="DeligneLusztigNames"/> is called with a record <A>obj</A>
## then this should have the components <C>isoc</C>, <C>isot</C>, <C>l</C>,
## and <C>q</C>,
## where <C>isoc</C> and <C>isot</C> are strings defining the isogeny class
## and isogeny type, and <C>l</C> and <C>q</C> are integers.
## <!-- which strings are supported? -->
## These components define a finite group of Lie type uniquely.
## Moreover this way one can choose Deligne-Lusztig names for a prescribed
## type in those cases where a group has more than one interpretation
## as a finite group of Lie type, see the example below.
## (The first call of <Ref Attr="DeligneLusztigNames"/> sets the attribute
## value in the character table.)
## <!-- be more precise here! -->
## <P/>
## <Example>
## gap> DeligneLusztigNames( "L2(7)" );
## [ [ 2 ],,,, [ 1, 1 ] ]
## gap> tbl:= CharacterTable( "L2(7)" );
## CharacterTable( "L3(2)" )
## gap> HasDeligneLusztigNames( tbl );
## true
## gap> DeligneLusztigNames( rec( isoc:= "A", isot:= "simple",
## > l:= 2, q:= 2 ) );
## [ [ 3 ],,, [ 2, 1 ],, [ 1, 1, 1 ] ]
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DeligneLusztigNames", IsCharacterTable );
DeclareAttribute( "DeligneLusztigNames", IsString );
DeclareAttribute( "DeligneLusztigNames", IsRecord );
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##
#A DeligneLusztigName( <chi> )
##
## <#GAPDoc Label="DeligneLusztigName">
## <ManSection>
## <Func Name="DeligneLusztigName" Arg="chi"/>
##
## <Description>
## For a unipotent character <A>chi</A>, <Ref Attr="DeligneLusztigName"/>
## returns the Deligne-Lusztig name of <A>chi</A>.
## For that, <Ref Func="DeligneLusztigNames"/> is called with the argument
## <C>UnderlyingCharacterTable( <A>chi</A> )</C>.
## <P/>
## <Example>
## gap> tbl:= CharacterTable( "F4(2)" );;
## gap> DeligneLusztigName( Irr( tbl )[9] );
## fail
## gap> HasDeligneLusztigNames( tbl );
## true
## gap> List( [ 1 .. 8 ], i -> DeligneLusztigName( Irr( tbl )[i] ) );
## [ "phi{1,0}", "[ [ 2 ], [ ] ]", "phi{2,4}''", "phi{2,4}'",
## "F4^II[1]", "phi{4,1}", "F4^I[1]", "phi{9,2}" ]
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareAttribute( "DeligneLusztigName", IsCharacter );
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##
#O UnipotentCharacter( <tbl>, <label> )
##
## <#GAPDoc Label="UnipotentCharacter">
## <ManSection>
## <Func Name="UnipotentCharacter" Arg="tbl, label"/>
##
## <Description>
## Let <A>tbl</A> be the ordinary character table of a finite group
## of Lie type in the &GAP; Character Table Library.
## <Ref Oper="UnipotentCharacter"/> returns the unipotent character with
## Deligne-Lusztig name <A>label</A>.
## <P/>
## The object <A>label</A> must be either
## a list of integers which describes a partition
## (if the finite group of Lie type is of the type <M>A_l</M> or
## <M>{}^2\!A_l</M>),
## a list of two lists of integers which describes a symbol
## (if the group is of classical type other than <M>A_l</M> and
## <M>{}^2\!A_l</M>) or a string (if the group is of exceptional type).
## <P/>
## A call of <Ref Oper="UnipotentCharacter"/> sets the attribute
## <Ref Func="DeligneLusztigNames"/> for <A>tbl</A>.
## <P/>
## <Example>
## gap> tbl:= CharacterTable( "U4(2).2" );;
## gap> UnipotentCharacter( tbl, [ [ 0, 1 ], [ 2 ] ] );
## Character( CharacterTable( "U4(2).2" ),
## [ 15, 7, 3, -3, 0, 3, -1, 1, 0, 1, -2, 1, 0, 0, -1, 5, 1, 3, -1, 2,
## -1, 1, -1, 0, 0 ] )
## </Example>
## </Description>
## </ManSection>
## <#/GAPDoc>
##
DeclareOperation( "UnipotentCharacter", [ IsCharacterTable, IsObject ] );
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##
#E
