Quelle unipot.gd Sprache: unbekannt

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##
#W unipot.gd Unipot package Sergei Haller
##
#Y Copyright (C) 2000-2004, Sergei Haller
#Y Arbeitsgruppe Algebra, Justus-Liebig-Universitaet Giessen
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## This is the declaration part of the package
##

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#C IsUnipotChevFamily
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## Category of families of elements of a unipotent subgroup of a Chevalley
## group
##

DeclareCategory("IsUnipotChevFamily", IsFamily);

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#C IsUnipotChevElem
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## Category of elements of a unipotent subgroup of a Chevalley group.
##

DeclareCategory( "IsUnipotChevElem", IsMultiplicativeElementWithInverse );
DeclareCategoryCollections( "IsUnipotChevElem" );

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#C IsUnipotChevSubGr
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## Category of unipotent subgroups of a Chevalley group.
##

DeclareCategory( "IsUnipotChevSubGr", IsGroup and IsUnipotChevElemCollection );

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#F UnipotChevFamily( <type>, <n>, <F> )
##
## This function creates a UnipotChevFamily of type <type> and of rank <n>
## over the ring <F>.
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## <type> must be one of A, B, C, D, E, F, G
## For the types A to D, <n> must be a positive integer.
## For the type E, <n> must be one of 6, 7, 8.
## For the type F, <n> must be 4.
## For the type G, <n> must be 2.
##

DeclareGlobalFunction( "UnipotChevFamily" );

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#F UnipotChevSubGr( <type>, <n>, <F> )
##
## `UnipotChevSubGr' returns the unipotent subgroup $U$ of the Chevalley group
## of type <type>, rank <n> over the ring <F>.
##
## <type> must be one of A, B, C, D, E, F, G
## For the types A to D, <n> must be a positive integer.
## For the type E, <n> must be one of 6, 7, 8.
## For the type F, <n> must be 4.
## For the type G, <n> must be 2.
##

DeclareGlobalFunction( "UnipotChevSubGr" );

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#O UnipotChevElem( , <list>, <list> ) . . `undocumented'
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#O UnipotChevElemByRootNumbers( , <list>, <list> )
#O UnipotChevElemByRN( , <list>, <list> )
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#O UnipotChevElemByFundamentalCoeffs( , <list>, <list> )
#O UnipotChevElemByFC( , <list>, <list> )
##
#O UnipotChevElemByRoots( , <list>, <list> )
#O UnipotChevElemByR( , <list>, <list> )
##
## Returns an element of a unipotent subgroup of a Chevalley group
##

DeclareOperation( "UnipotChevElem",
 [ IsUnipotChevSubGr, IsRecord, IS_OPERATION ] );
DeclareOperation( "UnipotChevElemByRootNumbers",
 [ IsUnipotChevSubGr, IsList, IsList ] );
DeclareOperation( "UnipotChevElemByFundamentalCoeffs",
 [ IsUnipotChevSubGr, IsList, IsList ] );
DeclareOperation( "UnipotChevElemByRoots",
 [ IsUnipotChevSubGr, IsList, IsList ] );

DeclareSynonym( "UnipotChevElemByRN", UnipotChevElemByRootNumbers );
DeclareSynonym( "UnipotChevElemByFC", UnipotChevElemByFundamentalCoeffs );
DeclareSynonym( "UnipotChevElemByR" , UnipotChevElemByRoots );

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#A CanonicalForm( <x> ) . . . . . . . . . canonical form of a `UnipotChevElem'
##
## `CanonicalForm' returns the canonical form of <x>.
## For more information on the canonical form see [Car72], Theorem 5.3.3 (ii).
##

DeclareAttribute( "CanonicalForm",
 IsUnipotChevElem
 ) ;

DeclareGlobalFunction( "ChevalleyCommutatorConstant" );

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#P IsRootElement( <x> ) . . . . . . . . . . . . . . . . . . for UnipotChevElem
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## `IsRootElement' returns `true' if and only if <x> is a <root element>,
## i.e $<x>=x_{r_i}(t)$ for some root $r_i$.
##
#N *Note:* the canonical form of <x> may be a root element even if <x> isn't
#N one.
##

DeclareProperty( "IsRootElement",
 IsUnipotChevElem
 ) ;

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#V UnipotChevInfo . . . . . . . . . . . . . . . . . . . . . . . . . InfoClass
##
## `UnipotChevInfo' is an `InfoClass' used in this package. `InfoLevel' of
## this `InfoClass' is set to 1 by default.
##

DeclareInfoClass("UnipotChevInfo");
SetInfoLevel(UnipotChevInfo, 1);

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##
#B Bibliography
##
#B [Car72] Roger W. Carter. Simple Groups of Lie Type.
#B John Wiley \& Sons Ltd., New York, 1972.
#B Wiley Classics Library Edition Published 1989.
##
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Quelle unipot.gd Sprache: unbekannt

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