%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W unipot.tex UNIPOT documentation Sergei.Haller %% %H $Id: unipot.tex,v 2.6 2002/07/25 08:39:33 gc1007 Exp $ %% %Y Copyright (C) 2000-2002, Sergei Haller %Y Arbeitsgruppe Algebra, Justus-Liebig-Universitaet Giessen %% %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{The GAP Package Unipot} This chapter describes the package {\Unipot}. Mainly, the package provides the ability to compute with elements of unipotent subgroups of Chevalley groups, but also some properties of this groups. In this chapter we will refer to unipotent subgroups of Chevalley groups as ``unipotent subgroups'' and to elements of unipotent subgroups as ``unipotent elements''. Specifically, we only consider unipotent subgroups generated by all positive root elements. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{General functionality} In this section we will describe the general functionality provided by this package. \>`UnipotChevInfo' V `UnipotChevInfo' is an `InfoClass' used in this package. `InfoLevel' of this `InfoClass' is set to 1 by default and can be changed to any level by `SetInfoLevel( UnipotChevInfo, <n> )'. Following levels are used throughout the package: \beginlist \itemitem{1.}%ordered{1} --- \itemitem{2.} When calculating the order of a finite unipotent subgroup, the power presentation of this number is printed. (See "Size!for `UnipotChevSubGr'" for an example) \itemitem{3.} When comparing unipotent elements, output, for which of them the canonical form must be computed. (See "Equality!for UnipotChevElem" for an example) \itemitem{4.} --- \itemitem{5.} While calculating the canonical form, output the different steps. \itemitem{6.} The process of calculating the Chevalley commutator constants is printed on the screen \endlist %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Unipotent subgroups of Chevalley groups} In this section we will describe the functionality for unipotent subgroups provided by this package. \>IsUnipotChevSubGr( <grp> ) C Category for unipotent subgroups. \>UnipotChevSubGr( <type>, <n>, <F> ) 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 type `"A"', <n> must be a positive integer. For the types `"B"' and `"C"', <n> must be a positive integer $\geq 2$. For the type `"D"', <n> must be a positive integer $\geq 4$. 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 $2$. \beginexample gap> U_G2 := UnipotChevSubGr("G", 2, Rationals); <Unipotent subgroup of a Chevalley group of type G2 over Rationals> gap> IsUnipotChevSubGr(U_G2); true \endexample \begintt gap> UnipotChevSubGr("E", 3, Rationals); Error, <n> must be one of 6, 7, 8 for type E called from UnipotChevFamily( type, n, F ) called from <function>( <arguments> ) called from read-eval-loop Entering break read-eval-print loop ... you can 'quit;' to quit to outer loop, or you can 'return;' to continue brk> \endtt \>PrintObj( <U> )!{for `UnipotChevSubGr'} M \>ViewObj( <U> )!{for `UnipotChevSubGr'} M Special methods for unipotent subgroups. (see {\GAP} Reference Manual, section "ref:View and Print" for general information on `View' and `Print') \beginexample gap> Print(U_G2); UnipotChevSubGr( "G", 2, Rationals )gap> View(U_G2); <Unipotent subgroup of a Chevalley group of type G2 over Rationals>gap> \endexample \>One( <U> )!{for `UnipotChevSubGr'} M \>OneOp( <U> )!{for `UnipotChevSubGr'} M Special methods for unipotent subgroups. Return the identity element of the group <U>. The returned element has representation `UNIPOT_DEFAULT_REP' (see "UNIPOT_DEFAULT_REP"). \>Size( <U> )!{for `UnipotChevSubGr'} M `Size' returns the order of a unipotent subgroup. This is a special method for unipotent subgroups using the result in Carter \cite{Carter72}, Theorem 5.3.3 (ii). \beginexample gap> SetInfoLevel( UnipotChevInfo, 2 ); gap> Size( UnipotChevSubGr("E", 8, GF(7)) ); #I The order of this group is 7^120 which is 25808621098934927604791781741317238363169114027609954791128059842592785343731\ 7437263620645695945672001 gap> SetInfoLevel( UnipotChevInfo, 1 ); \endexample \>RootSystem( <U> )!{for `UnipotChevSubGr'} M This method is similar to the method `RootSystem' for semisimple Lie algebras (see Section "ref:Semisimple Lie Algebras and Root Systems" in the {\GAP} Reference Manual for further information). `RootSystem' returns the underlying root system of the unipotent subgroup <U>. The returned object is from the category `IsRootSystem': \beginexample gap> R_G2 := RootSystem(U_G2); <root system of rank 2> gap> IsRootSystem(last); true gap> SimpleSystem(R_G2); [ [ 2, -1 ], [ -3, 2 ] ] gap> \endexample Additionally to the properties and attributes described in the Reference Manual, following attributes are installed for the Root Systems by the package {\Unipot}: \>PositiveRootsFC( <R> ) A \>NegativeRootsFC( <R> ) A The list of positive resp. negative roots of the root system <R>. Every root is represented as a list of coefficients of the linear combination in fundamental roots. E.g. let $r=\sum_{i=1}^l k_ir_i$, where $r_1, \dots, r_l$ are the fundamental roots, then $r$ is represented as the list $[k_1, \dots, k_l]$. \beginexample gap> U_E6 := UnipotChevSubGr("E",6,GF(2)); <Unipotent subgroup of a Chevalley group of type E6 over GF(2)> gap> R_E6 := RootSystem(U_E6); <root system of rank 6> gap> PositiveRoots(R_E6){[1..6]}; [ [ 2, 0, -1, 0, 0, 0 ], [ 0, 2, 0, -1, 0, 0 ], [ -1, 0, 2, -1, 0, 0 ], [ 0, -1, -1, 2, -1, 0 ], [ 0, 0, 0, -1, 2, -1 ], [ 0, 0, 0, 0, -1, 2 ] ] gap> PositiveRootsFC(R_E6){[1..6]}; [ [ 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ] ] gap> gap> PositiveRootsFC(R)[Length(PositiveRootsFC(R_E6))]; # the highest root [ 1, 2, 2, 3, 2, 1 ] \endexample \>GeneratorsOfGroup( <U> )!{for `UnipotChevSubGr'} M This is a special Method for unipotent subgroups of finite Chevalley groups. \>Representative( <U> ) M This method returns an element of the unipotent subgroup <U> with indeterminates instead of ring elements. Such an element could be used for symbolic computations (see "Symbolic Computation"). The returned element has representation `UNIPOT_DEFAULT_REP' (see "UNIPOT_DEFAULT_REP"). \beginexample gap> Representative(U_G2); x_{1}( t_1 ) * x_{2}( t_2 ) * x_{3}( t_3 ) * x_{4}( t_4 ) * x_{5}( t_5 ) * x_{6}( t_6 ) \endexample \>CentralElement( <U> ) M This method returns the representative of the center of <U> without calculating the center. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Elements of unipotent subgroups of Chevalley groups} In this section we will describe the functionality for unipotent elements provided by this package. \>IsUnipotChevElem( <elm> ) C Category for elements of a unipotent subgroup. \>IsUnipotChevRepByRootNumbers( <elm> ) R \>IsUnipotChevRepByFundamentalCoeffs( <elm> ) R \>IsUnipotChevRepByRoots( <elm> ) R `IsUnipotChevRepByRootNumbers', `IsUnipotChevRepByFundamentalCoeffs' and `IsUnipotChevRepByRoots' are different representations for unipotent elements. Roots of elements with representation `IsUnipotChevRepByRootNumbers' are represented by their numbers (positions) in `PositiveRoots(RootSystem(<U>))'. Roots of elements with representation `IsUnipotChevRepByFundamentalCoeffs' are represented by elements of `PositiveRootsFC(RootSystem(<U>))'. Roots of elements with representation `IsUnipotChevRepByRoots' are represented by roots themself, i.e. elements of `PositiveRoots(RootSystem(<U>))'. (See "UnipotChevElemByRootNumbers", "UnipotChevElemByFundamentalCoeffs" and "UnipotChevElemByRoots" for examples.) \>`UNIPOT_DEFAULT_REP' V This variable contains the default representation for newly created elements, e.g. created by `One' or `Random'. When {\Unipot} is loaded, the default representation is `IsUnipotChevRepByRootNumbers' and can be changed by assigning a new value to `UNIPOT_DEFAULT_REP'. \beginexample gap> UNIPOT_DEFAULT_REP := IsUnipotChevRepByFundamentalCoeffs;; \endexample *Note* that {\Unipot} doesn't check the type of this value, i.e. you may assign any value to `UNIPOT_DEFAULT_REP', which may result in errors in following commands: \begintt gap> UNIPOT_DEFAULT_REP := 3;; gap> One( U_G2 ); ... Error message ... \endtt \>UnipotChevElemByRootNumbers( <U>, <roots>, <felems> ) O \>UnipotChevElemByRootNumbers( <U>, <root>, <felem> ) O \>UnipotChevElemByRN( <U>, <roots>, <felems> ) O \>UnipotChevElemByRN( <U>, <root>, <felem> ) O `UnipotChevElemByRootNumbers' returns an element of a unipotent subgroup <U> with representation `Is\-UnipotChevRepByRootNumbers' (see "IsUnipotChevRepByRootNumbers"). <roots> should be a list of root numbers, i.e. integers from the range 1, ..., `Length(PositiveRoots(Root\-System(<U>)))'. And <felems> a list of corresponding ring elements or indeterminates over that ring (see {\GAP} Reference Manual, "ref:Indeterminate" for general information on indeterminates or section "Symbolic computation" of this manual for examples). The second variant of `UnipotChevElemByRootNumbers' is an abbreviation for the first one if <roots> and <felems> contain only one element. `UnipotChevElemByRN' is just a synonym for `UnipotChevElemByRootNumbers'. \beginexample gap> IsIdenticalObj( UnipotChevElemByRN, UnipotChevElemByRootNumbers ); true gap> y := UnipotChevElemByRootNumbers(U_G2, [1,5], [2,7] ); x_{1}( 2 ) * x_{5}( 7 ) gap> x := UnipotChevElemByRootNumbers(U_G2, 1, 2); x_{1}( 2 ) \endexample In this example we create two elements: $x_{r_1}( 2 ) . x_{r_5}( 7 )$ and $x_{r_1}( 2 )$, where $r_i, i = 1, \dots, 6$ are the positive roots in `PositiveRoots(RootSystem(<U>))' and $x_{r_i}(t), i = 1, \dots, 6$ the corresponding root elements. \>UnipotChevElemByFundamentalCoeffs( <U>, <roots>, <felems> ) O \>UnipotChevElemByFundamentalCoeffs( <U>, <root>, <felem> ) O \>UnipotChevElemByFC( <U>, <roots>, <felems> ) O \>UnipotChevElemByFC( <U>, <root>, <felem> ) O `UnipotChevElemByFundamentalCoeffs' returns an element of a unipotent subgroup <U> with representation `IsUnipotChevRepByFundamentalCoeffs' (see "IsUnipotChevRepByFundamentalCoeffs"). <roots> should be a list of elements of `PositiveRootsFC(Root\-System(<U>))'. And <felems> a list of corresponding ring elements or indeterminates over that ring (see {\GAP} Reference Manual, "ref:Indeterminate" for general information on indeterminates or section "Symbolic computation" of this manual for examples). The second variant of `UnipotChevElemByFundamentalCoeffs' is an abbreviation for the first one if <roots> and <felems> contain only one element. `UnipotChevElemByFC' is just a synonym for `UnipotChevElemByFundamentalCoeffs'. \beginexample gap> PositiveRootsFC(RootSystem(U_G2)); [ [ 1, 0 ], [ 0, 1 ], [ 1, 1 ], [ 2, 1 ], [ 3, 1 ], [ 3, 2 ] ] gap> y1 := UnipotChevElemByFundamentalCoeffs( U_G2, [[ 1, 0 ], [ 3, 1 ]], [2,7] ); x_{[ 1, 0 ]}( 2 ) * x_{[ 3, 1 ]}( 7 ) gap> x1 := UnipotChevElemByFundamentalCoeffs( U_G2, [ 1, 0 ], 2 ); x_{[ 1, 0 ]}( 2 ) \endexample In this example we create the same two elements as in "UnipotChevElemByRootNumbers": $x_{[ 1, 0 ]}( 2 ) . x_{[ 3, 1 ]}( 7 )$ and $x_{[ 1, 0 ]}( 2 )$, where $[ 1, 0 ] = 1r_1 + 0r_2 = r_1$ and $[ 3, 1 ] = 3r_1 + 1r_2=r_5$ are the first and the fifth positive roots of `PositiveRootsFC(RootSystem(<U>))' respectively. \>UnipotChevElemByRoots( <U>, <roots>, <felems> ) O \>UnipotChevElemByRoots( <U>, <root>, <felem> ) O \>UnipotChevElemByR( <U>, <roots>, <felems> ) O \>UnipotChevElemByR( <U>, <root>, <felem> ) O `UnipotChevElemByRoots' returns an element of a unipotent subgroup <U> with representation `IsUnipotChev\-RepByRoots' (see "IsUnipotChevRepByRoots"). <roots> should be a list of elements of `PositiveRoots(% Root\-System(<U>))'. And <felems> a list of corresponding ring elements or indeterminates over that ring (see {\GAP} Reference Manual, "ref:Indeterminate" for general information on indeterminates or section "Symbolic computation" of this manual for examples). The second variant of `UnipotChevElemByRoots' is an abbreviation for the first one if <roots> and <felems> contain only one element. `UnipotChevElemByR' is just a synonym for `UnipotChevElemByRoots'. \beginexample gap> PositiveRoots(RootSystem(U_G2)); [ [ 2, -1 ], [ -3, 2 ], [ -1, 1 ], [ 1, 0 ], [ 3, -1 ], [ 0, 1 ] ] gap> y2 := UnipotChevElemByRoots( U_G2, [[ 2, -1 ], [ 3, -1 ]], [2,7] ); x_{[ 2, -1 ]}( 2 ) * x_{[ 3, -1 ]}( 7 ) gap> x2 := UnipotChevElemByRoots( U_G2, [ 2, -1 ], 2 ); x_{[ 2, -1 ]}( 2 ) \endexample In this example we create again the two elements as in previous examples: $x_{[ 2, -1 ]}( 2 ) . x_{[ 3, -1 ]}( 7 )$ and $x_{[ 2, -1 ]}( 2 )$, where $[ 2, -1 ] = r_1$ and $[ 3, -1 ] = r_5$ are the first and the fifth positive roots of `PositiveRoots(RootSystem( <U>))' respectively. \>UnipotChevElemByRootNumbers( <x> )!{element conversion} O \>UnipotChevElemByFundamentalCoeffs( <x> )!{element conversion} O \>UnipotChevElemByRoots( <x> )!{element conversion} O These three methods are provided for converting a unipotent element to the respective representation. If <x> has already the required representation, then <x> itself is returned. Otherwise a *new* element with the required representation is generated. \beginexample gap> x; x_{1}( 2 ) gap> x1 := UnipotChevElemByFundamentalCoeffs( x ); x_{[ 1, 0 ]}( 2 ) gap> IsIdenticalObj(x, x1); x = x1; false true gap> x2 := UnipotChevElemByFundamentalCoeffs( x1 );; gap> IsIdenticalObj(x1, x2); true \endexample *Note:* If some attributes of <x> are known (e.g `Inverse' (see "Inverse!for `UnipotChevElem'") or `CanonicalForm' (see "CanonicalForm")), then they are ``converted'' to the new representation, too. \){\fmark}UnipotChevElemByRootNumbers( <U>, <list> ) O \){\fmark}UnipotChevElemByRoots( <U>, <list> ) O \){\fmark}UnipotChevElemByFundamentalCoeffs( <U>, <list> ) O *DEPRECATED* These are old versions of `UnipotChevElemByXX' (from {\Unipot} 1.0 and 1.1). They are deprecated now and exist for compatibility only. They may be removed at any time. \>CanonicalForm( <x> ) A `CanonicalForm' returns the canonical form of <x>. For more information on the canonical form see Carter \cite{Carter72}, Theorem 5.3.3 (ii). It says: Each element of a unipotent subgroup $U$ of a Chevalley group with root system $\Phi$ is uniquely expressible in the form $$ \prod_{r_i\in\Phi^+} x_{r_i}(t_i), $$ where the product is taken over all positive roots in increasing order. \beginexample gap> z := UnipotChevElemByFC( U_G2, [[0,1], [1,0]], [3,2]); x_{[ 0, 1 ]}( 3 ) * x_{[ 1, 0 ]}( 2 ) gap> CanonicalForm(z); x_{[ 1, 0 ]}( 2 ) * x_{[ 0, 1 ]}( 3 ) * x_{[ 1, 1 ]}( 6 ) * x_{[ 2, 1 ]}( 12 ) * x_{[ 3, 1 ]}( 24 ) * x_{[ 3, 2 ]}( -72 ) \endexample So if we call the positive roots $r_1,\dots,r_6$, we have $ z = x_{r_2}(3)x_{r_1}(2) = x_{r_1}( 2 ) x_{r_2}( 3 ) x_{r_3}( 6 ) x_{r_4}( 12 ) x_{r_5}( 24 ) x_{r_6}( -72 )$. \>PrintObj( <x> )!{for `UnipotChevElem'} M \>ViewObj( <x> )!{for `UnipotChevElem'} M Special methods for unipotent elements. (see {\GAP} Reference Manual, section "ref:View and Print" for general information on `View' and `Print'). The output depends on the representation of <x>. \beginexample gap> Print(x); UnipotChevElemByRootNumbers( UnipotChevSubGr( "G", 2, Rationals ), \ [ 1 ], [ 2 ] )gap> View(x); x_{1}( 2 )gap> \endexample \beginexample gap> Print(x1); UnipotChevElemByFundamentalCoeffs( UnipotChevSubGr( "G", 2, Rationals ), \ [ [ 1, 0 ] ], [ 2 ] )gap> View(x1); x_{[ 1, 0 ]}( 2 )gap> \endexample \>ShallowCopy( <x> )!{for `UnipotChevElem'} M This is a special method for unipotent elements. `ShallowCopy' creates a copy of <x>. The returned object is *not identical* to <x> but it is *equal* to <x> w.r.t. the equality operator `='. *Note* that `CanonicalForm' and `Inverse' of <x> (if known) are identical to `CanonicalForm' and `Inverse' of the returned object. (See {\GAP} Reference Manual, section "ref:Duplication of Objects" for further information on copyability) \>`<x> = <y>'{Equality!for UnipotChevElem}@{Equality!for `UnipotChevElem'} M \indextt{\\=} Special method for unipotent elements. If <x> and <y> are identical or are products of the *same* root elements then `true' is returned. Otherwise `CanonicalForm' (see "CanonicalForm") of both arguments must be computed (if not already known), which may be expensive. If the canonical form of one of the elements must be calculated and `InfoLevel' of `UnipotChevInfo' is at least 3, the user is notified about this: \beginexample gap> y := UnipotChevElemByRN( U_G2, [1,5], [2,7] ); x_{1}( 2 ) * x_{5}( 7 ) gap> z := UnipotChevElemByRN( U_G2, [5,1], [7,2] ); x_{5}( 7 ) * x_{1}( 2 ) gap> SetInfoLevel( UnipotChevInfo, 3 ); gap> y=z; #I CanonicalForm for the 1st argument is not known. #I computing it may take a while. #I CanonicalForm for the 2nd argument is not known. #I computing it may take a while. true gap> SetInfoLevel( UnipotChevInfo, 1 ); \endexample \>`<x> \< <y>'{Less than!for UnipotChevElem}@{Less than!for `UnipotChevElem'} M \indextt{\\\<} Special Method for `UnipotChevElem' This is needed e.g. by `AsSSortetList'. The ordering is computed in the following way: Let $x = x_{r_1}(s_1) ... x_{r_n}(s_n)$ and $y = x_{r_1}(t_1) ... x_{r_n}(t_n)$, then $$ x \< y \quad \Leftrightarrow \quad [ s_1, \dots, s_n ] \< [ t_1, \dots, t_n ], $$ where the lists are compared lexicographically. e.g. for $x = x_{r_1}(1)x_{r_2}(1) = x_{r_1}(1)x_{r_2}(1)x_{r_3}(0)$ (field elems: `[ 1, 1, 0 ]') and $y = x_{r_1}(1)x_{r_3}(1) = x_{r_1}(1)x_{r_2}(0)x_{r_3}(1)$ (field elems: `[ 1, 0, 1 ]') we have $y \< x$ (above lists ordered lexicographically). \>`<x> * <y>'{Multiplication!for UnipotChevElem}@{Multiplication!for `UnipotChevElem'} M \indextt{\\\*} Special method for unipotent elements. The expressions in the form $x_r(t)x_r(u)$ will be reduced to $x_r(t+u)$ whenever possible. \beginexample gap> y;z; x_{1}( 2 ) * x_{5}( 7 ) x_{5}( 7 ) * x_{1}( 2 ) gap> y*z; x_{1}( 2 ) * x_{5}( 14 ) * x_{1}( 2 ) \endexample *Note:* The representation of the product will be always the representation of the first argument. \beginexample gap> x; x1; x=x1; x_{1}( 2 ) x_{[ 1, 0 ]}( 2 ) true gap> x * x1; x_{1}( 4 ) gap> x1 * x; x_{[ 1, 0 ]}( 4 ) \endexample \>OneOp( <x> )!{for `UnipotChevElem'} M Special method for unipotent elements. `OneOp' returns the multiplicative neutral element of <x>. This is equal to `<x>^0'. \>Inverse( <x> )!{for `UnipotChevElem'} M \>InverseOp( <x> )!{for `UnipotChevElem'} M Special methods for unipotent elements. We are using the fact $$ \Bigl( x_{r_1}( t_1) . . . x_{r_m}(t_m) \Bigr)^{-1} = x_{r_m}(-t_m) . . . x_{r_1}(-t_1) \. $$ \>IsOne( <x> ) M Special method for unipotent elements. Returns `true' if and only if <x> is equal to the identity element. \>`<x> ^ <i>'{Powers!of UnipotChevElem}@{Powers!of `UnipotChevElem'} M Integral powers of the unipotent elements are calculated by the default methods installed in {\GAP}. But special (more efficient) methods are instlled for root elements and for the identity. \>`<x> ^ <y>'{Conjugation!of UnipotChevElem}@{Conjugation!of `UnipotChevElem'} M Conjugation of two unipotent elements, i.e. $x^y = y^{-1}xy$. The representation of the result will be the representation of <x>. \>Comm( <x>, <y> )!{for `UnipotChevElem'} M \>Comm( <x>, <y>, "canonical" )!{for `UnipotChevElem'} M Special methods for unipotent elements. `Comm' returns the commutator of <x> and <y>, i.e. $<x> ^{-1} . <y>^{-1} . <x> . <y>$. The second variant returns the canonical form of the commutator. In some cases it may be more efficient than `CanonicalForm( Comm( <x>, <y> ) )' \>IsRootElement( <x> ) P `IsRootElement' returns `true' if and only if <x> is a {\it root element}, i.e.\ $<x>=x_{r}(t)$ for some root $r$. We store this property immediately after creating objects. *Note:* the canonical form of <x> may be a root element even if <x> isn't one. \beginexample gap> x := UnipotChevElemByRN( U_G2, [1,5,1], [2,7,-2] ); x_{1}( 2 ) * x_{5}( 7 ) * x_{1}( -2 ) gap> IsRootElement(x); false gap> CanonicalForm(x); IsRootElement(CanonicalForm(x)); x_{5}( 7 ) true \endexample \>IsCentral( <U>, <z> ) Special method for a unipotent subgroup and a unipotent element. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Symbolic computation} In some cases, calculation with explicite elements is not enough. {\Unipot} povides a way to do symbolic calculations with unipotent elements for this purpose. This is done by using indeterminates (see {\GAP} Reference Manual, "ref:Indeterminates" for more information) over the underlying field instead of the field elements. \beginexample gap> U_G2 := UnipotChevSubGr("G", 2, Rationals);; gap> a := Indeterminate( Rationals, "a" ); a gap> b := Indeterminate( Rationals, "b", [a] ); b gap> c := Indeterminate( Rationals, "c", [a,b] ); c gap> x := UnipotChevElemByFC(U_G2, [ [3,1], [1,0], [0,1] ], [a,b,c] ); x_{[ 3, 1 ]}( a ) * x_{[ 1, 0 ]}( b ) * x_{[ 0, 1 ]}( c ) gap> CanonicalForm(x); x_{[ 1, 0 ]}( b ) * x_{[ 0, 1 ]}( c ) * x_{[ 3, 1 ]}( a ) * x_{[ 3, 2 ]}( a*c ) gap> CanonicalForm(x^-1); x_{[ 1, 0 ]}( -b ) * x_{[ 0, 1 ]}( -c ) * x_{[ 1, 1 ]}( b*c ) * x_{[ 2, 1 ]}( -b^2*c ) * x_{[ 3, 1 ]}( -a+b^3*c ) * x_{[ 3, 2 ]}( b^3*c^2 ) \endexample %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %E unipot.tex . . . . . . . . . . . . . . . . . . . . . . . . ends here