2.3-11 \=
2.3-12 \<
2.3-13 \*
2.3-17 \^
2.3-18 \^
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 these 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.
In this section we will describe the general functionality provided by this package.
‣ UnipotChevInfo | ( info class ) |
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 ).
The following levels are used throughout the package:
nothing is printed.
When calculating the order of a finite unipotent subgroup, the power presentation of this number is printed. (See Size (2.2-5) for an example.)
When comparing unipotent elements, report for which of them the canonical form must be computed. (See \= (2.3-11) for an example.)
nothing is printed.
While calculating the canonical form, the different steps are printed.
The process of calculating the Chevalley commutator constants is printed.
In this section we will describe the functionality for unipotent subgroups provided by this package.
‣ IsUnipotChevSubGr( grp ) | ( category ) |
Category for unipotent subgroups.
‣ UnipotChevSubGr( type, n, F ) | ( function ) |
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 \(2\).
gap> U_G2 := UnipotChevSubGr("G", 2, Rationals); <Unipotent subgroup of a Chevalley group of type G2 over Rationals> gap> IsUnipotChevSubGr(U_G2); true
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>
‣ PrintObj( U ) | ( method ) |
‣ ViewObj( U ) | ( method ) |
Special methods for unipotent subgroups. (see GAP Reference Manual, section Reference: View and Print for general information on View and Print)
gap> Print(U_G2); UnipotChevSubGr( "G", 2, Rationals )gap> View(U_G2); <Unipotent subgroup of a Chevalley group of type G2 over Rationals>gap>
‣ One( U ) | ( method ) |
‣ OneOp( U ) | ( method ) |
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 (2.3-3)).
‣ Size( U ) | ( method ) |
Size returns the order of a unipotent subgroup. This is a special method for unipotent subgroups using the result in Carter [Car89], Theorem 5.3.3 (ii).
gap> SetInfoLevel( UnipotChevInfo, 2 ); gap> Size( UnipotChevSubGr("E", 8, GF(7)) ); #I The order of this group is 7^120 which is 258086210989349276047917817413172383631691140276099547911280598425927853437317\ 437263620645695945672001 gap> SetInfoLevel( UnipotChevInfo, 1 );
‣ RootSystem( U ) | ( method ) |
This method is similar to the method RootSystem for semisimple Lie algebras (see Section Reference: 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:
gap> R_G2 := RootSystem(U_G2); <root system of rank 2> gap> IsRootSystem(last); true gap> SimpleSystem(R_G2); [ [ 2, -1 ], [ -3, 2 ] ]
In addition to the properties and attributes described in the Reference Manual, the following attributes are installed for the Root Systems by the package Unipot:
‣ PositiveRootsFC( R ) | ( attribute ) |
‣ NegativeRootsFC( R ) | ( attribute ) |
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 \([k_1, \dots, k_l]\).
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> PositiveRootsFC(R_E6)[Length(PositiveRootsFC(R_E6))]; # the highest root [ 1, 2, 2, 3, 2, 1 ]
‣ GeneratorsOfGroup( U ) | ( method ) |
This is a special Method for unipotent subgroups of finite Chevalley groups.
‣ Representative( U ) | ( method ) |
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 2.4). The returned element has representation UNIPOT_DEFAULT_REP (see UNIPOT_DEFAULT_REP (2.3-3)).
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 )
‣ CentralElement( U ) | ( attribute ) |
This method returns the representative of the center of U without calculating the center.
In this section we will describe the functionality for unipotent elements provided by this package.
‣ IsUnipotChevElem( elm ) | ( category ) |
Category for elements of a unipotent subgroup.
‣ IsUnipotChevRepByRootNumbers( elm ) | ( representation ) |
‣ IsUnipotChevRepByFundamentalCoeffs( elm ) | ( representation ) |
‣ IsUnipotChevRepByRoots( elm ) | ( representation ) |
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 themselves, i.e. elements of PositiveRoots(RootSystem(U)).
(See UnipotChevElemByRootNumbers (2.3-4), UnipotChevElemByFundamentalCoeffs (2.3-5) and UnipotChevElemByRoots (2.3-6) for examples.)
‣ UNIPOT_DEFAULT_REP | ( global variable ) |
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.
gap> UNIPOT_DEFAULT_REP := IsUnipotChevRepByFundamentalCoeffs;;
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 the following commands:
gap> UNIPOT_DEFAULT_REP := 3;; gap> One( U_G2 ); ... Error message ...
‣ UnipotChevElemByRootNumbers( U, roots, felems ) | ( operation ) |
‣ UnipotChevElemByRootNumbers( U, root, felem ) | ( operation ) |
‣ UnipotChevElemByRN( U, roots, felems ) | ( operation ) |
‣ UnipotChevElemByRN( U, root, felem ) | ( operation ) |
UnipotChevElemByRootNumbers returns an element of a unipotent subgroup U with representation IsUnipotChevRepByRootNumbers (see IsUnipotChevRepByRootNumbers (2.3-2)).
roots should be a list of root numbers, i.e. integers from the range 1, ..., Length(PositiveRoots(RootSystem(U))). And felems a list of corresponding ring elements or indeterminates over that ring (see GAP Reference Manual, Indeterminate (Reference: Indeterminate) for general information on indeterminates or section 2.4 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.
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 )
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 ) | ( operation ) |
‣ UnipotChevElemByFundamentalCoeffs( U, root, felem ) | ( operation ) |
‣ UnipotChevElemByFC( U, roots, felems ) | ( operation ) |
‣ UnipotChevElemByFC( U, root, felem ) | ( operation ) |
UnipotChevElemByFundamentalCoeffs returns an element of a unipotent subgroup U with representation IsUnipotChevRepByFundamentalCoeffs (see IsUnipotChevRepByFundamentalCoeffs (2.3-2)).
roots should be a list of elements of PositiveRootsFC(RootSystem(U)). And felems a list of corresponding ring elements or indeterminates over that ring (see GAP Reference Manual, Indeterminate (Reference: Indeterminate) for general information on indeterminates or section 2.4 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.
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 )
In this example we create the same two elements as in UnipotChevElemByRootNumbers (2.3-4): \(x_{[ 1, 0 ]}( 2 ) . x_{[ 3, 1 ]}( 7 )\) 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 ) | ( operation ) |
‣ UnipotChevElemByRoots( U, root, felem ) | ( operation ) |
‣ UnipotChevElemByR( U, roots, felems ) | ( operation ) |
‣ UnipotChevElemByR( U, root, felem ) | ( operation ) |
UnipotChevElemByRoots returns an element of a unipotent subgroup U with representation IsUnipotChevRepByRoots (see IsUnipotChevRepByRoots (2.3-2)).
roots should be a list of elements of PositiveRoots(RootSystem(U)). And felems a list of corresponding ring elements or indeterminates over that ring (see GAP Reference Manual, Indeterminate (Reference: Indeterminate) for general information on indeterminates or section 2.4 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.
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 )
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 ) | ( operation ) |
‣ UnipotChevElemByFundamentalCoeffs( x ) | ( operation ) |
‣ UnipotChevElemByRoots( x ) | ( operation ) |
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.
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
Note: If some attributes of x are known (e.g Inverse (see Inverse (2.3-15)) or CanonicalForm (see CanonicalForm (2.3-8))), 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 ) | ( attribute ) |
CanonicalForm returns the canonical form of x. For more information on the canonical form see Carter [Car89], 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.
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 )
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 ) | ( method ) |
‣ ViewObj( x ) | ( method ) |
Special methods for unipotent elements. (see GAP Reference Manual, section Reference: View and Print for general information on View and Print). The output depends on the representation of x.
gap> Print(x); UnipotChevElemByRootNumbers( UnipotChevSubGr( "G", 2, Rationals ), \ [ 1 ], [ 2 ] )gap> View(x); x_{1}( 2 )gap>
gap> Print(x1); UnipotChevElemByFundamentalCoeffs( UnipotChevSubGr( "G", 2, Rationals ), \ [ [ 1, 0 ] ], [ 2 ] )gap> View(x1); x_{[ 1, 0 ]}( 2 )gap>
‣ ShallowCopy( x ) | ( method ) |
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 Reference: Duplication of Objects for further information on copyability)
2.3-11 \=‣ \=( x, y ) | ( method ) |
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 (2.3-8)) 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:
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 );
2.3-12 \<‣ \<( x, y ) | ( method ) |
Special method for UnipotChevElem
This is needed e.g. by AsSSortedList.
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
\[ x < y \Leftrightarrow [ s_1, \dots, s_n ] < [ t_1, \dots, t_n ], \]
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).
2.3-13 \*‣ \*( x, y ) | ( method ) |
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.
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 )
Note: The representation of the product will be always the representation of the first argument.
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 )
‣ OneOp( x ) | ( method ) |
Special method for unipotent elements. OneOp returns the multiplicative neutral element of x. This is equal to x^0.
‣ Inverse( x ) | ( method ) |
‣ InverseOp( x ) | ( method ) |
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 ) | ( method ) |
Special method for unipotent elements. Returns true if and only if x is equal to the identity element.
2.3-17 \^‣ \^( x, i ) | ( method ) |
Integral powers of the unipotent elements are calculated by the default methods installed in GAP. But special (more efficient) methods are installed for root elements and for the identity.
2.3-18 \^‣ \^( x, y ) | ( method ) |
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[, "canonical"] ) | ( method ) |
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 ) | ( property ) |
IsRootElement returns true if and only if x is a 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.
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
‣ IsCentral( U, z ) | ( method ) |
Special method for a unipotent subgroup and a unipotent element.
In some cases, calculation with explicit elements is not enough. Unipot provides a way to do symbolic calculations with unipotent elements for this purpose. This is done by using indeterminates (see GAP Reference Manual, Reference: Indeterminates for more information) over the underlying field instead of the field elements.
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 ]}( b^3*c-a ) * x_{[ 3, 2 ]}( b^3*c^2 )
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