This chapter describes the package Unipot. This package provides the ability to compute with elements of unipotent subgroups of Chevalley 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''.
In this section we will describe the general functionality provided by this package.
UnipotChevInfo() I
UnipotChevInfo
is an InfoClass
used in this package. InfoLevel
of
this InfoClass
is set to 1 by default.
2.2 Unipotent subgroups of Chevalley groups
In this section we will describe the functionality for unipotent subgroups provided by this package.
IsUnipotChevSubGr 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 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.
gap> U_G2 := UnipotChevSubGr("G", 2, Rationals); <Unipotent subgroup of a Chevalley group of type G2 over Rationals>
gap> U_E3 := UnipotChevSubGr("E", 3, Rationals); Error <n> must be one of 6, 7, 8 for type E at Error( "<n> must be one of 6, 7, 8 for type E " ); 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 ) M
ViewObj(
U ) M
Special methods for unipotent subgroups.
(see GAP Reference Manual, section 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>
One(
U ) M
OneOp(
U ) M
Special methods for unipotent subgroups. Return the identity of U.
Size(
U ) M
Size
returns the size of a unipotent subgroup. This is a
special method for unipotent subgroups.
Size can be computed using the result in Carter Carter72, Theorem 5.3.3 (ii).
RootSystem(
U ) M
This method is similar to the method RootSystem
for semisimple Lie
algebras (see GAP4.1 Reference Manual, section 58.7 for further
information). RootSystem
calculates the root system of the unipotent
subgroup U. The output is a record with the following components:
fundroots
A set of fundamental roots
posroots
The set of positive roots of the root system.
The positive roots are listed according to increasing height.
gap> RootSystem(U_G2); rec( posroots := [ [ 2, -1 ], [ -3, 2 ], [ -1, 1 ], [ 1, 0 ], [ 3, -1 ], [ 0, 1 ] ], fundroots := [ [ 2, -1 ], [ -3, 2 ] ] ) gap>
2.3 Elements of unipotent subgroups of Chevalley groups
In this section we will describe the functionality for unipotent elements provided by this package.
IsUnipotChevElem C
Category for elements of a unipotent subgroup.
IsUnipotChevRepByRootNumbers R
IsUnipotChevRepByFundamentalCoeffs R
IsUnipotChevRepByRoots R
IsUnipotChevRepByRootNumbers
, IsUnipotChevRepByFundamentalCoeffs
and
IsUnipotChevRepByRoots
are different representations for unipotent
elements.
Roots of elements with representation IsUnipotChevRepByRootNumbers
are
represented by their numbers (positions) in RootSystem(
U).posroots
.
Roots of elements with representation IsUnipotChevRepByFundamentalCoeffs
are
represented by coefficients of linear combinations of fundamental roots
RootSystem(
U).fundroots
.
Roots of elements with representation IsUnipotChevRepByRoots
are
represented by roots themself.
(See UnipotChevElemByRootNumbers, UnipotChevElemByFundamentalCoeffs and UnipotChevElemByRoots for examples)
UnipotChevElemByRootNumbers(
U,
list ) O
UnipotChevElemByRootNumbers(
U,
r,
x ) O
UnipotChevElemByRN(
U,
list ) O
UnipotChevElemByRN(
U,
r,
x ) O
UnipotChevElemByRootNumbers
returns an element of a unipotent subgroup
U with representation IsUnipotChevRepByRootNumbers
(see IsUnipotChevRepByRootNumbers).
list should be a list of records with components r and x
representing the number of the root in RootSystem(
U).posroots
and a
ring element, respectively.
The second variant of UnipotChevElemByRootNumbers
is an abbreviation
for the first one if list contains only one record.
UnipotChevElemByRN
is a synonym for UnipotChevElemByRootNumbers
.
gap> IsIdenticalObj( UnipotChevElemByRN, UnipotChevElemByRootNumbers ); true gap> y := UnipotChevElemByRootNumbers(U_G2, [rec(r:=1, x:=2), rec(r:=5, x:=7)]); x_{1}( 2 ) * x_{5}( 7 ) gap> x := UnipotChevElemByRootNumbers(U_G2, 1, 2); x_{1}( 2 )
In this example we create two elements: xr1( 2 ) ·xr5( 7 ) and
xr1( 2 ), where ri, i = 1, ..., 6 are the positive roots in
RootSystem(
U).posroots
and xri(t), i = 1, ..., 6 the
corresponding root elements.
UnipotChevElemByFundamentalCoeffs(
U,
list ) O
UnipotChevElemByFundamentalCoeffs(
U,
coeffs,
x ) O
UnipotChevElemByFC(
U,
list ) O
UnipotChevElemByFC(
U,
coeffs,
x ) O
UnipotChevElemByFundamentalCoeffs
returns an element of a unipotent
subgroup U with representation IsUnipotChevRepByFundamentalCoeffs
(see IsUnipotChevRepByFundamentalCoeffs).
list should be a list of records with components coeffs and x
representing a root in RootSystem(
U).posroots
as coefficients of a
linear combination of fundamental roots RootSystem(
U).fundroots
and
a ring element, respectively.
The second variant of UnipotChevElemByFundamentalCoeffs
is an
abbreviation for the first one if list contains only one record.
UnipotChevElemByFC
is a synonym for UnipotChevElemByFundamentalCoeffs
.
gap> y1 := UnipotChevElemByFundamentalCoeffs( U_G2, > [ rec( coeffs := [ 1, 0 ], x := 2 ), > rec( coeffs := [ 3, 1 ], x := 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:
x[ 1, 0 ]( 2 ) ·x[ 3, 1 ]( 7 ) and x[ 1, 0 ]( 2 ),
where [ 1, 0 ] = 1r1 + 0r2 = r1 and [ 3, 1 ] = 3r1 + 1r2=r5
are the first and the fifth positive roots of RootSystem(
U).posroots
respectively.
UnipotChevElemByRoots(
U,
list ) O
UnipotChevElemByRoots(
U,
r,
x ) O
UnipotChevElemByR(
U,
list ) O
UnipotChevElemByR(
U,
r,
x ) O
UnipotChevElemByRoots
returns an element of a unipotent subgroup U
with representation IsUnipotChevRepByRoots
(see IsUnipotChevRepByRoots).
list should be a list of records with components r and x
representing the root in RootSystem(
U).posroots
and a ring
element, respectively.
The second variant of UnipotChevElemByRoots
is an abbreviation for the
first one if list contains only one record.
UnipotChevElemByR
is a synonym for UnipotChevElemByRoots
.
gap> y2 := UnipotChevElemByRoots( U_G2, > [ rec( r := [ 2, -1 ], x := 2 ), > rec( r := [ 3, -1 ], x := 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 ] = r1 and [ 3, -1 ] = r5
are the first and the fifth positive roots of RootSystem(
U).posroots
respectively.
UnipotChevElemByRootNumbers(
x ) O
UnipotChevElemByFundamentalCoeffs(
x ) O
UnipotChevElemByRoots(
x ) O
UnipotChevElemByRootNumbers
is provided for converting elements to the
representation IsUnipotChevRepByRootNumbers
. If x has already the
representation IsUnipotChevRepByRootNumbers
, then x itself is
returned. Otherwise a new element with representation
IsUnipotChevRepByRootNumbers
is generated.
UnipotChevElemByFundamentalCoeffs
and UnipotChevElemByRoots
do the
same for the representations IsUnipotChevRepByFundamentalCoeffs
and
IsUnipotChevRepByRoots
, respectively.
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!UnipotChevElem), CanonicalForm
(see CanonicalForm)),
then they are ``converted'' to the new representation, too.
CanonicalForm(
x ) A
CanonicalForm
returns the canonical form of x.
For more information on the canonical form see Carter Carter72,
Theorem 5.3.3 (ii). It says:
Each element of a unipotent subgroup U of a Chevalley group with
root system F is uniquely expressible in the form
|
gap> z := UnipotChevElemByFC( U_G2, > [ rec( coeffs := [0,1], x := 3 ), > rec( coeffs := [1,0], x := 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 )
PrintObj(
x ) M
ViewObj(
x ) M
Special methods for unipotent elements.
(see GAP Reference Manual, section View and Print for general
information on View
and Print
)
gap> Print(x); UnipotChevElemByRootNumbers( UnipotChevSubGr( "G", 2, Rationals ), [ rec( r := 1, x := 2 ) ] )gap> View(x); x_{1}( 2 )
gap> Print(x1); UnipotChevElemByFundamentalCoeffs( UnipotChevSubGr( "G", 2, Rationals ), [ rec( coeffs := [ 1, 0 ], x := 2 ) ] )gap> View(x1); x_{[ 1, 0 ]}( 2 )
ShallowCopy(
x ) 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 Duplication of Objects for further information on copyability)
x =
y M
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.
gap> y := UnipotChevElemByRootNumbers( U_G2, [ rec( > r := 1, > x := 2 ), rec( > r := 5, > x := 7 ) ] ); x_{1}( 2 ) * x_{5}( 7 ) gap> gap> z := UnipotChevElemByRootNumbers( U_G2, [ rec( > r := 5, > x := 7 ), rec( > r := 1, > x := 2 ) ] ); x_{5}( 7 ) * x_{1}( 2 ) 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>
x *
y M
Special method for unipotent elements. The expressions in the form xr(t)xr(u) will be reduced to xr(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: If both arguments have the same representation, the product will have it too. But if the representations are different, the representation of the first argument will become the representation of the product.
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 ) M
Special method for unipotent elements. OneOp
returns the
multiplicative neutral element of x. This is equal to x^0.
Inverse(
x ) M
InverseOp(
x ) M
Special methods for unipotent elements. We are using the fact
|
Comm(
x,
y ) M
Comm(
x,
y, "canonical" ) 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 root element,
i.e x =xr(t) for some root r.
We store this property just after creating objects.
Note: the canonical form of x may be a root element even if x isn't one.
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Unipot manual