- General functionality
- Unipotent subgroups of Chevalley groups
- Elements of unipotent subgroups of Chevalley groups
- Symbolic computation

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.

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:

- ---
- When calculating the order of a finite unipotent subgroup, the power presentation of this number is printed. (See Size!for `UnipotChevSubGr' for an example)
- When comparing unipotent elements, output, for which of them the canonical form must be computed. (See Equality!for UnipotChevElem for an example)
- ---
- While calculating the canonical form, output the different steps.
- The process of calculating the Chevalley commutator constants is printed on the screen

In this section we will describe the functionality for unipotent subgroups provided by this package.

`IsUnipotChevSubGr( `

` ) C`

Category for unipotent subgroups.

`UnipotChevSubGr( `

`, `

`, `

` ) 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 `geq2`.

For the type `"D"`

, `n` must be a positive integer `geq4`.

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( `

` ) M`

`ViewObj( `

` ) 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>gap>

`One( `

` ) M`

`OneOp( `

` ) 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( `

` ) M`

`Size`

returns the order of a unipotent subgroup. This is a
special method for unipotent subgroups using the result in
Carter Carter72, 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 25808621098934927604791781741317238363169114027609954791128059842592785343731\ 7437263620645695945672001 gap> SetInfoLevel( UnipotChevInfo, 1 );

`RootSystem( `

` ) M`

This method is similar to the method `RootSystem`

for semisimple Lie
algebras (see Section 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 ] ] gap>

Additionally to the properties and attributes described in the Reference Manual, following attributes are installed for the Root Systems by the package Unipot:

`PositiveRootsFC( `

` ) A`

`NegativeRootsFC( `

` ) 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_{i}r_{i}`, where

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 ]

`GeneratorsOfGroup( `

` ) M`

This is a special Method for unipotent subgroups of finite Chevalley groups.

`Representative( `

` ) 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).

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( `

` ) M`

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( `

` ) 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
`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`

.

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
following commands:

gap> UNIPOT_DEFAULT_REP := 3;; gap> One( U_G2 ); ... Error message ...

`UnipotChevElemByRootNumbers( `

`, `

`, `

` ) O`

`UnipotChevElemByRootNumbers( `

`, `

`, `

` ) O`

`UnipotChevElemByRN( `

`, `

`, `

` ) O`

`UnipotChevElemByRN( `

`, `

`, `

` ) O`

`UnipotChevElemByRootNumbers`

returns an element of a unipotent subgroup
`U` with representation `IsUnipotChevRepByRootNumbers`

(see
IsUnipotChevRepByRootNumbers).

`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 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`

.

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

`PositiveRoots(RootSystem(`

`))`

and `UnipotChevElemByFundamentalCoeffs( `

`, `

`, `

` ) O`

`UnipotChevElemByFundamentalCoeffs( `

`, `

`, `

` ) O`

`UnipotChevElemByFC( `

`, `

`, `

` ) O`

`UnipotChevElemByFC( `

`, `

`, `

` ) O`

`UnipotChevElemByFundamentalCoeffs`

returns an element of a unipotent
subgroup `U` with representation `IsUnipotChevRepByFundamentalCoeffs`

(see IsUnipotChevRepByFundamentalCoeffs).

`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 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`

.

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: `x _{[ 1, 0 ]}( 2 ) . x_{[ 3, 1 ]}( 7 )`
and

`PositiveRootsFC(RootSystem(`

`))`

respectively.
`UnipotChevElemByRoots( `

`, `

`, `

` ) O`

`UnipotChevElemByRoots( `

`, `

`, `

` ) O`

`UnipotChevElemByR( `

`, `

`, `

` ) O`

`UnipotChevElemByR( `

`, `

`, `

` ) O`

`UnipotChevElemByRoots`

returns an element of a unipotent subgroup `U`
with representation `IsUnipotChevRepByRoots`

(see
IsUnipotChevRepByRoots).

`roots` should be a list of elements of ```
PositiveRoots(
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`

.

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

`PositiveRoots(RootSystem( `

`))`

respectively.
`UnipotChevElemByRootNumbers( `

` ) O`

`UnipotChevElemByFundamentalCoeffs( `

` ) O`

`UnipotChevElemByRoots( `

` ) 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.

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!for `UnipotChevElem') or `CanonicalForm`

(see
CanonicalForm)), then they are ``converted'' to the new representation,
too.

` UnipotChevElemByRootNumbers( `

`U``, `

`list`` ) O`

` UnipotChevElemByRoots( `

`U``, `

`list`` ) O`

` 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( `

` ) 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 `Phi` is uniquely expressible in the form

`
prod _{r_iinPhi^+} 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},...,r_{6}`, we have

`PrintObj( `

` ) M`

`ViewObj( `

` ) M`

Special methods for unipotent elements. (see GAP Reference Manual,
section 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( `

` ) 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)

` = `

` 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. 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 );

` < `

` M`

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

` x < y Leftrightarrow [ s _{1}, ..., s_{n} ] < [ t_{1}, ..., 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 `[ 1, 0, 1 ]`

)
we have

` * `

` M`

Special method for unipotent elements. The expressions in the form
`x _{r}(t)x_{r}(u)` will be reduced to

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( `

` ) M`

Special method for unipotent elements. `OneOp`

returns the multiplicative
neutral element of `x`. This is equal to `x``^0`

.

`Inverse( `

` ) M`

`InverseOp( `

` ) 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( `

` ) M`

Special method for unipotent elements. Returns `true`

if and only if `x`
is equal to the identity element.

` ^ `

` 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.

` ^ `

` M`

Conjugation of two unipotent elements, i.e. `x ^{y} = y^{-1}xy`. The
representation of the result will be the representation of

`Comm( `

`, `

` ) M`

`Comm( `

`, `

`, "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(
```

`, `

` ) )`

`IsRootElement( `

` ) P`

`IsRootElement`

returns `true`

if and only if `x` is a *root*
element, i.e. ` x=x_{r}(t)` for some root

**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( `

`, `

` )`

Special method for a unipotent subgroup and a unipotent element.

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, 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 ]}( -a+b^3*c ) * x_{[ 3, 2 ]}( b^3*c^2 )

Unipot manual

Oktober 2004