Let us fix an algebraically closed field *K* and let * G* be a connected
reductive algebraic group over

Then * G* is determined up to isomorphism by the

This is obtained by a slight generalization of our setup for a Coxeter
group *W*. This time we assume the canonical basis of the vector space *V*
on which *W* acts is a *ℤ*-basis of *X( T)*, and

`W.simpleRoots`

whose lines are the simple roots expressed in this
basis of `W.simpleCoroots`

whose lines are the simple coroots in the basis of
`x*y`

. Thus,
we must have the relation
`W.simpleCoroots*TransposedMat(W.simpleRoots)=CartanMat(W)`

.
We get that by a new form of the function `CoxeterGroup`

, where the
arguments are the two matrices `W.simpleRoots`

and `W.simpleCoroots`

described above. The roots need not generate *V*, so the matrices need not
be square. For instance, the root datum of the linear group of rank 3 can
be specified as:

gap> W := CoxeterGroup( [ [ -1, 1, 0], [ 0, -1, 1 ] ], > [ [ -1, 1, 0], [ 0, -1, 1 ] ] ); CoxeterGroup([[-1,1,0],[0,-1,1]],[[-1,1,0],[0,-1,1]]) gap> MatXPerm( W, W.1); [ [ 0, 1, 0 ], [ 1, 0, 0 ], [ 0, 0, 1 ] ]

here the symmetric group on 3 letters acts by permutation of the basis
vectors of *V*. The dimension of *V* (the length of the vectors in
`.simpleRoots`

) is called the **rank** and the dimension of the subspace
generated by the roots (the length of `.simpleroots`

) the **semi-simple
rank**. In the example the rank is 3 and the semisimple rank is 2; the
integral elements of *V* correspond to the characters of a maximal torus,
and the integral elements of *V ^{∨}* to one-parameter subgroups of that
torus.

The default form `W:=CoxeterGroup("A",2)`

defines the adjoint algebraic
group (the group with a trivial center). In that case *Φ* is a basis of
*X( T)*, so

`W.simpleRoots`

is the identity matrix and `W.simpleCoroots`

is the Cartan matrix `CartanMat(W)`

of the root system. The form
`CoxeterGroup("A",2,"sc")`

constructs the semisimple simply connected
algebraic group, where `W.simpleRoots`

is the transposed of `CartanMat(W)`

and `W.simpleCoroots`

is the identity matrix.
There is also a function `RootDatum`

which understands some familiar names
for the algebraic groups and gives the results that could be obtained by
giving the appropriate matrices `W.simpleRoots`

and `W.simpleCoroots`

:

gap> RootDatum("sl",3); # same as CoxeterGroup("A",2,"sc") RootDatum("sl",3)

It is also possible to compute with finite order elements of * T*. Over an
algebraically closed field, finite order elements of

`Mod1`

whenever the need arises, where
`Mod1`

is the function which replaces the numerator of a fraction with the
numerator `mod`

the denominator; the fraction `Mod1`

of
rationals and raising to the power
Here is an example of computations with semisimple-elements given as list
in *ℚ/ℤ*.

gap> G:=RootDatum("sl",4); RootDatum("sl",4) gap> L:=ReflectionSubgroup(G,[1,3]); ReflectionSubgroup(RootDatum("sl",4), [ 1, 3 ]) gap> AlgebraicCentre(L); rec( Z0 := SubTorus(ReflectionSubgroup(RootDatum("sl",4), [ 1, 3 ]),[ [ 1, 2, \ 1 ] ]), AZ := Group( <0,0,1/2> ), descAZ := [ [ 1, 2 ] ] ) gap> SemisimpleSubgroup(last.Z0,3); Group( <1/3,2/3,1/3> ) gap> e:=Elements(last); [ <0,0,0>, <1/3,2/3,1/3>, <2/3,1/3,2/3> ]

First, the group * G=SL_{4}* is constructed, then the Levi subgroup

`AlgebraicCentre`

returns a record with : the neutral component `.descAZ`

, see AlgebraicCentre for an explanation.
Finally the semi-simple elements of order 3 in

gap> e[2]^G.2; <1/3,0,1/3> gap> Orbit(G,e[2]); [ <1/3,2/3,1/3>, <1/3,0,1/3>, <2/3,0,1/3>, <1/3,0,2/3>, <2/3,0,2/3>, <2/3,1/3,2/3> ]

Since over an algebraically closed field *K* the points of * T* are in
bijection with

`Rank(T)`

non-zero elements of
`GF(4)`

:

gap> s:=SemisimpleElement(G,List([1,2,1],i->Z(4)^i)); <Z(2^2),Z(2^2)^2,Z(2^2)> gap> s^G.2; <Z(2^2),Z(2)^0,Z(2^2)> gap> Orbit(G,s); [ <Z(2^2),Z(2^2)^2,Z(2^2)>, <Z(2^2),Z(2)^0,Z(2^2)>, <Z(2^2)^2,Z(2)^0,Z(2^2)>, <Z(2^2),Z(2)^0,Z(2^2)^2>, <Z(2^2)^2,Z(2)^0,Z(2^2)^2>, <Z(2^2)^2,Z(2^2),Z(2^2)^2> ]

We can compute the centralizer *C _{G}(s)* of a semisimple element in

gap> G:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> Centralizer(G,s); Extended((A1xA1)<1,3>.(q+1))

The result is an extended reflection group; the reflection group part is
the Weyl group of *C _{G}^{0}(s)* and the extended part are representatives of

- CoxeterGroup (extended form)
- RootDatum
- Torus
- FundamentalGroup for algebraic groups
- IntermediateGroup
- SemisimpleElement
- Operations for Semisimple elements
- Centralizer for semisimple elements
- SubTorus
- Operations for Subtori
- AlgebraicCentre
- SemisimpleSubgroup
- IsIsolated
- IsQuasiIsolated
- QuasiIsolatedRepresentatives
- SemisimpleCentralizerRepresentatives

`CoxeterGroup( `

`simpleRoots`, `simpleCoroots` )

`CoxeterGroup( `

`C`[, "sc"] )

`CoxeterGroup( `

`type1`, `n1`, ... , `typek`, `nk`[, "sc"] )

The above are extended forms of the function `CoxeterGroup`

allowing to
specify more general root data. In the first form a set of roots is given
explicitly as the lines of the matrix `simpleRoots`, representing vectors
in a vector space `V`, as well as a set of coroots as the lines of the
matrix `simpleCoroots` expressed in the dual basis of *V ^{∨}*. The product

`C`=`simpleCoroots`*TransposedMat(`simpleRoots`)

must be a valid Cartan
matrix. The dimension of `Length(``C`)

. The length
of
In the second form `C` is a Cartan matrix, and the call `CoxeterGroup(`

is equivalent to `C`)`CoxeterGroup(IdentityMat(Length(`

. When the
optional `C`)),`C`)`"sc"`

argument is given the situation is reversed: the simple
coroots are given by the identity matrix, and the simple roots by the
transposed of `C` (this corresponds to the embedding of the root system in
the lattice of characters of a maximal torus in a **simply connected**
algebraic group).

The argument `"sc"`

can also be given in the third form with the same
effect.

The following fields in a Coxeter group record complete the description of
the corresponding **root datum**:

`simpleRoots`

:

the matrix of simple roots

`simpleCoroots`

:

the matrix of simple coroots

`matgens`

:

the matrices (in row convention --- that is, the matrices operate**from the right**) of the simple reflections of the Coxeter group.

`RootDatum(`

`type`,`rank`)

This function returns the root datum for the algebraic group described by
`type` and `rank`. The types understood as of now are: `"gl"`

, `"sl"`

,
`"pgl"`

, `"sp"`

, `"so"`

, `"psp"`

, `"pso"`

, `"halfspin"`

,
`"spin"`

, `"F4"`

and `"G2"`

.

gap> RootDatum("spin",8);# same as CoxeterGroup("D",4,"sc") RootDatum("spin",8)

`Torus(`

`rank`)

This function returns the **CHEVIE** object corresponding to the notion of a
torus of dimension `rank`, a Coxeter group of semisimple rank 0 and given
`rank`. This corresponds to a split torus; the extension to Coxeter cosets
is more useful (see Torus for Coxeter cosets).

gap> Torus(3); CoxeterGroup() gap> ReflectionName(last); "(q-1)^3"

`FundamentalGroup(`

`W`)

This function returns the fundamental group of the algebraic group defined
by the Coxeter group record `W`. This group is returned as a group of
diagram automorphisms of the corresponding affine Weyl group, that is as a
group of permutations of the set of simple roots enriched by the lowest
root of each irreducible component. The definition we take of the
fundamental group of a (not necessarily semisimple) reductive group is
*(P∩ Y( T))/Q* where

gap> W:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> FundamentalGroup(W); Group( ( 1, 2, 3,12) ) gap> W:=CoxeterGroup("A",3,"sc"); CoxeterGroup("A",3,"sc") gap> FundamentalGroup(W); Group( () )

`IntermediateGroup(`

`W`, `indices`)

This computes a Weyl group record representing a semisimple algebraic group
intermediate between the adjoint group --- obtained by a call like
`CoxeterGroup("A",3)`

--- and the simply connected semi-simple group ---
obtained by a call like `CoxeterGroup("A",3,"sc")`

. The group is
specified by specifying a subset of the **minuscule weights**, which are
weights whose scalar product with every coroot is in *-1,0,1*. The
non-trivial elements of the (algebraic) center of a semi-simple simply
connected algebraic group are in bijection with the minuscule weights; this
set is also in bijection with *P/Q* where *P* is the coweight lattice and
*Q* is the coroot lattice. The minuscule weights are specified, if `W` is
irreducible, by the list `indices` of their position in the Dynkin diagram
(see PrintDiagram). The constructed group has lattice *Y( T)* generated
by the sum of the coroot lattice and the weights with the given indices. If

gap> W:=CoxeterGroup("A",3);; gap> IntermediateGroup(W,[]); # adjoint CoxeterGroup("A",3) gap> FundamentalGroup(last); Group( ( 1, 2, 3,12) ) gap> IntermediateGroup(W,[2]);# intermediate CoxeterGroup([[2,0,-1],[0,1,0],[0,0,1]],[[1,-1,0],[-1,2,-1],[1,-1,2]]) gap> FundamentalGroup(last); Group( ( 1, 3)( 2,12) )

`SemisimpleElement(`

`W`,`v`[,`additive`])

`W` should be a root datum, given as a Coxeter group record for a Weyl
group, and `v` a list of length `W.rank`

. The result is a semisimple
element record, which has the fields:

`.v`

:- the given list, taken
`Mod1`

if its elements are rationals.

`.group`

:- the parent of the group
`W`.

gap> G:=CoxeterGroup("A",3);; gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0>

If all elements of *v* are rational numbers, they are converted by `Mod1`

to fractions between *0* and *1* representing roots of unity, and these
roots of unity are multiplied by adding `Mod1`

the fractions. In this way
any semisimple element of finite order can be represented.

If the entries are not rational numbers, they are assumed to represent
elements of a field which are multiplied or added normally. To explicitly
control if the entries are to be treated additively or not, a third
argument can be given: if `true`

the entries are treated additively, or
not if `false`

. For entries to be treated additively, they must belong to a
domain for which the method `Mod1`

had been defined.

The arithmetic operations `*`

, `/`

and `^`

work for semisimple elements.
They also have `Print`

and `String`

methods. We first give an element with
elements of *ℚ/ℤ* representing roots of unity.

gap> G:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> t:=SemisimpleElement(G,[1/2,1/3,1/7]); <1/2,1/3,1/7> gap> s*t; <1/2,5/6,1/7> gap> t^3; <1/2,0,3/7> gap> t^-1; <1/2,2/3,6/7> gap> t^0; <0,0,0> gap> String(t); "<1/2,1/3,1/7>"

- then a similar example with elements of
`GF(5)`

:

gap> s:=SemisimpleElement(G,Z(5)*[1,2,1]); <Z(5),Z(5)^2,Z(5)> gap> t:=SemisimpleElement(G,Z(5)*[2,3,4]); <Z(5)^2,Z(5)^0,Z(5)^3> gap> s*t; <Z(5)^3,Z(5)^2,Z(5)^0> gap> t^3; <Z(5)^2,Z(5)^0,Z(5)> gap> t^-1; <Z(5)^2,Z(5)^0,Z(5)> gap> t^0; <Z(5)^0,Z(5)^0,Z(5)^0> gap> String(t); "<Z(5)^2,Z(5)^0,Z(5)^3>"

The operation `^`

also works for applying an element of its defining Weyl
group to a semisimple element, which allows orbit computations:

gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> s^G.2; <1/2,1/2,1/2> gap> Orbit(G,s); [ <0,1/2,0>, <1/2,1/2,1/2>, <1/2,0,1/2> ]

The operation `^`

also works for applying a root to a semisimple element:

gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> s^G.roots[4]; 1/2 gap> s:=SemisimpleElement(G,Z(5)*[1,1,1]); <Z(5),Z(5),Z(5)> gap> s^G.roots[4]; Z(5)^2

`Frobenius(`

:`WF`)

If`WF`

is a Coxeter coset associated to the Coxeter group`W`, the function`Frobenius`

returns the associated automorphism which can be applied to semisimple elements, see Frobenius.

gap> W:=CoxeterGroup("D",4);;WF:=CoxeterCoset(W,(1,2,4));; gap> s:=SemisimpleElement(W,[1/2,0,0,0]); <1/2,0,0,0> gap> F:=Frobenius(WF); function ( arg ) ... end gap> F(s); <0,1/2,0,0> gap> F(s,-1); <0,0,0,1/2>

`Centralizer( `

`W`, `s`)

`W` should be a Weyl group record or and extended reflection group and `s`
a semisimple element for `W`. This function returns the stabilizer of the
semisimple element `s` in `W`, which describes also *C _{G}(s)*, if

gap> G:=CoxeterGroup("A",3); CoxeterGroup("A",3) gap> s:=SemisimpleElement(G,[0,1/2,0]); <0,1/2,0> gap> Centralizer(G,s); Extended((A1xA1)<1,3>.(q+1))

`SubTorus(`

`W`,`Y`)

The function returns the subtorus * S* of the maximal torus

`S.generators`

of the returned subtorus object. Here, adapted means
that there is a set of integral vectors, stored in `S.complement`

, such
that `M:=Concatenation(S.generators,S.complement)`

is a basis of

gap> W:=CoxeterGroup("A",4);; gap> SubTorus(W,[[1,2,3,4],[2,3,4,1],[3,4,1,2]]); Error, not a pure sublattice in SubTorus( W, [ [ 1, 2, 3, 4 ], [ 2, 3, 4, 1 ], [ 3, 4, 1, 2 ] ] ) called from main loop brk> gap> SubTorus(W,[[1,2,3,4],[2,3,4,1],[3,4,1,1]]); SubTorus(CoxeterGroup("A",4),[ [ 1, 0, 3, -13 ], [ 0, 1, 2, 7 ], [ 0, 0, 4, -3 ] ])

The operation `in`

can test if a semisimple elements belongs to a subtorus:

gap> W:=RootDatum("gl",4);; gap> r:=AlgebraicCentre(W); rec( Z0 := SubTorus(RootDatum("gl",4),[ [ 1, 1, 1, 1 ] ]), AZ := Group( <0,0,0,0> ), descAZ := [ [ 1 ] ] ) gap> SemisimpleElement(W,[1/4,1/4,1/4,1/4]) in r.Z0; true

The operation `Rank`

gives the rank of the subtorus:

gap> Rank(r.Z0); 1

`AlgebraicCentre( `

`W` )

`W` should be a Weyl group record, or an extended Weyl group record. This
function returns a description of the centre *Z* of the algebraic group
defined by `W` as a record with the following fields:

`Z0`

:- the neutral component
*Z*of^{0}*Z*as a subtorus of.**T**

`AZ`

:- representatives of
*A(Z):=Z/Z*given as a group of semisimple elements.^{0}

`descAZ`

:- if
`W`is not an extended Weyl group, describes the inclusion of*A(Z)*in the center`pi`

of the corresponding simply connected group. It contains a list elements given as words in the generators of`pi`

which generate*A(Z)*.

gap> G:=CoxeterGroup("A",3,"sc");; gap> L:=ReflectionSubgroup(G,[1,3]); ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]) gap> AlgebraicCentre(L); rec( Z0 := SubTorus(ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]),[ [\ 1, 2, 1 ] ]), AZ := Group( <0,0,1/2> ), descAZ := [ [ 1, 2 ] ] ) gap> G:=CoxeterGroup("A",3);; gap> s:=SemisimpleElement(G,[0,1/2,0]);; gap> Centralizer(G,s); (A1xA1)<1,3>.(q+1) gap> AlgebraicCentre(last); rec( Z0 := SubTorus(ReflectionSubgroup(CoxeterGroup("A",3), [ 1, 3 ]),), AZ := Group( <0,1/2,0> ) )

Note that in versions of **CHEVIE** prior to april 2017, the field `Z0`

was
no a subtorus but what is now `Z0.generators`

, and there was a field
`complement`

which is now `Z0.complement`

.

`SemisimpleSubgroup( `

`S`, `n` )

This function returns the subgroup of semi-simple elements of order
dividing `n` in the subtorus *S*.

gap> G:=CoxeterGroup("A",3,"sc");; gap> L:=ReflectionSubgroup(G,[1,3]);; gap> z:=AlgebraicCentre(L);; gap> z.Z0; SubTorus(ReflectionSubgroup(CoxeterGroup("A",3,"sc"), [ 1, 3 ]),[ [ 1,\ 2, 1 ] ]) gap> SemisimpleSubgroup(z.Z0,3); Group( <1/3,2/3,1/3> ) gap> Elements(last); [ <0,0,0>, <1/3,2/3,1/3>, <2/3,1/3,2/3> ]

`IsIsolated(`

`W`,`s`)

`s` should be a semi-simple element for the algebraic group * G* specified
by the Weyl group record

gap> W:=CoxeterGroup("E",6);; gap> QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/6,1/6,0,1/6,0>, <0,1/2,0,0,0,0>, <1/3,0,0,0,0,1/3> ] gap> Filtered(last,x->IsIsolated(W,x)); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/2,0,0,0,0> ]

`IsQuasiIsolated(`

`W`,`s`)

`s` should be a semi-simple element for the algebraic group * G* specified
by the Weyl group record

gap> W:=CoxeterGroup("E",6);; gap> QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/6,1/6,0,1/6,0>, <0,1/2,0,0,0,0>, <1/3,0,0,0,0,1/3> ] gap> Filtered(last,x->IsQuasiIsolated(ReflectionSubgroup(W,[1,3,5,6]),x)); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/2,0,0,0,0> ]

`QuasiIsolatedRepresentatives(`

`W`[,`p`])

`W` should be a Weyl group record corresponding to an algebraic group
* G*. This function returns a list of semisimple elements for

gap> W:=CoxeterGroup("E",6);;QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <0,0,0,1/3,0,0>, <0,1/6,1/6,0,1/6,0>, <0,1/2,0,0,0,0>, <1/3,0,0,0,0,1/3> ] gap> List(last,x->IsIsolated(W,x)); [ true, true, false, true, false ] gap> W:=CoxeterGroup("E",6,"sc");;QuasiIsolatedRepresentatives(W); [ <0,0,0,0,0,0>, <1/3,0,2/3,0,1/3,2/3>, <1/2,0,0,1/2,0,1/2>, <2/3,0,1/3,0,1/3,2/3>, <2/3,0,1/3,0,2/3,1/3>, <2/3,0,1/3,0,2/3,5/6>, <5/6,0,2/3,0,1/3,2/3> ] gap> List(last,x->IsIsolated(W,x)); [ true, true, true, true, true, true, true ] gap> QuasiIsolatedRepresentatives(W,3); [ <0,0,0,0,0,0>, <1/2,0,0,1/2,0,1/2> ]

`SemisimpleCentralizerRepresentatives(`

`W` [,`p`])

`W` should be a Weyl group record corresponding to an algebraic group
* G*. This function returns a list giving representatives

`ReflectionSubgroup(W,h)`

. If a second argument

gap> W:=CoxeterGroup("G",2); CoxeterGroup("G",2) gap> l:=SemisimpleCentralizerRepresentatives(W); [ [ ], [ 1 ], [ 1, 2 ], [ 1, 5 ], [ 2 ], [ 2, 6 ] ] gap> List(last,h->ReflectionName(ReflectionSubgroup(W,h))); [ "(q-1)^2", "A1.(q-1)", "G2", "A2<1,5>", "~A1<2>.(q-1)", "~A1<2>xA1<6>" ] gap> SemisimpleCentralizerRepresentatives(W,2); [ [ ], [ 1 ], [ 1, 2 ], [ 1, 5 ], [ 2 ] ]

gap3-jm

08 Sep 2017