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In Magma a group of Lie type over a field is defined by generators which satisfy
Steinberg relations (see section The Steinberg Presentation). In particular the
unipotent elements xα(a) are parametrised by the field and the torus
elements are parametrised by the non-zero elements of the field. However, the
number of generators and relations can be reduced considerably using a form of
the Curtis--Steinberg--Tits (CST) presentation [BGK+97].
This section describes basic functions to compute with highest weight
representations of finite groups of Lie type defined by CST presentations:
see the previous section and Chapter REPRESENTATIONS OF LIE GROUPS AND ALGEBRAS for more functions for
highest weight representations.
Currently this functionality is available only for algebraically simple
finite reductive groups with a simply connected root datum: all untwisted
types and the twisted groups of types ()2An (n odd),
()2Dn, ()3D4 and ()2E6.
Let G(q) be a simply connected group defined over the field Fq, let
Δ = {α1, ..., αd} be a base of simple roots and let B
be a basis for Fq regarded as a vector space over its prime field.
The CST presentation can be described as follows.
For i < j, let Φij be the subsystem spanned by
αi and αj, put Ψ = bigcupi, jΦij and Υ
= bigcupi, j{ (α, β) ∈Φij x Φij | α≠∓ β }. Then G(q) has a presentation with generators
xα(a) for α ∈Ψ and a∈B. It is enough to require
relations xα(a)xα(b) = xα(a + b) for α∈Ψ and
relations
[xα(a), xβ(b)] =
∏i, j > 0 xiα + jβ(Cijαβaibj)
for (α, β)∈Υ and a, b∈B. (If G(q) were not simply connected we
would need additional generators for the torus.)
In Magma, the CST generators are represented by a pair of sequences X, Y
eqalignno(
X &= [[xα(a) | a ∈B] | α ∈Ψ ]quadand
Y &= [[x - α(a) | a ∈B] | α ∈Ψ],
)
where X (resp. Y) may be regarded as a matrix whose rows are indexed by
positive (resp. negative) roots and whose columns are indexed by basis elements
of Fq.
For many functions there are optional parameters OnlySimple and GS.
If OnlySimple is true, only the CST generators xα(a) where
∓α is a simple root are used. (The function ExtendGeneratorList
can be used to extend the simple generators to the full collection of CST
generators.)
The functions generally use the default signs for the extraspecial pairs
(Section Constructing Root Data). However, if GS is true,
the root order and signs used by Gilkey and Seitz [GS88]
are used; this only applies to groups of types F4 and G2.
GS: BoolElt Default: false
OnlySimple: BoolElt Default: false
Weyl: BoolElt Default: false
Signs: Any Default: 1
The Curtis--Steinberg--Tits generators for the
group of Lie type t and rank r over the field Fq in the
irreducible representation with highest weight w. If OnlySimple is
true, only generators for the simple roots are returned. If
Weyl is true, the Weyl representation of highest weight w
is returned. If w is the empty sequence, the standard representation
is used.
The parameter Signs can take the values described in
Section Constructing Root Data.
GS: BoolElt Default: false
The Curtis--Steinberg--Tits relations for the simply connected
group of Lie type t and rank r over the field Fq. The function
returns an SLP-group G and a sequence containing the relations
as straight-line programs in G.
If GS is true, the Gilkey--Seitz structure constants are used
and the simple roots for groups of type G2 are swapped.
GS: BoolElt Default: false
Given Curtis--Steinberg--Tits generators X, Y for a simply connected
group of Lie type t and rank r over the field Fq, verify that
the generators satisfy the relations. Set GS to true if the
Gilkey--Seitz conventions hold for X and Y.
Verify that the CST generators for the 273-dimensional representation
of (F) 4(5) satisfy the CST relations but not the relations for
(F) 4(5). When the relations are not satisfied, the index of the
first relation which fails is returned.
> X,Y := CST_Generators("F",4,5,[0,0,1,0]);
> CST_VerifyPresentation("F",4,5,X,Y);
true
> CST_VerifyPresentation("F",4,3,X,Y);
false 1
> G, rels := CST_Presentation("F",4,3);
> rels[1];
function(G)
w1 := G.1^3; return w1;
end function
GS: BoolElt Default: false
UseMap: BoolElt Default: false
Given Curtis--Steinberg--Tits generators X, Y which satisfy the
presentation for a group of Lie type t and rank r over the field of
q elements, return a function f from the group they generate to the
standard Magma copy obtained from ChevalleyGroup(t,r,q).
The function f will be a homomorphism up to a scalar multiple.
Set GS to true if the Gilkey--Seitz conventions hold for X and
Y. If UseMap is true, the function f is returned as a Magma
Map, otherwise the type is UserProgram.
Construct a map from the 28-dimensional representation of the simply
connected version of the twisted group () 3(D) 4(3) to its standard
8-dimensional representation.
> X,Y := CST_Generators("3D",4,3,[0,1,0,0]);
> f := CSTtoChev("3D",4,3,X,Y : UseMap);
> G := Domain(f); G:Minimal;
MatrixGroup(28, GF(5^3))
> L := Codomain(f); L;
GL(8, GF(5, 3))
> C := ChevalleyGroup("3D",4,3);
> Order(C);
20560831566912
> forall{ x : x in Generators(G) | f(x) in C };
true
GS: BoolElt Default: false
Given matrix generators (for the simple roots and their negatives)
for a simply connected group of Lie type t and rank k over the
field of q elements, return the Curtis--Steinberg--Tits generators.
The function CST_Generators returns the Curti--Steinberg--Tits
generators either for a Weyl module of weight w or its irreducible
quotient. This function returns the corresponding irreducible
representation as a Map.
OnlySimple: BoolElt Default: false
For a simply connected finite group G over the field of q elements
and a q-restricted weight w return Curtis--Steinberg--Tits generators
X, Y for the irreducible G-module of weight w for the group G.
If OnlySimple is true, return generators for just the
simple roots.
If varpi1, varpi2, ...,varpik are the fundamental weights,
then w = a1varpin + a2varpi2 + ... + ak varpik is
q-restricted if 0≤ai < q for 1 ≤i≤k.
This is a version of IrreducibleHighestWeightGenerators which returns
the homomorphism from G to the matrix representation of weight w.
For a finite group G of Lie type and a weight w this function
returns a module M of highest weight w such that every highest
weight module of weight w is a quotient of M.
This function returns a homomorphism from G into GL(M) and
Curtis-Steinberg--Tits generators for the image, where M is the
module returned by the previous function.
Let L be a complex semisimple Lie algebra with root system
Φ, simple roots Δ, Cartan subalgebra H, one-dimensional
root spaces (L)α and Cartan decomposition (L) =
H direct-sum bigoplusα ∈Φ(L)α.
We choose basis vectors eα∈(L)α such that
[eα, eβ] = cα, β, and the structure constants
cα, β are integers ∓ (r + 1), where r is the greatest
integer such that β - rα is a root.
For all α∈Δ, we have hα = [e - α, eα]∈H
and the Chevalley basis of L is the set
{eα}α∈Φ∪{hα}α∈Δ. The
Z-span of the Chevalley basis is the Lie algebra (L)Z.
Given a field F, define (L)F = (L)Z tensor F. For
all roots α there is a homomorphism xα from the additive
group of F to GL((L)F) given by
xα(ξ) = 1 + ξ ad eα +
(ξ2/2!)(ad eα)2 + ... .
Then
G_(egtrm ad)(F) = < xα(ξ) | ξ∈F,
α∈Φ >.
is the adjoint Chevalley group. Other than a few exceptions of rank 1
or 2 over fields of at most 3 elements these groups are simple. However,
G_(egtrm ad)(F) is generally not the adjoint group of Lie
type in the sense of linear algebraic groups (see [Car93, p. 39]).
More generally, given a representation varphi : (L)Z to GL(M),
where M is a Z-module, we may define root elements
xα(ξ) = 1 + ξvarphi(eα) +
(ξ2/2!)varphi(eα)2 + ... .
and set
Gvarphi(F) = < xα(ξ) | ξ∈F, α∈Φ >.
This is also called a Chevalley group.
Suppose that Δ = {α1, ..., αn} and that Φ' is a
root subsystem of Φ with simple roots Δ'⊂Δ such that
Δ - Δ' = {αi} for some i. The restriction of the
adjoint action of (L)Z to the Lie subalgebra (L)Z'
corresponding to Φ' preserves the Z-submodule V of (L)Z
whose basis is the set X of elements eβ such that the coefficient of
αi is 1 when β is expressed as a sum of simple roots.
For root data of types A, B, C, D and rank n and types (E)6
and (E)7, the embedding in the root datum of rank n + 1 adds an extra
node to the Dynkin diagram and the construction of the previous paragraph
produces the "standard module" for the corresponding Chevalley group.
For groups of type (E)8 the "standard module" is the adjoint
representation.
The construction of the "standard modules" for groups of types (F)4
and (G)2 is more complicated. In order to defined them we identify
the Lie algebra of type (F)4 with the algebra of fixed points of the
graph automorphism order 2 of (E)6 and identify the Lie algebra of
type (G)2 with the fixed points of a graph automorphism of order 3
of (D)4.
This function returns two sequences of lower triangular integer matrices
defining the action of the Z-form (L)Z of the simple Lie algebra
of type t and rank r on its "standard module". The first sequence
represents the simple roots and the second sequence represents the negatives
of the simple roots.
This function returns the adjoint Chevalley group of type t and rank r
over the field of q elements as a matrix group. The generators are
Curtis--Steinberg--Tits generators.
The adjoint Chevalley group of type (B) n(q) is isomorphic
to the permutation group (P)Ω(2n + 1, q).
> n := 2;
> q := 5;
> G := AdjointChevalleyGroup("B",n,q);
> Type(G),Dimension(G);
GrpMat 10
> H := POmega(2*n+1,q);
> Type(H), Degree(H);
GrpPerm 156
> flag, _ := IsIsomorphic(G,H);
> flag;
true
The matrix of ad (eα) acting (on the right) on the module
with basis B, where B must be a subset of the positive or the
negative roots of the root datum R, as outlined in the construction above.
The argument α is a vector representing a root in the root basis.
Negative: BoolElt Default: false
The matrix of ad (eα) where α is the r-th root acting
(on the right) on the subspace of the Lie algebra of R spanned by the
roots indexed by X. The elements of X are indices of positive roots
unless Negative is true, in which case they they are indices of
negative roots.
LieTypeGenerators(t,k,K) : MonStgElt, RngIntElt, FldFin -> SeqEnum,SeqEnum
LieTypeGenerators(G) : GrpLie -> SeqEnum,SeqEnum
GS: BoolElt Default: false
The Curtis--Steinberg--Tits generators of a simply connected group G of
Lie type or the simply connected group of Lie type t and rank r over
the finite field K or Fq. This function is available for both twisted
and untwisted groups. If GS is true, the Gilkey--Seitz structure
constants and root order are used.
GS: BoolElt Default: false
The Curtis--Steinberg--Tits generators of the simply connected group of
Lie type t and rank r over the field of q elements, returned as
straight-line programs. If GS is true, the Gilkey--Seitz structure
constants and root order are used.
From an irreducible quasisimple matrix group H of known Lie
type t and rank r over the field of q elements, the work of
[LO16] produces Curtis--Steinberg--Tits generators
X, Y as part of the constructive recognition algorithm. This
section describes some functions to construct homomorphisms
ρ : G to H and their inverses from such generators, where
G is the simply connected group of Lie type t, rank r over Fq.
From the homomorphism ρ : G to H and a matrix A∈H, an
element g∈G such that ρ(g) = A can be constructed using
the Chevalley normal form of the Bruhat decomposition of A.
That is, we write g = uh/dot wu', where u, h, /dot w
and u' have the properties described in Subsection Twisted Groups of Lie type.
This uses the "row reduction" algorithms for twisted [CT] and
untwisted [CMT04] groups (a generalisation of Gaussian
row reduction of matrices to groups of Lie type).
OnlySimple: BoolElt Default: false
GS: BoolElt Default: false
Verify: BoolElt Default: false
Given an algebraically simple, simply connected group G of Lie
type and Curtis--Steinberg--Tits generators for a representation,
return the homomorphism from G to the group generated by X and Y.
If generators X and Y are available only for the simple roots
and their negatives, set OnlySimple to true. If the
generators follow the Gilkey--Seitz conventions, set GS to true.
If Verify is true, the function first checks that X and Y
satisfy the appropriate CST presentation. This function applies to
both twisted and untwisted groups.
Given a homomorphism ρ : G to H from a simply connected group G of
Lie type to a matrix group H and a matrix A, this function returns a
sequence s and a field element z. If A is not in the image of ρ
(modulo scalars) then s = [ ], otherwise the elements of
s = [u, h, /dot w, u'] are the components of the Chevalley normal
form of an element g = uh/dot wu' such that A = zρ(g).
Choose a random element in a twisted group of Lie type, get a scalar
multiple of its image in an irreducible highest weight representation
and then check the Chevalley normal form.
> G := TwistedGroupOfLieType("2E",6,3);
> RootDatum(G);
Twisted simply connected root datum of dimension 6 of type 2E6,4
> Dimension(G);
78
> X,Y := CST_Generators("2E",6,3,[0,1,0,0,0,0]);
> rho := Morphism(G,X,Y);
> L := Codomain(rho);
> Dimension(L);
77
> F<t> := BaseRing(L);
> I := sub<L | &cat X, &cat Y>;
> g := Random(G);
> A := L!ScalarMatrix(77,t)*rho(g);
> s,z := ChevalleyForm(rho,A);
> z;
t
> &* s eq g;
true
TwistedPrepareRewrite(t,r,q,X,Y) : MonStgElt,RngIntElt,RngIntElt, SeqEnum,SeqEnum -> UserProgram, Map
OnlySimple: BoolElt Default: false
GS: BoolElt Default: false
This function constructs the group G of Lie type t and rank r over the
field of q elements and the homomorphism f : G to H, where H is the
matrix group generated by the CST generators X and Y. In addition to f
this function returns a map varphi : H to G such that
f varphi = idH.
If generators X and Y are available only for the simple roots
and their negatives, set OnlySimple to true. If the
generators follow the Gilkey--Seitz conventions, set GS to true.
TwistedLieTypeRewrite(t,r,q,X,Y,g) : MonStgElt,RngIntElt,RngIntElt,SeqEnum,SeqEnum,GrpMatElt -> BoolElt, GrpSLPElt
OnlySimple: BoolElt Default: false
GS: BoolElt Default: false
Given a finite (untwisted or twisted) matrix group H with generators X,
Y in CST format and an element g∈H, return a boolean flag b and, if b
is true, an SLP π that expresses g as a word in the given generators.
If the parameter OnlySimple is true, the return value π is an SLP in
the generators corresponding to the simple roots and their negatives. Set
GS to true if the generators follow the Gilkey--Seitz conventions.
Check that the SLP returned by TwistedLieTypeRewrite
evaluates to the correct matrix when evaluated on the CST generators.
> X,Y := CST_Generators("3D",4,5,[]);
> H := sub< Parent(X[1,1]) | &cat X, &cat Y>;
> g := Random(H);
> flag, s := TwistedLieTypeRewrite("3D",4,5,X,Y,g);
> flag;
true
> gens := &cat X cat &cat Y;
> g eq Evaluate(s,gens);
true
TwistedRowReductionMap(ρ) : Map[GrpLie,GrpMat] -> UserProgram
Given an irreducible representation ρ : G(q)to GL(M) of an
untwisted (respectively twisted) finite group G of Lie type, this function
returns a function f such that ρ(f(A)) = A for all A in the
image of A.
More precisely, given A in the codomain of ρ, the application of f to
A returns two values: a sequence w of length 0 or 1, and an element z.
If A is a scalar multiple of an element of the image of ρ, then
w[1] is a Steinberg word in the domain of ρ and z is a field element
such that zρ(w[1]) = A; otherwise w is empty and z is a message
indicating the reason for failure. In particular, if A is in the image of
ρ, then z is 1.
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