The roots are stored as an indexed set
{@ α1, ..., αN, αN + 1, ..., α2N @},
where α1, ..., αN are the positive roots in an order compatible with height; and αN + 1, ..., α2N are the corresponding negative roots (i.e. αi + N= - αi). The simple roots are α1, ..., αn where n is the rank.
Many of these functions have an optional argument Basis which may take one of the following values
The lattice containing the (co)roots of the group of Lie type G.
The simple (co)roots of the group of Lie type G as the rows of a matrix.
The number of positive roots of the group of Lie type G.
Basis: MonStgElt Default: "Standard"
An indexed set containing the (co)rootsof the group of Lie type G.
Basis: MonStgElt Default: "Standard"
An indexed set containing the positive (co)rootsof the group of Lie type G.
Basis: MonStgElt Default: "Standard"
The rth (co)root of the group of Lie type G.
Basis: MonStgElt Default: "Standard"
If v is a (co)root of the group of Lie type G, this returns its position; otherwise it returns 0.
> G := GroupOfLieType("A3", 25 : Isogeny := 2);
> Roots(G);
{@
(1 0 0),
(0 1 0),
(1 0 2),
(1 1 0),
(1 1 2),
(2 1 2),
(-1 0 0),
(0 -1 0),
(-1 0 -2),
(-1 -1 0),
(-1 -1 -2),
(-2 -1 -2)
@}
> PositiveCoroots(G);
{@
(2 -1 -1),
(-1 2 0),
(0 -1 1),
(1 1 -1),
(-1 1 1),
(1 0 0)
@}
> #Roots(G) eq 2*NumPosRoots(G);
true
> Coroot(G, 4);
(1 1 -1)
> Coroot(G, 4 : Basis := "Root");
(1 1 0)
> CorootPosition(G, [1,1,-1]);
4
> CorootPosition(G, [1,1,0] : Basis := "Root");
4
Basis: MonStgElt Default: "Standard"
The unique (long) root of greatest height in the root datum of the group of Lie type G.
Basis: MonStgElt Default: "Standard"
The unique short root of greatest heightin the root datum of the group of Lie type G.
> G := GroupOfLieType("G2", RealField());
> HighestRoot(G);
(3 2)
> HighestLongRoot(G);
(3 2)
> HighestShortRoot(G);
(2 1)
The reflections in the Weyl group have representatives in the group of Lie type.
The sequence of representatives of reflections in the group of Lie type G.
The representative of the reflections in the rth root in the group of Lie type G.
> G := GroupOfLieType("A2", Rationals());
> Reflections(G);
[ n1 , n2 , n1 n2 n1 ]
The height of the rth (co)root of the group of Lie type G, i.e. the sum of the coefficients of αr (resp. αrstar) with respect to the simple (co)roots.
The sequence of squares of the lengthsof the (co)roots of the group of Lie type G.
The square of the length of the rth (co)root of the group of Lie type G.
Returns true if, and only if, the rth root of the group of Lie type G is long, i.e. the rth coroot is short.
Returns true if, and only if, the rth root of the group of Lie type G is short, i.e. the rth coroot is long.
An additive orderon the positive roots of the group of Lie type G, i.e. a sequence containing the numbers 1, ..., N in some order so that αr + αs=αt implies t is between r and s. This is computed using the techniques of Papi [Pap94].
> G := GroupOfLieType("A5", GF(3));
> a := AdditiveOrder(G);
> Position(a, 2);
6
> Position(a, 3);
10
The (co)weight latticeof the group of Lie type G.
Basis: MonStgElt Default: "Standard"
The fundamental (co)weightsof the group of Lie type G as the rows of a matrix.
Basis: MonStgElt Default: "Standard"
The unique dominant weight in the same W-orbit as v, where W is the Weyl group of G and v is a weight given as a vector or a sequence representing a vector. The second value returned is a Weyl group element taking v to the dominant weight.