Roots, Coroots and Weights#

The roots are stored as an indexed set

\[\{@\; \alpha_1,\dots,\alpha_N,\alpha_{N+1},\dots,\alpha_{2N} \; @\},\]

where \(\alpha_1,\dots,\alpha_N\) are the positive roots in an order compatible with height; and \(\alpha_{N+1},\dots,\alpha_{2N}\) are the corresponding negative roots (i.e. \(\alpha_{i+N}=-\alpha_i\)). The simple roots are \(\alpha_1,\dots,\alpha_n\) where \(n\) is the rank.

Many of these functions have an optional argument Basis which may take one of the following values

  1. "Standard": the standard basis for the (co)root space. This is the default.

  2. "Root": the basis of simple (co)roots.

  3. "Weight": the basis of fundamental (co)weights (see Subsection Weights below).

Accessing Roots and Coroots#

RootSpace(G): GrpLie -> Lat#
CorootSpace(G): GrpLie -> Lat#

The lattice containing the (co)roots of the group of Lie type \(G\).

SimpleRoots(G): GrpLie -> Mtrx#
SimpleCoroots(G): GrpLie -> Mtrx#

The simple (co)roots of the group of Lie type \(G\) as the rows of a matrix.

NumberOfPositiveRoots(G): GrpLie -> RngIntElt#
NumPosRoots(G): GrpLie -> RngIntElt#

The number of positive roots of the group of Lie type \(G\).

Roots(G): GrpLie -> SetIndx#
Coroots(G): GrpLie -> SetIndx#
Basis: MonStgElt                    Default: "Standard"

An indexed set containing the (co)roots of the group of Lie type \(G\).

PositiveRoots(G): GrpLie -> SetIndx#
PositiveCoroots(G): GrpLie -> SetIndx#
Basis: MonStgElt                    Default: "Standard"

An indexed set containing the positive (co)roots of the group of Lie type \(G\).

Root(G, r): GrpLie, RngIntElt -> SetIndx#
Coroot(G, r): GrpLie, RngIntElt -> SetIndx#
Basis: MonStgElt                    Default: "Standard"

The \(r\)th (co)root of the group of Lie type \(G\).

RootPosition(G, v): GrpLie, . -> SetIndx#
CorootPosition(G, v): GrpLie, . -> SetIndx#
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.

Example: Roots Coroots (ex-ac3db4)#
> 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

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HighestRoot(G): GrpLie -> LatElt#
HighestLongRoot(G): GrpLie -> LatElt#
Basis: MonStgElt                    Default: "Standard"

The unique (long) root of greatest height in the root datum of the group of Lie type \(G\).

HighestShortRoot(G): GrpLie -> LatElt#
Basis: MonStgElt                    Default: "Standard"

The unique short root of greatest height in the root datum of the group of Lie type \(G\).

Example: Heighest Roots (ex-69e0c7)#
> G := GroupOfLieType("G2", RealField());
> HighestRoot(G);
(3 2)
> HighestLongRoot(G);
(3 2)
> HighestShortRoot(G);
(2 1)

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Reflections#

The reflections in the Weyl group have representatives in the group of Lie type.

Reflections(G): GrpLie -> GrpLieElt#

The sequence of representatives of reflections in the group of Lie type \(G\).

Reflection(G, r): GrpLie, RngIntElt -> GrpLieElt#

The representative of the reflections in the \(r\)th root in the group of Lie type \(G\).

Example: Reflections (ex-ff84ed)#
> G := GroupOfLieType("A2", Rationals());
> Reflections(G);
[ n1 , n2 , n1 n2 n1  ]

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Operations and Properties for Root and Coroot Indices#

RootHeight(G, r): GrpLie, RngIntElt -> RngIntElt#
CorootHeight(G, r): GrpLie, RngIntElt -> RngIntElt#

The height of the \(r\)th (co)root of the group of Lie type \(G\), i.e. the sum of the coefficients of \(\alpha_r\) (resp. \(\alpha_r^\star\)) with respect to the simple (co)roots.

RootNorms(G): GrpLie -> [RngIntElt]#
CorootNorms(G): GrpLie -> [RngIntElt]#

The sequence of squares of the lengths of the (co)roots of the group of Lie type \(G\).

RootNorm(G, r): GrpLie, RngIntElt -> RngIntElt#
CorootNorm(G, r): GrpLie, RngIntElt -> RngIntElt#

The square of the length of the \(r\)th (co)root of the group of Lie type \(G\).

IsLongRoot(G, r): GrpLie, RngIntElt -> BoolElt#

Returns true if, and only if, the \(r\)th root of the group of Lie type \(G\) is long, i.e. the \(r\)th coroot is short.

IsShortRoot(G, r): GrpLie, RngIntElt -> BoolElt#

Returns true if, and only if, the \(r\)th root of the group of Lie type \(G\) is short, i.e. the \(r\)th coroot is long.

AdditiveOrder(G): GrpLie -> SeqEnum#

The additive order on the positive roots of the group of Lie type \(G\) equal to the Papi order of the longest word \(w_0\) of the Weyl group of \(G\); it corresponds to the order of roots in a reduced expression for \(w_0\). If \(\alpha_r\), \(\alpha_s\) and \(\alpha_t\) are positive roots and \(\alpha_r+\alpha_s=\alpha_t\), then \(t\) lies between \(r\) and \(s\). It is computed using the techniques of [Papi, 1994].

Example: Additive Order (ex-dabc7d)#
> G := GroupOfLieType("A5", GF(3));
> a := AdditiveOrder(G);
> Position(a, 2);
6
> Position(a, 3);
10

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Weights#

WeightLattice(G): GrpLie -> Lat#
CoweightLattice(G): GrpLie -> Lat#

The (co)weight lattice of the group of Lie type \(G\).

FundamentalWeights(G): GrpLie -> Mtrx#
FundamentalCoweights(G): GrpLie -> Mtrx#
Basis: MonStgElt                    Default: "Standard"

The fundamental (co)weights of the group of Lie type \(G\) as the rows of a matrix.

DominantWeight(G, v): GrpLie, . -> ModTupFldElt, GrpFPCoxElt#
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.

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