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 and Coroots of an Isogenous A3 (ex-ac3db4)

Run in calculator (opens in a new tab)

> 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

Run in calculator (opens in a new tab)

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

Run in calculator (opens in a new tab)

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 of a Group of Lie Type (ex-ff84ed)
> G := GroupOfLieType("A2", Rationals());
> Reflections(G);
[ n1 , n2 , n1 n2 n1  ]

Run in calculator (opens in a new tab)

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 [Pap94].

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

Run in calculator (opens in a new tab)

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.

On this page