Root Actions

The functions in this section give access to the action on the underlying root system (or datum) of a permutation Coxeter group. These functions do not apply to finitely presented Coxeter groups

In the following functions, the optional parameter Basis determines which basis the roots are given with respect to: "Standard" for the standard basis of the root space; "Root" for the basis of simple (co)roots; "Weight" for the basis of simple (co)weights.

RootGSet(W): GrpPermCox -> GSet

CorootGSet(W): GrpPermCox -> GSet

Basis: MonStgElt                    Default: "Standard"

The \(G\)-set of the Coxeter group \(W\) acting on the (co)roots.

Example: Root G-Sets (ex-90d28a)

> W := CoxeterGroup("B3");
> X := RootGSet(W);
> r := Root(W, 5);
> r;
(0 1 1)
> Image(W.1, X, r);
(1 1 1)

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RootAction(W): GrpPermCox -> Map

CorootAction(W): GrpPermCox -> Map

Basis: MonStgElt                    Default: "Standard"

The map \(X\times W \to X\) giving the action of the Coxeter group \(W\) on the (co)root space \(X\).

Example: Coroot Action (ex-1fa494)

> W := CoxeterGroup("B3");
> act := CorootAction(W);
> act([1,-2,1], W.1);
(-1 -1  1)

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ReflectionGroup(W): GrpPermCox -> GrpMat, Map

CoreflectionGroup(W): GrpPermCox -> GrpMat, Map

Basis: MonStgElt                    Default: "Standard"

The Coxeter group \(W\) as a real reflection group (ie. as a matrix group over some subfield of R) acting on the (co)root space, and the isomorphism from \(W\) to the (co)reflection group.

Example: Reflection Group of a Coxeter Group (ex-2596af)

> W := CoxeterGroup("B3");
> _, h := ReflectionGroup(W);
> W.1*W.3;
(1, 10)(2, 8)(3, 12)(4, 7)(5, 6)(11, 17)(13, 16)(14, 15)
> h(W.1*W.3);
[-1  0  0]
[ 1  1  2]
[ 0  0 -1]

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