Standard Action

Every finite Coxeter group \(W\) has a standard action. For example, the standard action group of a Coxeter group of type \(A_n\) is the symmetric group of degree \(n+1\) acting on \(\{1,\dots,n\}\).

StandardAction(W): GrpFPCox → Map

StandardAction(W): GrpPermCox → Map

The standard action of the finite Coxeter group \(W\).

StandardActionGroup(W): GrpFPCox → GrpPerm, Map

\name{IntrStandardActionGroup}{StandardActionGroup}(W)(): GrpPermCox → GrpPerm, Map

The group \(G\) of the standard action of the finite Coxeter group \(W\), together with an isomorphism \(W\to G\).

Example: Standard Action

> W := CoxeterGroup("A3");
> G, h := StandardActionGroup(W);
> IsSymmetric(G);
true
%%a> assert $1;
> h(W.1); h(W.2); h(W.3);
(1, 2)
(2, 3)
(3, 4)