See Chapter MATRIX GROUPS OVER GENERAL RINGS for general functions for matrix groups.
Strict: BoolElt Default: true
The default action is to return true if every generator of G is a reflection. If Strict is false, the function checks if G can be generated by some of its reflections, not necessarily those returned by Generators(G).
Returns the orders of the reflections, the roots and the coroots of the reflection group G.
Returns true if and only if the matrix group G is a real reflection group. If true, the simple orders, roots, and coroots are also returned.
> W := ComplexReflectionGroup("A", 4);
> IsReflectionGroup(W);
true
> IsRealReflectionGroup(W);
true
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -1 2 -1]
[ 0 0 -1 2]
> W := ComplexReflectionGroup("M", 3);
> IsReflectionGroup(W);
true
> IsRealReflectionGroup(W);
^
Runtime error in 'IsRealReflectionGroup': The group must be defined over the
reals
Returns true if and only if the real reflection group W is crystallographic; i.e., its Cartan matrix has integral entries.
Returns true if and only if the real reflection group W is simply laced; i.e., its Coxeter graph has no labels.
> W := ReflectionGroup("A~2 D4");
> IsFinite(W);
false
> IsCrystallographic(W);
true
> IsSimplyLaced(W);
true
The dual of the reflection group G, ie, the reflection group gotten by swapping roots with coroots.
The overgroup of H, ie. the reflection group whose roots are permuted by the elements of the reflection subgroup H.
The root datum whose roots are permuted by the elements of the reflection subgroup H.Every Coxeter group W has a standard action. For example, the standard action group of a Coxeter group of type An is the symmetric group of degree n + 1 acting on {1, ..., n}.
The standard action of the reflection group W.
The group G of the standard action of the reflection group W, together with an isomorphism W to G.