Properties of Reflection Groups

See Chapter ChapGrpMat for general functions for matrix groups.

IsReflectionGroup(G): GrpMat → BoolElt

Strict : BoolElt : 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).

RootsAndCoroots(G): GrpMat → [RngIntElt], [ModTupRngElt], [ModTupRngElt]

Returns the orders of the reflections, the roots and the coroots of the reflection group \(G\).

IsRealReflectionGroup(G): GrpMat → BoolElt, [], []

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.

Example: Is Reflection Group

> W := ComplexReflectionGroup("A", 4);
> IsReflectionGroup(W);
true
%%a> assert $1;
> IsRealReflectionGroup(W);
true
%%a> assert $1;
\bln
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
\bln
[ 2 -1  0  0]
[-1  2 -1  0]
[ 0 -1  2 -1]
[ 0  0 -1  2]
> W := ComplexReflectionGroup("M", 3);
> IsReflectionGroup(W);
true
%%a> assert $1;
\< IsRealReflectionGroup(W);
                        ^
Runtime error in 'IsRealReflectionGroup': The group must be defined over the
reals

IsCrystallographic(W): GrpMat → BoolElt

Returns true if and only if the real reflection group \(W\) is crystallographic; i.e., its Cartan matrix has integral entries.

IsSimplyLaced(W): GrpMat → BoolElt

Returns true if and only if the real reflection group \(W\) is simply laced; i.e., its Coxeter graph has no labels.

Example: Properties

> W := ReflectionGroup("A~2 D4");
> IsFinite(W);
false
%%a> assert not $1;
> IsCrystallographic(W);
true
%%a> assert $1;
> IsSimplyLaced(W);
true
%%a> assert $1;

Dual(G): GrpMat → BoolElt

The dual of the reflection group \(G\), ie, the reflection group gotten by swapping roots with coroots.

Overgroup(H): GrpMat → GrpMat

The overgroup of \(H\), i.e. the reflection group whose roots are permuted by the elements of the reflection subgroup \(H\).

Overdatum(H): GrpMat → RootDtm

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 \(A_n\) is the symmetric group of degree \(n+1\) acting on \(\{1,\dots,n\}\).

StandardAction(W): GrpMat → Map

The standard action of the reflection group \(W\).

StandardActionGroup(W): GrpMat → GrpPerm, Map

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