Properties of Reflection Groups

See Chapter MATRIX GROUPS for general functions for matrix groups.

IsReflectionGroup( G ) : GrpMat -> BoolElt, [RngIntElt], [ModTupRngElt], [ModTupRngElt]

Returns true if, and only if, the matrix group G is a reflection group. If true, the simple orders, roots, and coroots are also returned.

IsRealReflectionGroup( G ) : GrpMat -> BoolElt, [], []

Returns true if, and only if, the matrix group G is a real reflection group.

Example GrpRfl_IsReflectionGroup (H85E16)

> W := ComplexReflectionGroup( "A", 4 );
> IsReflectionGroup( W );
true
> IsRealReflectionGroup( W );
true
> W := ComplexReflectionGroup( "M", 4 );
> IsReflectionGroup( W );
true

> IsRealReflectionGroup( W );

>> IsRealReflectionGroup( W );
                        ^
Runtime error in 'IsRealReflectionGroup': The group must be defined over the
reals

IsCrystallographic( W ) : GrpPermCox -> BoolElt

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

IsSimplyLaced( W ) : GrpFPCox -> BoolElt

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

Example GrpRfl_Properties (H85E17)

> W := ReflectionGroup( "A~2 D4" );
> IsFinite( W );
false
> IsCrystallographic( W );
true
> IsSimplyLaced( W );
true