A reflection group is a group generated by a finite set of pseudoreflections. The simple roots (resp. simple coroots, simple orders) of a reflection group are the roots (resp. coroots, orders) of its generators. The roots (resp. coroots, orders) of a reflection group are the roots (resp. coroots, orders) of all the reflections contained in the group.
See also Section Constructing Real Reflection Groups on the construction of real reflection groups and Section Constructing Finite Complex Reflection Groups on the construction of finite complex reflection groups.
Returns true if, and only if, G is a reflection group with the given generators. If G is a reflection group, the orders, roots and coroots are also returned.
The reflection group with simple roots given by the rows of A, simple coroots given by the rows of B, and simple orders m=(m_1, ..., m_n).
The reflection group with simple roots given by the rows of A and simple coroots given by the rows of B. The orders are all taken to be 2.
> F<z> := CyclotomicField( 7 ); > M := MatrixAlgebra( F, 2 ); > A := M!1; > B := M![1,1,-1,1]; > G := ReflectionGroup( A, B, [2,7] ); > IsReflectionGroup(G); true [ 2, 7 ] [1 0] [0 1] [ 1 1] [-1 1]