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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 Construction of Real Reflection Groups on the construction of
real reflection groups and Section Construction of Finite Complex Reflection Groups on the
construction of finite complex reflection groups.
Returns true if, and only if, the matrix group 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 the matrix A,
simple coroots given by the rows of the matrix B,
and simple orders given by the sequence m=[m1, ..., mn].
If < A[i], B[i] >ne2, then B[i] is multiplied by the appropriate constant.
The reflection group with
simple roots given by the sequence A,
simple coroots given by the sequence B,
and simple orders given by the sequence m=[m1, ..., mn].
If < A[i], B[i] >ne2, then B[i] is multiplied by the appropriate constant.
The reflection group with
simple roots given by the rows of the matrix A and
simple coroots given by the rows of the matrix B.
The orders are all taken to be 2.
If < A[i], B[i] >ne2, then B[i] is multiplied by the appropriate constant.
The reflection group with
simple roots given by the sequence A and
simple coroots given by the sequence B.
The orders are all taken to be 2.
If < A[i], B[i] >ne2, then B[i] is multiplied by the appropriate constant.
> 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]
[ 2 2]
[-2 2]
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