|
In Magma a pseudo-reflection group is a group generated by a
finite set of invertible pseudo-reflections. A convenient way to provide
the generators for a pseudo-reflection group W is via a finite collection
of roots and coroots. In this context the roots and coroots of
the generators are called the basic roots and basic coroots
of W.
In the most general case, even when the pseudo-reflection group W is
generated by reflections, there are no known distinguished generating
reflections whose roots have properties analogous to simple roots in
Weyl groups or Coxeter groups. Therefore, one should be careful to
distinguish between the basic roots as defined here and the simple (or
fundamental) roots of real reflection groups
See Section Construction of Real Reflection Groups for the construction of
real reflection groups and Section Construction of Finite Complex Reflection Groups for the
construction of finite complex reflection groups.
The pseudo-reflection group with the basic roots and corresponding coroots
given by the rows of the matrices A and B.
A direct construction of the Shephard and Todd group G(14, 1, 2) with user supplied
roots and coroots.
> F<z> := CyclotomicField(7);
> A := Matrix(F,2,3,[[z,0,1],[0,1,0]]);
> B := Matrix(F,2,3,[[1,1,1],[1,2,1]]);
> G<x,y> := PseudoReflectionGroup(A,B);
> IsReflectionGroup(G);
true
> Order(x),Order(y),Order(x*y);
14 2 28
> #G;
392
[Next][Prev] [Right] [Left] [Up] [Index] [Root]
|