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.
> 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