Finite Complex Reflection Groups

sage.categories.finite_complex_reflection_groups.FiniteComplexReflectionGroups

The category of finite complex reflection groups.

See ComplexReflectionGroups for the definition of complex reflection group. In the finite case, most of the information about the group can be recovered from its degrees and codegrees, and to a lesser extent to the explicit realization as subgroup of \(GL(V)\). Hence the most important optional methods to implement are:

  • ComplexReflectionGroups.Finite.ParentMethods.degrees(),
  • ComplexReflectionGroups.Finite.ParentMethods.codegrees(),
  • ComplexReflectionGroups.Finite.ElementMethods.to_matrix().

Finite complex reflection groups are completely classified. In particular, if the group is irreducible, then it’s uniquely determined by its degrees and codegrees and whether it’s reflection representation is primitive or not (see [LT2009] Chapter 2.1 for the definition of primitive).

EXAMPLES:

sage: from sage.categories.complex_reflection_groups import ComplexReflectionGroups
sage: ComplexReflectionGroups().Finite()
Category of finite complex reflection groups
sage: ComplexReflectionGroups().Finite().super_categories()
[Category of complex reflection groups,
 Category of finite groups,
 Category of finite finitely generated semigroups]

An example of a finite reflection group:

sage: W = ComplexReflectionGroups().Finite().example(); W       # optional - gap3
Reducible real reflection group of rank 4 and type A2 x B2

sage: W.reflections()                              # optional - gap3
Finite family {1: (1,8)(2,5)(9,12), 2: (1,5)(2,9)(8,12),
               3: (3,10)(4,7)(11,14), 4: (3,6)(4,11)(10,13),
               5: (1,9)(2,8)(5,12), 6: (4,14)(6,13)(7,11),
               7: (3,13)(6,10)(7,14)}

W is in the category of complex reflection groups:

sage: W in ComplexReflectionGroups().Finite()      # optional - gap3
True