Coxeter Groups [HB34]

Finite Coxeter groups are implemented as a subclass of permutation groups so that they inherit all the operations for permutation groups as well as having many specialized functions. This module was implemented by Don Taylor and Scott Murray.

• Cartan matrix corresponding to a given Dynkin diagram
• Construction of a Coxeter group from a root datum or Cartan matrix
• Dynkin diagram of a Cartan matrix or Coxeter group
• Root datum for a Coxeter group
• Element as a reduced word in the standard generators
• Element of maximal length
• Unique long (short) root of greatest height
• Long word
• Short root of maximal height
• Reflections in Coxeter group
• Reflection subgroup
• Reduced representatives for cosets of the reflection subgroup
• Actions on roots and co-roots
• Coxeter group as a real reflection group
• Coxeter and parabolic subgroups; Transversals
• Braid group, pure braid group and Coxeter group presentation