page_title: Coxeter notation
section_title: Complex reflections
hierarchical_section_title: Coxeter notation / Complex reflections
caption_attribution_description: English: Rank 2 infinite Shephard subgroup relations
context_page_description: In geometry, Coxeter notation is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, with modifiers to indicate certain subgroups. The notation is named after H. S. M. Coxeter, and has been more comprehensively defined by Norman Johnson.
context_section_description: Coxeter notation has been extended to Complex space, Cⁿ where nodes are unitary reflections of period greater than 2. Nodes are labeled by an index, assumed to be 2 for ordinary real reflection if suppressed. Complex reflection groups are called Shephard groups rather than Coxeter groups, and can be used to construct complex polytopes.
  In , a rank 1 shephard group , order p, is represented as ₚ[], []ₚ or ]ₚ[. It has a single generator, representing a 2π/p radian rotation in the Complex plane: .
  Coxeter writes the rank 2 complex group, ₚ[q]ᵣ represents Coxeter diagram . The p and r should only be suppressed if both are 2, which is the real case [q]. The order of a rank 2 group ₚ[q]ᵣ is .
  The rank 2 solutions that generate complex polygons are: ₚ[4]₂ (p is 2,3,4,...), ₃[3]₃, ₃[6]₂, ₃[4]₃, ₄[3]₄, ₃[8]₂, ₄[6]₂, ₄[4]₃, ₃[5]₃, ₅[3]₅, ₃[10]₂, ₅[6]₂, and ₅[4]₃ with Coxeter diagrams , , , , , , , , , , , , . 
  Infinite groups are ₃[12]₂, ₄[8]₂, ₆[6]₂, ₃[6]₃, ₆[4]₃, ₄[4]₄, and ₆[3]₆ or , , , , , , .
  Index 2 subgroups exists by removing a real reflection: ₚ[2q]₂ → ₚ[q]ₚ. Also index r subgroups exist for 4 branches: ₚ[4]ᵣ → ₚ[r]ₚ.
  For the infinite family ₚ[4]₂, for any p = 2, 3, 4,..., there are two subgroups: ₚ[4]₂ → [p], index p, while and ₚ[4]₂ → ₚ[]×ₚ[], index 2.
caption: Some subgroup relations among infinite Shephard groups
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