Complex reflection group

In mathematics, a complex reflection group is a finite group acting on a finite-dimensional complex vector space that is generated by complex reflections: non-trivial elements that fix a complex hyperplane pointwise.

Complex reflection groups arise in the study of the invariant theory of polynomial rings. In the mid-20th century, they were completely classified in work of Shephard and Todd. Special cases include the symmetric group of permutations, the dihedral groups, and more generally all finite real reflection groups (the Coxeter groups or Weyl groups, including the symmetry groups of regular polyhedra).

A (complex) reflection r (sometimes also called pseudo reflection or unitary reflection) of a finite-dimensional complex vector space V is an element ${\displaystyle r\in GL(V)}$ of finite order that fixes a complex hyperplane pointwise. I.e., the fixed-space ${\displaystyle \operatorname {Fix} (r):=\operatorname {ker} (r-\operatorname {Id} _{V})}$ has codimension 1.

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