Konrotation

Konrotation is a term used in some niche mathematical discussions to denote a symmetry operation that combines a rotation with a conjugation or reflection. The term does not have a single standard definition; usage varies by author or discipline.

Definition and framing: Broadly, a konrotation on a vector space V is a map K: V → V defined as K = C ∘ R, where R is a rotation operator and C is a conjugation or reflection operator that may be linear or anti-linear depending on the framework. In complex spaces, C may be complex conjugation; in real spaces, C can be a sign flip or a reflection accompanying a rotation. The precise form and domain of K depend on the ambient structure and the chosen operators R and C.

Properties: Depending on the choice of R and C, konrotation can be linear or anti-linear, may be an involution, and can preserve or reverse certain norms or inner products under suitable conditions. When C is an isometric, orientation-changing reflection and R is an orthogonal rotation, K often behaves as a composite symmetry that intertwines rotation and reflection.

Examples: In the complex plane, with R_theta(z) = e^{i theta} z and C as complex conjugation, the konrotation is K_theta(z) = overline{e^{i theta} z} = e^{-i theta} overline{z}. In higher dimensions, K can be defined on R^n by composing a standard orthogonal rotation with a fixed reflection, producing a family of parameterized transformations.

Applications and context: The concept appears in theoretical discussions of generalized symmetry groups and in some quantum-mechanical or computer-graphics contexts as a hypothetical transformation that combines rotation with a conjugation-like operation. Definitions are not standardized, so researchers specify R and C when introducing konrotation.

See also: Rotation, Reflection, Conjugation, Anti-unitary operator.