RoTNUN

RoTNUN is a fictional mathematical construct introduced to illustrate interactions between continuous rotational symmetries and discrete permutation symmetries. In this article, rotnun is treated as a hypothetical object that combines a spatial rotation with a permutation of labeled components. It is not a standard term in mathematics, but serves as an example for teaching group actions and semidirect products.

Definition: A rotnun is an ordered pair (R, P) where R ∈ SO(3) is a three-dimensional rotation and P ∈ Sym(n) is a permutation of n labeled slots. The operation of a rotnun on a labeled configuration (v; i) is defined by (R, P)·(v; i) = (R v; P(i)). The composition of two rotnuns is (R1, P1) ∘ (R2, P2) = (R1R2, P1P2).

Group structure: The set of rotnuns on fixed n with the operation ∘ forms a semidirect product SO(3) ⋉ Sym(n) under the natural action of Sym(n) by permuting coordinates. The identity rotnun is (I, e) with I the identity rotation and e the identity permutation; inverses exist, given by (R, P)^{-1} = (R^{-1}, P^{-1}).

Examples and interpretation: If P is the identity permutation, a rotnun reduces to a pure rotation. If R is the identity, it reduces to a permutation of components. Nontrivial rotnuns encode how a rotation can induce a rearrangement of labeled components and, conversely, how a permutation can affect the result of applying a rotation to a labeled structure.

Applications and limitations: In teaching contexts, rotnun is used to illustrate how continuous and discrete symmetries interact in a single framework. It is not standard notation, and there is no established literature outside this illustrative context.

See also: Rotation group, Symmetric group, Semidirect product, Group action, Permutation representation.