The permutation $pi$ of $1,2,ldots,n$ that satisfies> $$0 < { pi(1) alpha } < { pi(2) alpha } < cdots < { pi(n)alpha} < 1$$ ($alpha$ is any irrational) has been studied from a combinatorialviewpoint (S—s, Boyd) and from an analytic viewpoint (Schoi§engeier). I will present some results on algebraic properties of this permutation, the most significant being a mysterious appearance in the study of nearly periodic binary words (a.k.a., Sturmian words) with the representation theory of symmetric groups.