Antipodal Sets and Designs on Unitary Groups

The unitary group U(n) is a symmetric space and has the point-symmetry for every point $$x\in U(n)$$ . A great antipodal set on U(n) is a “good” finite subset of U(n) related to the point-symmetries. On the other hand, a great antipodal set on U(n) is an analogue of a pair of antipodal points on spheres. It is known that a finite subset of a sphere is a tight spherical 1-design if and only if it is a pair of antipodal points. In this paper, we investigate a relation between great antipodal sets on U(n) and design theory on U(n). Moreover, we give a relation between a great antipodal set on U(n) and a Hamming cube graph $${\mathcal {Q}}_n$$ .