Great antipodal sets on complex Grassmannian manifolds as designs with the smallest cardinalities

Antipodal sets on compact symmetric spaces are defined in terms of point symmetries. Chen–Nagano (1988) [10] introduced an invariant “2-number” on compact symmetric spaces as the largest cardinalities of antipodal sets, and an antipodal set with the largest cardinality is said to be great. Sánchez (1997) [34] and Tanaka–Tasaki (2013) [40] proved that any two great antipodal sets on a symmetric R-space are congruent. In particular, great antipodal sets on a complex Grassmannian manifold are unique up to the natural action of the unitary group. The aim of this paper is to give a characterization of great antipodal sets on complex Grassmannian manifolds as certain designs with the smallest cardinalities. To this, we extend the definition of designs on complex Grassmannian manifolds introduced by Roy (2010) [31].