Reproducing kernels for the irreducible components of polynomial spaces on unions of Grassmannians

Abstract

The decomposition of polynomial spaces on unions of Grassmannians ${\mathcal{G}}_{k_1,d}\cup \dots \cup {\mathcal{G}}_{k_r,d}$ into irreducible orthogonally invariant subspaces and their reproducing kernels are investigated. We also generalize the concepts of cubature points and $t$-designs from single Grassmannians to unions. We derive their characterization as minimizers of a suitable energy potential to enable $t$-design constructions by numerical optimization. We also present new analytic families of $t$-designs for $t=1,2,3$.