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The decomposition of polynomial spaces on unions of Grassmannians $\mathcalG_{{k_1},d}\cup\ldots\cup \mathcal G_{{k_r},d}$ into irreducible orthogonallyinvariant subspaces and their reproducing kernels are investigated. We alsogeneralize the concepts of cubature points and $t$-designs from singleGrassmannians to unions. We derive their characterization as minimizers of asuitable energy potential to enable $t$-design constructions by numericaloptimization. We also present new analytic families of $t$-designs for$t=1,2,3$.

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