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

TL;DR

This paper studies the decomposition of polynomial spaces on unions of Grassmannians into irreducible components, explores their reproducing kernels, and introduces new methods for constructing t-designs through energy minimization.

Contribution

It generalizes the concepts of cubature points and t-designs to unions of Grassmannians and provides new analytic families of t-designs for t=1,2,3.

Findings

Characterization of irreducible components and their kernels.

Development of energy-based optimization for t-designs.

New explicit t-designs for low t values.

Abstract

The decomposition of polynomial spaces on unions of Grassmannians $\mathcal G_{{k_1},d}\cup\ldots\cup \mathcal G_{{k_r},d}$ 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 $t=1,2,3$.