Designs on the Tautological bundle

TL;DR

This paper develops a generalized design framework for linear operators, constructs such a design on the tautological bundle over the complex projective line, and uses invariant theory and representation analysis to realize specific projections.

Contribution

It introduces a new generalized design concept for linear operators and applies it to the tautological bundle, utilizing invariant theory and representation decomposition.

Findings

Constructed a finite sum design on the tautological bundle

Realized the projection onto the lowest-dimensional summand as a finite sum

Utilized invariant theory for the binary icosahedral group

Abstract

In this paper, we introduce the framework of a generalized design, which represents any linear operator as a finite sum of local linear maps attached to finitely many points, thereby abstracting the core of design theory without employing integration. We then construct such a design on the space of sections of the tautological bundle over the complex projective line. By using the irreducible decomposition of this space as an SU(2)-representation, we show that the projection onto its lowest-dimensional summand can be realized as a finite sum of these local maps. Our construction relies on invariant theory for the binary icosahedral group and an analysis of fixed-point subspaces in symmetric tensor representations.