Designs on the Tautological bundle
Ikeda Yuya
TL;DR
This work reinterprets design theory in a measure-free, finite-averaging framework called a $\tau$-design and applies it to the tautological bundle $T_{2,1}$ over $\mathbb{C}P^{1}$. It leverages the $\mathrm{SU}(2)$-decomposition of $\Gamma(T_{2,1})$ into irreducibles $\Gamma^{2k}(T_{2,1}) \cong S^{2k-1}(\mathbb{C}^{2})$ and analyzes the invariant theory of the binary icosahedral group $G_{\mathrm{icosa}}$, whose Hilbert series is $P_{2}^{G_{\mathrm{icosa}}}(t)=\frac{1+t^{30}}{(1-t^{12})(1-t^{20})}$. The main result shows that if $X$ and $\lambda$ yield $\operatorname{tr}_{\mathbb{C}}(\Psi_{(X,\lambda)})=2$, then the orbit-averaged pair $Y=\frac{1}{|G_{\mathrm{icosa}}|} G_{\mathrm{icosa}}\cdot(X,\lambda)$ forms a $\tau$-design, with the projection on each irreducible summand behaving as $\Psi_{(Y,\lambda_Y)}|_{\Gamma^{2k}(T_{2,1})}=\mathrm{id}$ for $k=1$ and $0$ for $k=2,3,4,5$. The paper culminates in an explicit 12-point $\tau$-design constructed from the $G_{\mathrm{icosa}}$-orbit, illustrating a concrete finite averaging mechanism in a geometric representation-theoretic setting. This work tightly interweaves design theory, SU(2) representation theory, and invariant theory to produce finite, algebraic designs on complex projective spaces.
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.