A Clebsch-Gordan formula for SL_3 and applications to rationality

TL;DR

This paper provides an explicit basis for equivariant maps between SL_3 representations and applies it to prove the rationality of certain moduli spaces of plane curves, advancing understanding in invariant theory.

Contribution

It introduces a new explicit basis for equivariant maps in SL_3 representations and applies it to solve the rationality problem for specific moduli spaces.

Findings

Explicit basis for equivariant maps from R⊗S to T in SL_3 representations

Proved the rationality of the moduli space of plane curves of degree 34

Established a criterion for stable rationality of quotients of Grassmannians by SL-actions

Abstract

If R, S, T are irreducible SL_3-representations, we give an easy and explicit description of a basis of the space of equivariant maps from R tensor S to T. We apply this method to the rationality problem for invariant function fields. In particular, we prove the rationality of the moduli space of plane curves of degree 34. This uses a criterion which ensures the stable rationality of some quotients of Grassmannians by an SL-action.