Self–self-dual spaces of polynomials

Abstract

A space of polynomials V of dimension 7 is called self-dual if the divided Wronskian of any 6-subspace is in V. A self-dual space V has a natural inner product. The divided Wronskian of any isotropic 3-subspace of V is a square of a polynomial. We call V self–self-dual if the square root of the divided Wronskian of any isotropic 3-subspace is again in V. We show that the self–self-dual spaces have a natural non-degenerate skew-symmetric 3-form defined in terms of Wronskians. We show that the self–self-dual spaces correspond to G 2-populations related to the Bethe Ansatz of the Gaudin model of type G 2 and prove that a G 2-population is isomorphic to the G 2 flag variety.