Hyperdeterminants from the E8 discriminant

Abstract We find expressions of the polynomials defining the dual varieties of Grassmannians Gr ( 3 , 9 ) and Gr ( 4 , 8 ) both in terms of the fundamental invariants and in terms of a generic semi-simple element. We restrict the polynomial defining the dual of the adjoint orbit of E 8 and obtain the polynomials of interest as factors. To find an expression of the Gr ( 4 , 8 ) discriminant in terms of fundamental invariants, which has 15 , 942 terms, we perform interpolation with mod-p reductions and rational reconstruction. From these expressions for the discriminants of Gr ( 3 , 9 ) and Gr ( 4 , 8 ) we also obtain expressions for well-known hyperdeterminants of formats 3 × 3 × 3 and 2 × 2 × 2 × 2 .