Hyperdeterminants from the $E_8$ 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 project 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-$p$ reduction 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 $3\times 3\times 3$ and $2\times 2\times 2\times 2$.

23 pages. Replaced the Corollary 2.4 from v.2 with Proposition 2.6 and Corollary 2.7. Added Theorem 3.1. Included example the dual of a nodal curve. Made minor grammatical and structural revisions

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Cited by in corpus (2)

• Restricted Secant Varieties of Grassmannians
• Toward Jordan Decompositions of Tensors