Hyperdeterminants from the $E_8$ Discriminant

Frédéric Holweck; Luke Oeding

doi:10.1016/j.jalgebra.2021.10.017

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arXiv · 1810.05857

Hyperdeterminants from the $E_8$ Discriminant

Frédéric Holweck·Luke Oeding

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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 $2\times 2\times 2\times 2$.

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BibTeX RIS

Frédéric Holweck, Luke Oeding. 2018-10-13. Hyperdeterminants from the $E_8$ Discriminant. https://doi.org/10.1016/j.jalgebra.2021.10.017

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