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Hyperdeterminants from the $E_8$ Discriminant

Frédéric Holweck, Luke Oeding

TL;DR

This paper develops a geometric framework to relate dual varieties and discriminants via projections, enabling the expression of Gr(3,9) and Gr(4,8) discriminants through restricted E8/E7 discriminants and fundamental invariants. By employing Vinberg's theta representations and a tangency-based restriction theorem, the authors derive divisibility relations such as Δ_Gr(3,9) | Res(Δ_E8, 𝔤_1^*) and Δ_Gr(4,8) | Res(Δ_E7, 𝔤_1^*), and compute explicit semi-simple evaluations. They produce complete expansions of the Gr(3,9) discriminant as a degree-120 polynomial in the fundamental invariants f_{12}, f_{18}, f_{24}, f_{30} and the Gr(4,8) discriminant as a degree-126 polynomial in f_2,f_6,f_8,f_{10},f_{12},f_{18}, with 15,942 monomials; these are obtained via modular interpolation and rational reconstruction, aided by extensive computational scripts. In addition, the work recovers well-known hyperdeterminants for formats 3×3×3 and 2×2×2×2, tying classical invariant theory to modern multilinear algebra through a unifying Proj/Restrict framework. The results provide practical, computable expressions for high-dimensional discriminants and advance the understanding of how E8/E7 discriminants govern Grassmannian duals in Vinberg’s theta-representation setting.

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

Hyperdeterminants from the $E_8$ Discriminant

TL;DR

This paper develops a geometric framework to relate dual varieties and discriminants via projections, enabling the expression of Gr(3,9) and Gr(4,8) discriminants through restricted E8/E7 discriminants and fundamental invariants. By employing Vinberg's theta representations and a tangency-based restriction theorem, the authors derive divisibility relations such as Δ_Gr(3,9) | Res(Δ_E8, 𝔤_1^*) and Δ_Gr(4,8) | Res(Δ_E7, 𝔤_1^*), and compute explicit semi-simple evaluations. They produce complete expansions of the Gr(3,9) discriminant as a degree-120 polynomial in the fundamental invariants f_{12}, f_{18}, f_{24}, f_{30} and the Gr(4,8) discriminant as a degree-126 polynomial in f_2,f_6,f_8,f_{10},f_{12},f_{18}, with 15,942 monomials; these are obtained via modular interpolation and rational reconstruction, aided by extensive computational scripts. In addition, the work recovers well-known hyperdeterminants for formats 3×3×3 and 2×2×2×2, tying classical invariant theory to modern multilinear algebra through a unifying Proj/Restrict framework. The results provide practical, computable expressions for high-dimensional discriminants and advance the understanding of how E8/E7 discriminants govern Grassmannian duals in Vinberg’s theta-representation setting.

Abstract

We find expressions of the polynomials defining the dual varieties of Grassmannians and 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 , and obtain the polynomials of interest as factors. To find an expression of the discriminant in terms of fundamental invariants, which has terms, we perform interpolation with mod- reduction and rational reconstruction. From these expressions for the discriminants of and we also obtain expressions for well-known hyperdeterminants of formats and .

Paper Structure

This paper contains 17 sections, 9 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Decomposing restrictions of dual varieties
  3. Examples
  4. Restrictions of discriminants of Lie algebras
  5. Expressions of the restriction of $\text{E}_n$-discriminants to semi-simple elements
  6. Restricting $\Delta_{\text{E}_{8}}$ to semi-simple elements of $\bigwedge^3 \mathbb{C}^9$
  7. Restricting $\Delta_{\text{E}_8}$ and $\Delta_{\text{E}_6}$ to semi-simple elements of $\mathbb{C}^3\otimes\mathbb{C}^3\otimes \mathbb{C}^3$
  8. Restricting $\Delta_{\text{E}_7}$ to semi-simple elements of $\bigwedge^4 \mathbb{C}^8$
  9. Restricting $\Delta_{\text{E}_7}$ to semi-simple elements of $\mathbb{C}^2\otimes \mathbb{C}^2\otimes \mathbb{C}^2\otimes \mathbb{C}^2$
  10. The $\operatorname{Gr}(3,9)$ and $\operatorname{Gr}(4,8)$ discriminants in fundamental invariants
  11. $\operatorname{Gr}(3,9)$
  12. $\operatorname{Gr}(4,8)$
  13. Evaluation of the $\operatorname{Gr}(3,9)$ and $\operatorname{Gr}(4,8)$ discriminants
  14. Warm-up with $\operatorname{Gr}(3,6)$
  15. Evaluating the $\operatorname{Gr}(3,9)$-discriminant
  16. ...and 2 more sections

Key Result

Theorem 2.2

Consider a non-trivial vector space splitting $V=A\oplus B$. Let $X\subset \mathbb{P} V$ and $Y\subset \mathbb{P} A$ be projective varieties. Let $\pi_B$ denote rational map $\mathbb{P} V \dashrightarrow \mathbb{P} A$ induced from the projection $V \to A$. If for each smooth point $[y] \in Y$ there Moreover if $X^{\vee}$ and $Y^{\vee}$ are hypersurfaces defined respectively by polynomials $\Delta

Figures (3)

  • Figure 1: Division relations for a sequence of discriminants starting from the $\text{E}_8$-discriminant. The first row comes from the inclusion $\mathfrak{so}_8\subset \mathfrak{e}_6\subset \mathfrak{e}_7\subset \mathfrak{e}_8$, and an application of TevelevJMS. The second row comes from Examples \ref{['ex:e8']}, \ref{['ex:e7']}, \ref{['ex:e6']}, and \ref{['ex:so8']}. The last two rows will be explicitly given in Section \ref{['sec:gr39gr48']}.
  • Figure 2: Terms up to symmetry by $\langle\alpha, \beta \rangle$ occurring in expressions of fundamental invariants for $\mathbb{C}[{\bigwedge^{\space3}}\mathbb{C}^{9}]^{\operatorname{SL}(9)}$ restricted to semi-simple parts.
  • Figure 3: Terms up to symmetry occurring in expressions of fundamental invariants for $\mathbb{C}[{\bigwedge^{\space4}}\mathbb{C}^{8}]^{\operatorname{SL}(8)}$ restricted to semi-simple parts.

Theorems & Definitions (23)

  • Remark 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Proposition 2.6: GKZ
  • Corollary 2.7
  • proof
  • Remark 2.8
  • ...and 13 more