Free resolutions of orbit closures for the representations associated to gradings on Lie algebras of type $E_6$, $F_4$ and $G_2$

TL;DR

This paper develops an interactive Macaulay2-based method to explicitly construct free resolutions of orbit closures in representations associated with gradings on exceptional Lie algebras, confirming their expected geometric and algebraic properties.

Contribution

It introduces a new computational approach for explicitly constructing free resolutions of orbit closures in exceptional Lie algebra representations.

Findings

Confirmed the shape of expected resolutions for $E_6$, $F_4$, and $G_2$.

Validated geometric properties of the orbit closures.

Provided explicit resolutions using Macaulay2.

Abstract

The irreducible representations of complex semisimple algebraic groups with finitely many orbits are parametrized by graded simple Lie algebras. For the exceptional Lie algebras, Kraskiewicz and Weyman exhibit the Hilbert polynomials and the expected minimal free resolutions of the normalization of the orbit closures. We present an interactive method to construct explicitly these and related resolutions in Macaulay2. The method is then used in the cases of the Lie algebras of type $E_6$, $F_4$ and $G_2$, to confirm the shape of the expected resolutions as well as some geometric properties of the orbit closures.