ExteriorExtensions: A package for Macaulay2
Luke Oeding
TL;DR
The paper presents a Macaulay2 package, ExteriorExtensions, that constructs a graded Lie-extension $\mathfrak{a} = \mathfrak{sl}_n(\mathbb{F}) \oplus M$ where $M$ is a sum of exterior powers, under the Jordan decomposition consistent with the $G$-action (GJD). It develops an explicit equivariant bracket via a structure tensor $B$ with components constrained by $k \equiv i+j \pmod n$, employing exterior products, the Hodge star, contractions, and projection to $\mathfrak{sl}_n$ to realize the full bracket. The toolset includes adjoint representations, the Killing form, and block-structured invariants (blockRanks) to study invariants of $\operatorname{Ad}_x$, enabling investigations in tensor invariant theory, quantum information, and algebraic geometry. Concrete examples demonstrate identifications with simple Lie algebras (e.g., $\mathfrak{sl}_{4} \oplus \bigwedge^{2}\mathbb{C}^{4} \cong \mathfrak{sp}_6$ and $\mathfrak{sl}_{8} \oplus \bigwedge^{4}\mathbb{C}^{8} \cong \mathfrak{e}_7$), showcasing Lie-algebra tests, Killing form evaluations, Cartan-subalgebra constructions, and invariant computations, while highlighting field-definition issues and future extensions to larger tensor formats. This work provides a computational pathway to analyze tensor orbits and graded-Extensor structures with practical implications for invariant theory and geometry.
Abstract
We explain a Macaulay2 implementation of a construction, which appeared in [Holweck-Oeding arXiv:2206.13662], of a graded algebra structure on the direct sum of a Lie algebra $\mathfrak{g}$ (typically $\mathfrak{sl}_n$) and a $\mathfrak{g}$-module (typically a subspace of an exterior algebra $\bigwedge^{\bullet}\mathbb{C}^n$). We implement brackets, a Killing form, matrix representations of adjoint operators and ranks of blocks of their powers.