Equations for GL invariant families of polynomials

TL;DR

This paper introduces an algorithm that computes equations for GL-invariant families of homogeneous polynomials, aiding in understanding varieties like symmetroids, with a focus on cubic and quartic cases.

Contribution

The paper presents a new algorithm and a supporting database for finding equations of GL-invariant polynomial families, including a Julia implementation for broader applicability.

Findings

Algorithm successfully computes ideals for invariant polynomial families.

Database of Young tableaux and highest weight polynomials supports the algorithm.

Implementation enables future exploration of polynomial varieties.

Abstract

We provide an algorithm that takes as an input a given parametric family of homogeneous polynomials, which is invariant under the action of the general linear group, and an integer $d$. It outputs the ideal of that family intersected with the space of homogeneous polynomials of degree $d$. Our motivation comes from open problems, which ask to find equations for varieties of cubic and quartic symmetroids. The algorithm relies on a database of specific Young tableaux and highest weight polynomials. We provide the database and the implementation of the database construction algorithm. Moreover, we provide a julia implementation to run the algorithm using the database, so that more varieties of homogeneous polynomials can easily be treated in the future.