  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.
