What is a concrete example of a perpetuant (in classical invariant theory)?

Question

I am trying to determine whether an object of my recent research is actually a "perpetuant" in the sense of Sylvester and classical invariant theory. There are a few papers on the topic, including Kraft and Procesi's very recent one, but the language and notation is far removed from what I am used to.

Primarily I just need to know the "natural habitat" of perpetuants. Let ${\rm SL}(2,\mathbb{C})$ act naturally on the space $\mathbb{C}^2$, and consider the entire symmetric algebra $S(\mathbb{C^2})$, rather than the usual space $S^m(\mathbb{C}^2)$ of binary $m$-forms. Then are perpetuants certain ${\rm SL}(2,\mathbb{C})$-invariant elements of $S(S(\mathbb{C^2}))$? Or do they live in a different space altogether?

Answer 1

In a simplified form, omitting some details, the situation is as follows. Consider the derivative $D_n$: $$ D_n = x_0 \frac{\partial}{\partial x_1} + x_1 \frac{\partial}{\partial x_2} + \cdots + x_{n-1} \frac{\partial}{\partial x_n} $$ of the ring of polynomials on $n+1$ variables $x_0, x_1, \ldots, x_n$.

Suppose element $a$ belongs to the kernel $\ker D_n$ (which is a finitely generated algebra) and, furthermore, let it belong to the minimal generating system of this kernel.

It's easy to see that $a$ also belongs to the kernel $\ker D_m$ for all $m > n$. The question now arises - does $a$ now belong to the minimal generating system of $\ker D_m$ for $m > n$? There are such elements, for example, $x_0$ or $x_1^2 - 2x_0x_2$, but in general, this is not the case.

Suppose there exists an element $a \in \ker D_n$ which belongs to the minimal generating systems of $\ker D_m$ for all $m > n$.

There is a classical isomorphism, due to Roberts, between $\ker D_n$ and the algebra of covariants of a binary form of order $n$.

The image of element $a$ under this isomorphism is called a perpetuant. For example, the binary form itself is a perpetuant.

So, they live in the same space.