Before I explain myself better, I would like readers to keep in mind I am a physicist and not a mathematician.
Let $G$ be an infinite reductive group such as $\mathrm{SL}(n,\mathbb{C})$ and a k-dimensional representation of that group (might be a direct sum of irreducible representations). Then we can find its ungraded Hilbert series $H(t)$ by using the Molien-Weyl formula.
Now assume that I already have the closed form of $H(t)$. It should be given by,
$H(t)=\frac{P(t)}{Q(t)}=\frac{1+\cdots+t^r}{(1-t^{d_1})^{a_1} \cdots (1-t^{d_n})^{a_n}}$,
where the Krull dimension is given by $\sum_i a_i$, the ring is Gorenstein and thus $P(t)$ is palindromic, and that $\mathrm{deg}(Q) - \mathrm{deg}(P) = k$.
Now comes the question. I know that $H(t)$ is not unique and that I can find another $\tilde{H}(t)$ such that the properties above stated are kept. But how do I find the minimal form of the Hilbert series such that all properties remain the same? Is there an algorithm? In "Computational Invariant Theory" by Derksen and Kemper the authors mention that in finite groups they search form this minimal form with $d_1 \cdots d_n$ and $d_1 + \cdots + d_n$.
I ask this question because I have an enormous $H(t)$ and I have been doing trial and error which doesn't seem to work out very well.