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Let $n$ be a positive integer, and let $G$ denote a given fixed subgroup of the full permutation group $\Sigma_n$. Consider $n$ variables $z_1,\cdots,z_n$, then $\Sigma_n$ acts naturally on these $n$ variables, and so, by restriction, so does $G$. Assume also given a group homomorphism $\rho: G \to U(1)$. Let $V$ denote the space of all complex $n$-ary forms of degree $d$, where $d$ is some given positive integer. Then $G$ acts naturally on $V$. We define an element $f \in V$ to be a $\rho$-invariant if the following holds:

$g.f = \rho(g).f$ for all $g \in G$.

Given the above, is there some formula that gives the dimension of the subspace of $V$ consisting of all $\rho$-invariants?

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    for a start, are you familiar with the Molien formula ? – 2017-02-26
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    @mercio "merci beaucoup!" Thank you so much! I did not know this formula. I guess one can probably extend $G$ so that the relative invariants of $G$ are the absolute invariants of the extended $G$ perhaps. Thank you! Please post your comment as an answer, and I will accept it. It is a very nice formula by the way. – 2017-02-26

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As mercio pointed it out, at least for the case of absolute invariants, this is precisely what the Molien formula does. It gives a formula for the generating function for the $m_d$, where $m_d$ is the dimension of the space of invariant $n$-ary forms of degree $d$.

There is also a variant of the Molien formula for relative/weighted invariants, as well for covariants.