Modules of invariants? Let $G \subset SL(V, \mathbb{C})$ be a finite group and $R=(\operatorname{Sym}\[V\])^G$ is the ring of polynomial invariants, $W$ some irreducible complex representation of $G$. I want to know is there any methods (or at least examples) of computing generators and relations of the $R$ module $M=(\operatorname{Sym}\[V\] \otimes_\mathbb{C} W)^G$?
 A: The polynomial ring Sym(V) is naturally graded:  $Sym(V) = \oplus Sym(V)_d$  Suppose you have can compute the isotypic decomposition of these graded components 
Sym(V$)_d$ =
${\oplus_{\chi \in A_d} U_\chi}$
where the $U_\chi$ are irreducible representations of $G$.  Then $M = \oplus_d \oplus_{\chi \in A_d} (U_\chi \otimes W)^G$.  Now $(U_\chi \otimes W)^G = 0$ unless $W \cong U_\chi^*$ in which case
$(U_\chi \otimes W)^G$ is one dimensional.  Thus your problem really amounts to decomposing $Sym(V)$ as a $G$-representation.  So far none of these depends on $G$ being finite (but it should be reductive).
$M$ is called the module of $W$-covariants of $V$.  The Hilbert Series of $M$ may be computed using Molien's Theorem.  A minimal generating set for $M$ contains only elements of degree less than or equal to the order of $G$.   Lots of other things are known but I suggest you read about it.  One good reference for this is {\it Invariants of Finite Groups and Their Applications to Combinatorics} R. Stanley Bull. A.M.S. 1979.
