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$?



Modules of invariants?
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.


