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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$?

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$M$ is often free. It is locally free, so if $R$ has no nontrivial vector bundles, say if $R$ is a free ring, then it is free. This gets you the answer instantly in a lot of cases. –  Will Sawin Nov 9 '12 at 17:12
@Will $M$ is not locally free even for $\operatorname{dim} V=2$. In this case $M$ is reflexive module over Kleinian singularity. Moreover, projective dimension of such module is infinite in this example. –  Sasha Pavlov Nov 9 '12 at 18:47
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up vote 2 down vote accepted

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.

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