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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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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