# Basic semi-invariants

Let $G$ be a (finite) group and $\chi$ be a linear character corresponding to an irreducible representation. A polynomial $f_{\chi}$ is called semi-invariant (of type $\chi$) if $\sigma\circ f=\chi(\sigma) \cdot f$ for any $\sigma \in G$. The semi-invariants form a finitely generated module over the ring of invariant polynomials. For finite reflection groups the generators of this module (the basic semi-invariants) seem to be factors of the Jacobian of the generators of the invariant ring, i.e. $det(\frac{\partial f_i}{\partial x_j})$, where $C[X]^{G}=C[f_1,\ldots,f_n]$. Is there a similar description in the case when $G$ is a general finite group?

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