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Grothendieck groups of invariant rings: Linear actions of finite groups. (English) Zbl 0905.19001
Let $$k$$ be a field and let $$G$$ be a finite group whose order is nonzero in $$k$$. Let $$V$$ be a finite-dimensional vector space over $$k$$ with basis $$\{X_1, \dots, X_n\}$$ and suppose that the elements of $$G$$ act as $$k$$-linear automorphisms of $$V$$. Then $$G$$ acts as automorphisms on the symmetric algebra $$S(V)= k[X_1, \dots, X_n]$$ with fixed ring $$R=S (V)^G$$.
The aim of this paper is to give a presentation of the Grothendieck group $$G_0(R)$$ of finitely generated $$R$$-modules, together with some explicit calculations. The presentation has the form $$G_0(R)= G_0(kG)/ {\mathcal U}$$, where $${\mathcal U}$$ is a sum of subgroups that are induced from certain subgroups of the Grothendieck groups of the normalizers of the nontrivial subgroups of $$G$$. The intricate derivation of this result occupies the bulk of the paper.
The presentation leads to a direct sum decomposition $$G_0(R) \cong \mathbb{Z} \oplus G_0 (R)_{tors}$$, in which the torsion subgroup $$G_0 (R)_{tors}$$ is finite of exponent dividing $$| G|^n$$. Some exact computations of the torsion part are given, including the cases where $$G$$ is of prime order $$p$$ or dihedral of order $$2p$$. When $$G$$ has prime order and $$k$$ does not contain a primitive $$p$$-th root of unity, $$G_0 (R)$$ is torsion-free. Thus the condition of the Sheppard-Todd-Chevalley theorem for $$G_0 (R)$$ to be torsion-free, that $$G$$ is generated by pseudo-reflections, is sufficient but not necessary. If $$k$$ does contain a primitive $$p$$-th root of unity, the torsion part is a nonzero sum of cyclic $$p$$-groups.
The authors consider the effect of factoring out the pseudo-reflections. They also remark that their results continue to hold when the symmetric algebra is replaced by the corresponding power series ring.

##### MSC:
 19A31 $$K_0$$ of group rings and orders 16W20 Automorphisms and endomorphisms 16E20 Grothendieck groups, $$K$$-theory, etc.
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