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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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