## Asymptotics of dimensions of invariants for finite groups.(English)Zbl 0693.20005

Let G be a finite group, $$A=\sum^{\infty}_{m=0}A_ n$$ be a finitely generated graded algebra over a base field k of characteristic zero. The group G acts on A by graded k-algebra automorphisms. Then $$A^ G=\sum^{\infty}_{m=0}A^ G_ m$$ is the subalgebra of G-invariants. Let PK$$\subseteq G$$ be the subgroup of G which acts on each $$A_ m$$ by sclars. Suppose that A is an integral domain and g.c.d.$$\{$$ m: $$A_ m\neq 0\}=1$$. It is shown that (a) $$A^ G_ m=0$$ if m is not divisible by #(PK); (b) if #(PK)$$=c$$, then $$\lim_{\ell \to \infty}(\dim A^ G_{\ell c}/\dim A_{\ell c})=1/\#(G/PK)$$.
Reviewer: Chen Zhijie

### MSC:

 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 16W50 Graded rings and modules (associative rings and algebras) 15A72 Vector and tensor algebra, theory of invariants
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### References:

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