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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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[1] Atiyah, M; Macdonald, I, Introduction to commutative algebra, (1969), Addison-Wesley Reading, MA · Zbl 0175.03601
[2] Burrow, M, Representation theory of finite groups, (1965), Academic Press New York/London · Zbl 0192.12303
[3] Howe, R, “the classical groups” and invariants of binary forms, (), 133-166
[4] Howe, R, (GLn, glm) duality and symmetric plethysm, (), 85-109
[5] Stanley, R, Invariants of finite groups and their applications to combinatorics, Bull. amer. math. soc. (N.S.), 1, 475-511, (1979) · Zbl 0497.20002
[6] Weil, A, Basic number theory, () · Zbl 0823.11001
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