Platonov, V. P.; Benyash-Krivets, V. V. Character rings of \(n\)-dimensional representations of finitely generated groups. (English. Russian original) Zbl 0669.20004 Sov. Math., Dokl. 34, 83-87 (1987); translation from Dokl. Akad. Nauk SSSR 289, 293-297 (1986). Let \(\Gamma\) be a group on \(m\) generators. For any field \(K\) and any linear algebraic \(K\)-group \(G\), the set \(\operatorname{Hom}(\Gamma,G(K))\) of all \(n\)-dimensional representations can be identified with the \(K\)-points of some algebraic variety. Let \(g\in\Gamma\). Define a function \(\tau_ g\) on \(\operatorname{Hom}(\Gamma,G(K))\) with values in \(K\): \(\tau_ g(\rho)=tr(\rho (g))\), \(\rho\in\operatorname{Hom}(\Gamma,G(K))\). Let \(T(\Gamma,G(K))\) be the ring generated by the functions \(\tau_ g\), \(g\in\Gamma\).The main results of the paper under review are the following theorems: Theorem 1. Suppose that the group \(\Gamma\) has an infinite cyclic factor- group. The following assertions hold for a field \(K\) of characteristic zero: 1) for every \(n\geq 2\), \(T(\Gamma,GL_ n(K))\) is not finitely generated; 2) for every \(n\geq 4\), \(T(\Gamma,SL_ n(K))\) is not finitely generated; 3) for \(n=3\), \(T(\Gamma,SL_ 3(K))\) is finitely generated for every \(\Gamma\). Theorem 2. Let \(K\) be an infinite field of characteristic \(p>0\). The following assertions hold: 1) for an arbitrary group \(\Gamma\), the ring \(T(\Gamma,GL_ n(K))\) is finitely generated when \(n<p\); and the ring \(T(\Gamma,SL_ n(K))\) is finitely generated when \(n<2p\); 2) if \(\Gamma\) has an infinite cyclic factor-group, then \(T(\Gamma,GL_ n(K))\) and \(T(\Gamma,SL_ n(K))\) are not finitely generated when \(n\geq p\) and \(n\geq 2p\), respectively. Reviewer: V.L.Popov Cited in 1 Document MSC: 20C15 Ordinary representations and characters 20G05 Representation theory for linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields Keywords:finitely generated rings; trace; linear algebraic \(K\)-group; \(n\)-dimensional representations; \(K\)-points; algebraic variety × Cite Format Result Cite Review PDF