Ring of invariants of general linear group over local ring \(\mathbb Z_{p^{m} }\)
Let \(\mathbb{F}_q\) be the finite field with \(q\) elements, and \(G=\mathrm{GL}_n(\mathbb{F}_q)\) be the general linear group. The group \(G\) acts naturally on the polynomial ring \(\mathbb{F}_q[x_1, \dots, x_n]\). The ring of invariants \(\mathbb{F}_q[x_1, \dots, x_n]^G\) is a polynomial ring over \(\mathbb{F}_q\); in fact, it is generated by the Dickson invariants (see [\textit{L. E. Dickson}, American M. S. Trans. 12, 75--98 (1911; JFM 42.0136.01); \textit{C. Wilkerson}, Contemp. Math. 19, 421--434 (1983; Zbl 0525.55013)]. The main result of this paper is to find the ring of invariants for an analogous situation by replacing the finite field \(\mathbb{F}_q\) by the finite ring \(\mathbb{Z}_{p^m}\) of characteristic \(p^m\) where \(p\) is a prime number. Thus the authors determine the ring of invariants of \(\mathbb{Z}_{p^m}[x_1, \ldots, x_n]\) under the action of \(G=\mathrm{GL}_n(\mathbb{Z}_{p^m})\).
scientific article; zbMATH DE number 5968164