Modular Invariants of Finite Affine Linear Groups

Article

Abstract

We study modular invariants of finite affine linear groups over a finite field \(\mathbb {F}_{q}\) under affine actions and linear actions. We generalize a result of Chuai (J Algebra 318:710–722, 2007, Theorem 4.2) to any m-folds affine actions. Suppose \(G\leqslant \mathrm{GL}(n,\mathbb {F}_{q})\) is a subgroup and W denotes the canonical module of \(\mathrm{GL}(n,\mathbb {F}_{q})\). We denote by \(\mathbb {F}_{q}[W]^{G}\) the invariant ring of G acting linearly on W and denote by \(\mathbb {F}_{q}[W_{n+1}]^{AG(W^{*})}\) the invariant ring of the affine group \(AG(W^{*})\) of G acting canonically on \(W_{n+1}:=W\oplus \mathbb {F}_{q}\). We show that if \(\mathbb {F}_{q}[W]^{G}=\mathbb {F}_{q}[f_{1},f_{2},\ldots ,f_{s}]\), then \(\mathbb {F}_{q}[W_{n+1}]^{AG(W^{*})}=\mathbb {F}_{q}[f_{1},f_{2},\ldots ,f_{s},h_{n+1}]\), where \(h_{n+1}\) denotes the \((n+1)\)-th Mui’s invariant of degree \(q^{n}\). Let \(\mathrm{AGL}_{1}(\mathbb {F}_{p})\) be the 1-dimensional affine general linear groups over the prime field \(\mathbb {F}_{p}\). We find a generating set for the ring of vector invariants \(\mathbb {F}_{p}[mW_{2}]^{\mathrm{AGL}_{1}(\mathbb {F}_{p})}\) and determine the Noether’s number \(\upbeta _{mW_{2}}(\mathrm{AGL}_{1}(\mathbb {F}_{p}))\) for any \(m\in \mathbb {N}^{+}\).

Keywords

Modular invariants Finite affine group Noether’s number 

Mathematics Subject Classification

13A50 

Notes

Acknowledgements

This research was supported by NSF of China (No. 11401087) and the Fundamental Research Funds for the Central Universities (2412017FZ001) at NENU. The author would like to thank the referee for careful reading and helpful comments, especially generalizing the results in Sects. 2 and 3 from the prime field \(\mathbb {F}_{p}\) to any finite field \(\mathbb {F}_{q}\). The author also thanks David L. Wehlau for his patience, help and encouragement.

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Copyright information

© Sociedade Brasileira de Matemática 2017

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsNortheast Normal UniversityChangchunPeople’s Republic of China

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