Ring of invariants of general linear group over local ring $\mathbb{Z}_{p^m } $

Abstract

Let $\mathbb{Z}_{p^m } $ be the ring of integers modulo p ^m, where p is a prime and m ⩾ 1. The general linear group GL _n($\mathbb{Z}_{p^m } $) acts naturally on the polynomial algebra A _n:= $\mathbb{Z}_{p^m } $[x ₁, …, x _n]. Denote by $A_n^{GL_2 (\mathbb{Z}_{p^m } )} $ the corresponding ring of invariants. The purpose of the present paper is to calculate this invariant ring. Our results also generalize the classical Dickson’s theorem.

Keywords

Dickson’s theorem / invariant / finite local ring

References

[1] Benson D. Polynomial Invariants of Finite Groups, 1993, Cambridge: Cambridge University Press

[2] Chen Y. Modular Invariants of Finite Classical Groups, 2009, Dalian: Dalian University of Technology.

[3] Derksen H., Kemper G. Computatinal Invariant Theory, 2002, Berlin: Springer-Verlag.

[4] Dickson L. E. A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans Am Math Soc, 1911, 12, 75-98

[5] Hu S. J., Kang M. C. Efficient generation of the ring of invariants. J Algebra, 1996, 180, 341-363

[6] Mead D. G. Generators for the algebra of symmetric polynomials. Am Math Monthly, 1993, 100, 391-397

[7] Nan J., Chen Y. Rational invariants of certain classical similitude groups over finite fields. Indiana Univ Math J, 2008, 57 4 1947-1957

[8] Neusel M. D. Invariant Theory, 2006, Providence: Am Math Soc.

[9] Richman D. Explicit generators of the invariants of finite groups. Adv Math, 1996, 124, 49-76

[10] Smith L. Polynomial Invariants of Finite Groups, 1995, Natick: A K Peters, Ltd.

[11] Smith L. Polynomial invariants of finite groups: a survey of recent developments. Bull of Am Math Soc, 1997, 34 3 211-250

[12] Wilkerson C. A primer on the Dickson invariants. Contemp Math, 1983, 19, 421-434