Invariants of 2×2-Matrices over Finite Fields

✍ Scribed by Larry Smith

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 114 KB
• Volume: 8
• Category: Article
• ISSN: 1071-5797
• DOI: 10.1006/ffta.2002.0359

✦ Synopsis

Let F q be the finite field with q elements, q ¼ p n ; p 2 N a prime, and Mat 2:2 ðF q Þ the vector space of 2 Â 2-matrices over F. The group GLð2; FÞ acts on Mat 2;2 ðF q Þ by conjugation. In this note, we determine the invariants of this action. In contrast to the case of an infinite field, where the trace and determinant generate the ring of invariants, several new invariants appear in the case of finite fields. # 2002 Elsevier Science (USA)

This manuscript was motivated by review [12] of paper [1], which piqued my interest once again in what is after all one of the oldest and most basic problems in invariant theory: namely, determine all the polynomials in the entries of a generic n  n-matrix that are constant on similarity classes. In the language of modern invariant theory 1 this may be expressed as follows: let F be a field and Mat n;n ðFÞ the vector space of n  n-matrices over F. The general linear group GLðn; FÞ acts on Mat n;n ðFÞ by conjugation, and hence also on the polynomial algebra F½Mat n;n ðFÞ on the dual vector space Mat n;n ðFÞ * . What is being asked for is a description of the ring of invariants F½Mat n;n ðFÞ GLðn;FÞ .

As is well known, if the ground field is an algebraically closed field, such as the complex numbers, the only such functions are symmetric polynomials of the eigenvalues of the matrix entries (see e.g., [10, Chap. 19] or 1 We refer to [9] for basic material on invariant theory.