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Zbl 0813.16015
Aslaksen, Helmer; Tan, Eng-Chye; Zhu, Chen-bo
Generators and relations of invariants of $2\times 2$ matrices. (English)
[J] Commun. Algebra 22, No.5, 1821-1832 (1994). ISSN 0092-7872; ISSN 1532-4125

Let $G$ be a classical group of $n \times n$ matrices over a field $F$, where $F$ is equal to $\bbfR$ or $\bbfC$ and let $G$ act on $m$-tuples of $n \times n$ matrices by simultaneous conjugation. The main purpose of the paper under review is to find minimal systems of generators of the algebra of polynomial invariants under the action of $G$ in the case of $2\times 2$ matrices. This was done for $\text{GL}(2,F)$, $U(2)$ and $O(2,F)$ by {\it K. S. Sibirskij} [Sib. Mat. Zh. 9, No. 1, 152-164 (1968); translated in Sib. Math. J. 9, 115-124 (1968; Zbl 0273.15024)]; and for $\text{SO}(2,F)$ by {\it H. Aslaksen} [Math. Scand. 65, 59-66 (1989; Zbl 0679.15029)]. Now the authors solve this problem for the remaining classical groups of $2 \times 2$ matrices. They also find algebraically independent sets of invariants of $\text{GL}(2,F)$, $\text{O}(2,F)$ and $\text{SO}(2,F)$ and some syzygies between the invariants of these groups.
[V.Drensky (Sofia)]

MSC 2000:: *16R30 Trace rings and invariant theory (assoc. rings and algebras)
16R50 Other kinds of identities of assoc. rings
15A72 Vector and tensor algebra

Keywords: minimal systems of generators; algebra of polynomial invariants; action; classical groups; syzygies

Citations: Zbl 0693.15018; Zbl 0679.15029; Zbl 0273.15024

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