Defining relations for the algebra of invariants of \(2\times 2\) matrices
Let \(G\) be a subgroup of the general linear group \(\text{GL}_2\) over the complex numbers, and assume that \(G\) acts by conjugation on the \(m\)-tuples of \(2\times 2\) matrices. The invariants of such actions are of significant importance in the study of the polynomial identities satisfied by algebras related to \(M_2\). The paper under review describes the defining relations of the algebras of invariants of this action when \(G\) is either \(\text{GL}_2\), or \(\text{O}_2\), the orthogonal group, or \(\text{SO}_2\), the special orthogonal group. The main techniques used in the proofs of these results are derived from the classical invariant theory as well as from the representation theory of the general linear group. The results of the paper will be of interest for the specialists in invariant theory, and especially for those involved in PI theory.
scientific article; zbMATH DE number 1925057