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On automorphisms of matrix invariants induced from the trace ring. (English) Zbl 0802.16017
Assume \(M^ m_ n\) is the affine space of all \(m\)-tuples of \(n\) by \(n\) matrices over a fixed algebraically closed field of characteristic zero and let the group \(PGL_ n\) act on this space by (simultaneous) conjugations. Denote the corresponding quotient space by \(Q_{m,n}\), and \(A = \text{Aut}(Q_{m,n})\) be the group of automorphisms of \(Q_{m,n}\) that preserve the representation type of \(Q_{m,n}\). The main results proved in the paper under review are the following. (i) If \(m \geq n + 1\) then for any positive integer \(s\) the group \(A\) acts \(s\)-transitively on \(Q_{m,n}\) (considered as a stratified affine variety). (ii) Let \(T_{m,n}\) be the trace ring of the ring of \(m\) generic \(n\) by \(n\) matrices. Then there exist elements in \(A\) (for \(n \geq 3\) and \(m \geq 2\)) that cannot be induced by any automorphism of \(T_{m,n}\).

MSC:
16R30 Trace rings and invariant theory (associative rings and algebras)
16R10 \(T\)-ideals, identities, varieties of associative rings and algebras
15A72 Vector and tensor algebra, theory of invariants
16W20 Automorphisms and endomorphisms
16S50 Endomorphism rings; matrix rings
16G60 Representation type (finite, tame, wild, etc.) of associative algebras
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