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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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