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On automorphisms of matrix invariants. (English) Zbl 0820.16021
Let \(M_ n(k)^ m\) be the set of \(m\)-tuples of \(n \times n\) matrices over an algebraically closed field \(k\) of characteristic 0 and let \(Q_{m,n}\) be the algebraic quotient for the action by (simultaneous) conjugation of \(\text{PSL}_ n(k)\) on \(M_ n(k)^ m\). Every point \(x \in Q_{m,n}\) defines a semisimple representation of the free algebra \(k \langle u_ 1, \ldots, u_ m \rangle\) in \(M_ n(k)\). Its representation type \(\tau\) gives the multiplicities and the degrees of the irreducible components.
The purpose of this interesting paper is to show that \(Q_{m,n}\) has a very large group \(\text{Aut}(Q_{m,n})\) of algebraic automorphisms preserving the representation type. Among the main results are the following. If \(m > n\) then \(\text{Aut}(Q_{m,n})\) acts transitively on each set \(Q_{m,n}(\tau)\) of the points of fixed representation type \(\tau\). When \(\tau\) corresponds to an irreducible representation of degree \(n\) and \(m > n\) then \(\text{Aut}(Q_{m,n})\) acts \(s\)-transitively on \(Q_{m,n}(\tau)\) for every integer \(s \geq 1\). As a consequence for \(n = 1\) the author obtains that the group of algebraic automorphisms of \(k^ m\), \(m > 1\), acts \(s\)-transitively on \(k^ m\) for every \(s \geq 1\).
Reviewer: V.Drensky (Sofia)

MSC:
16R30 Trace rings and invariant theory (associative rings and algebras)
14L30 Group actions on varieties or schemes (quotients)
16W20 Automorphisms and endomorphisms
14H37 Automorphisms of curves
16S10 Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
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