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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

On automorphisms of matrix invariants


Author: Zinovy Reichstein
Journal: Trans. Amer. Math. Soc. 340 (1993), 353-371
MSC: Primary 16R30
MathSciNet review: 1124173
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Abstract: Let $ {Q_{m,n}}$ be the space of $ m$-tuples of $ n \times n$-matrices modulo the simultaneous conjugation action of $ PG{L_n}$. Let $ {Q_{m,n}}(\tau)$ be the set of points of $ {Q_{m,n}}$ of representation type $ \tau$. We show that for $ m \geq n + 1$ the group $ \operatorname{Aut}({Q_{m,n}})$ of representation type preserving algebraic automorphisms of $ {Q_{m,n}}$ acts transitively on each $ {Q_{m,n}}(\tau)$. Moreover, the action of $ \operatorname{Aut}({Q_{m,n}})$ on the Zariski open subset $ {Q_{m,n}}(1,n)$ of $ {Q_{m,n}}$ is $ s$-transitive for every positive integer $ s$. We also prove slightly weaker analogues of these results for all $ m \geq 3$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1993-1124173-1
PII: S 0002-9947(1993)1124173-1
Keywords: Matrix invariants, representation type, algebraic automorphisms
Article copyright: © Copyright 1993 American Mathematical Society