Rationality of certain field of invariant functions. (English) Zbl 0712.15034
Let \(M_ n\) be the space of \(n\times n\)-matrices over an algebraically closed field k, the group \(GL_ n\) of invertible matrices acts on \((M_ n)^ r\) diagonally via conjugation. Only in a few cases is it known that the corresponding field of invariant rational functions is rational [cf. E. Formanek, J. Algebra 62, 304-319 (1980; Zbl 0437.16013)]. The author gives a short elegant proof of the fact that, if one replaces a copy of \(M_ n\) by its subvariety of singular matrices, the field of invariants is rational.
Reviewer: I.V.Dolgachev
MSC:
15A72 | Vector and tensor algebra, theory of invariants |
16R30 | Trace rings and invariant theory (associative rings and algebras) |
14M20 | Rational and unirational varieties |
14L24 | Geometric invariant theory |
Keywords:
invariants; conjugation; field of invariant rational functions; subvariety of singular matricesCitations:
Zbl 0437.16013References:
[1] | DOI: 10.1080/03081087908817278 · Zbl 0419.16010 · doi:10.1080/03081087908817278 |
[2] | DOI: 10.1016/0021-8693(80)90184-2 · Zbl 0437.16013 · doi:10.1016/0021-8693(80)90184-2 |
[3] | Procesi C., Atti Accad. Naz. Ser. 6 pp 239– (1967) |
[4] | Shafarevich I. R., Basic Algebraic Geometry (1977) · Zbl 0362.14001 |
[5] | Zariski O., Commutative Algebra 1 (1979) |
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