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Stable rationality of certain \(PGL_ n\)-quotients. (English) Zbl 0741.14032
Consider the action of \(PGL_ n\) on the space \(M_ n\times M_ n\) of pairs of complex \(n\times n\) matrices by simultaneous conjugation, and let \(\mathbb{C}(M_ n\times M_ n)^{PGL_ n}\) be the field of \(PGL_ n\)- invariant complex rational functions on this space. It is known that this field is rational over \(\mathbb{C}\) if \(n\leq 4\) [case \(n=2\) dates back to the last century, cases \(n=3,4\) were considered by E. Formanek, Linear Multilinear Algebra 7, 203-212 (1979; Zbl 0419.16010) and J. Algebra 62, 304-319 (1980; Zbl 0437.16013), respectively]. For \(n\geq 5\) the question whether this field is (stably) rational is open.
The main result of the paper under review is the affirmative answer to this question for \(n=5\) and \(n=7\) (the stable rationality is proved in these cases). — In combination with some known results this shows that \(\mathbb{C}(V)^{PGL_ n}\) is stably rational whenever \(V\) is an almost free representation of \(PGL_ n\) and \(n\) divides \(420=2^ 2\cdot 3\cdot 5\cdot 7\).
Reviewer: V.L.Popov (Moskva)

14M20 Rational and unirational varieties
14L24 Geometric invariant theory
14L30 Group actions on varieties or schemes (quotients)
14M17 Homogeneous spaces and generalizations
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