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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)

##### MSC:
 14M20 Rational and unirational varieties 14L24 Geometric invariant theory 14L30 Group actions on varieties or schemes (quotients) 14M17 Homogeneous spaces and generalizations
##### Citations:
Zbl 0419.16010; Zbl 0437.16013
Full Text:
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