Matrix invariants of composite size (Q1188143)

Let \(k\) be any field and \(n\geq 1\). Then \(\text{PGL}_ n(k)\) has a natural embedding in \(GL_{n^ 2} (k)\) as the group of automorphisms of \(M_ n(k)\), and one can consider the homogeneous space \(H(n)=GL_{n^ 2}(k)/ \text{PGL}_ n(k)\). This space \(H(n)\) has been proved to be rational for \(n=2,3\) and 4 by \textit{E. Formanek} [J. Algebra 62, 304-319 (1980; Zbl 0437.16013); cf. also \textit{C. Procesi}, ``Rings with polynomial identities'' (New York 1973; Zbl 0262.16018)] and stably rational for \(n=5\) and 7 by \textit{C. Bessenrodt} and \textit{L. Le Bruyn} [Invent. Math. 104, No. 1, 179-199 (1991; Zbl 0741.14032)]. The author proves here that for coprime \(m\) and \(n\), \(H(mn)\) is stably birational to \(H(m) \times H(n)\). It was also known that \(H(p)\) is retract rational for \(p\) prime [cf. \textit{D. J. Saltman}, Isr. J. Math. 47, 165-215 (1984; Zbl 0546.14013)]; hence by the results proved here, \(H(n)\) is retract rational whenever \(n\) is squarefree or twice a squarefree number.