[For the entire collection see Zbl 0668.00005.]
Let \(\Delta_{m,n}\) be the generic division algebra related to the algebra of m generic \(n\times n\) matrices over any field. One of the main open problems for \(\Delta_{m,n}\) is the question for the rationality (or stable rationality of the centre \(K_{m,n}\) of \(\Delta_{m,n}\). The answer affirmatively is known for \(n\leq 4\) only (see [C. Procesi, Rings with Polynomial Identities (1973; Zbl 0262.16018)] for \(m=2\) and [E. Formanek, Linear Multilinear Algebra 7, 203-212 (1979; Zbl 0419.16010), J. Algebra 69, 105-112 (1983; Zbl 0459.16013)], \(n=3,4.)\) The purpose of the paper under review is to survey some results supporting the rationality conjecture. As a consequence of the methods exposed, new results are obtained as well.
Since there exists a reduction of the problem to the case \(m=2\) due to Procesi, the main object to study is the algebra \(K_{2,n}\). First, the authors give relations with some recent results in algebraic geometry on moduli spaces. Second, there is a cellular decomposition conjecture of V. Kac in representation theory of finite-dimensional hereditary algebras which would imply the rationality of \(K_{2,n}\). Calculating the zeta- functions of the parametrizing varieties up to dimension 8, the results obtained are compatible with the Kac conjecture. Finally, the authors sketch a systematic approach to the problem based on the theory of tori- invariants.