On stability of diagonal actions and tensor invariants (Q2901894)

It was proved in [\textit{I. V. Arzhantsev}, Math. Notes 71, No. 6, 735--738 (2002); translation from Mat. Zametki 71, No. 6, 803--806 (2002; Zbl 1018.14017)] that for any semisimple algebraic group \(G\) acting on an affine normal variety \(X\) the diagonal action of \(G\) on \(X^{\oplus n}\) is \textit{stable} (i.e. the generic orbits are closed) for \(n\) sufficiently large. The smallest such \(n\) is denoted by \(s_m(G)\) while \(s(G)\) denotes the smallest integer such that \(X^{\oplus n}\) is stable for every \(n \geq s(G)\). The paper gives bounds on these two quantities in terms of \(m(G)\), which is defined as the smallest number \(k\) such that the tensor power \(V^{\otimes k}\) contains a nonzero \(G\)-invariant for any nonzero rational \(G\)-module \(V\). Namely, it is proved [see Theorems 3, 4] that \(m(G) \leq s_m(G) \leq s(G) \leq \dim(G) m(G)\). The precise value of \(m(G)\) is then determined for every simple algebraic group [see Theorem 1] by constructing so-called \textit{balanced collections} of elements in their Weyl groups. This allows in the end the calculation of \(m(G)\) for any connected simply connected semisimple algebraic group \(G\) [see Propositions 1, 2].