On the Morita Frobenius numbers of blocks of finite reductive groups (Q342848)

Let \(G\) be a finite group, \(p\) be a prime, and \(B\) a block of \(\overline {\mathbb F}_pG\). The Morita Frobenius number \(\mathrm{mf}(B)\), as defined by \textit{R. Kessar} [J. Algebra 186, No. 3, 872--933 (1996; Zbl 0894.20015)], is the smallest positive integer \(a\) such that \(B\) is Morita equivalent to its \(p^a\)-Frobenius twist. The author shows that \(\mathrm{mf}(B)\) is \(1\) if \(G\) is a covering group of an alternating group or a finite simple group of Lie type in characteristic \(p\). In the case \(G\) is a covering group of a finite simple group \(S\) in characteristic \(\neq p\) and \(B\) dominates a unipotent block \(b\) of \(S\), she proves that \(\mathrm{mf}(B) = 1\), unless possibly \(S = E_8(q)\) and \(b\) is one of a couple of explicitly described unipotent blocks, in which case one has \(\mathrm{mf}(B) \leq 2\).