Representations of infinitesimal groups of characteristic 2
This short paper continues the authors' earlier work [\textit{R. Farnsteiner} and \textit{D. Voigt}, Adv. Math. 155, No. 1, 1--22 (2000; Zbl 0981.16030)]. In the previous work, it was shown that, for \(k\) an algebraically closed field of characteristic \(p\geq 3\) an infinitesimal \(k\)-group \(G\) has a principal block of finite representation type if and only if \(G/M(G)\) is a semidirect product of the \(n\)-th Frobenius kernel of \(\mu_k\) with a \(V\)-uniserial normal subgroup, which is true if and only if \(H(G)\) is a Nakayama algebra, which in turn is true if and only if \(\dim(\Hom({\pmb\alpha}_{p^2},G))\leq 1\), which in turn is true if and only if \(H(G_2)\) is a Nakayama algebra, where \(G_2=\ker F^2\colon G\to G\) and \(H(G)\) is the distribution algebra of \(G\). In this paper, the main result stated above, an analogue of Higman's theorem, is extended to the case \(p=2\). Specifically, a lemma from the previous paper is extended to the case \(p=2\), from which the main result follows.
scientific article; zbMATH DE number 1753063