Parameterization of irreducible characters for \(p\)-solvable groups. (Q2880061)

Let \(p\) be a prime number, let \(k\) be an algebraically closed field of characteristic \(p\), and let \(G\) be a finite group. The weights for \(G\) with respect to \(p\) were introduced by Alperin in 1987 to formulate his celebrated ``weight conjecture.'' In 1992, Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters, together with their defects. However, this refinement is formulated in terms of a vanishing alternating sum of cardinals of suitable sets of irreducible characters, without giving any possible refinement for the weights.NEWLINENEWLINE In 2000, \textit{G. R. Robinson} proved Dade's conjecture for \(p\)-solvable groups [J. Algebra 229, No. 1, 234-248 (2000; Zbl 0955.20006)]. The goal of the paper under review is to show that, in the case of \(p\)-solvable groups, there is, up to the choice of a polarization \(\omega\), a natural bijection between the sets of absolutely irreducible characters of \(G\) and of \(G\)-conjugacy classes of suitable inductive weights, preserving blocks and defects.NEWLINENEWLINE The main idea of the proof is to refine the methods the author developed in [\textit{L.~Puig}, ``Weight parameterization of simple modules for \(p\)-solvable groups'', \url{ArXiv:1005.3748}] to provide the desired natural bijection, which is compatible with the action of the group of outer automorphisms of \(G\). Suppose \(\mathcal O\) is a complete discrete valuation ring of characteristic 0 whose residue field is \(k\). As in the previous paper (cited above), the inductive arguments the author uses lead to the consideration of a more general situation including central extensions of the involved finite groups. But in the present paper, the central \(k^*\)-extensions that the author considered in his previous paper (cited above) need to be replaced by the central \(\mathcal O^*\)-extensions, called \(\mathcal O^*\)-groups. Since \(\mathcal O^*\)-groups are not quite as nicely behaved as \(k^*\)-groups, much of the present paper is used to develop the general theory of \(\mathcal O^*\)-groups.