A reduction theorem for the blockwise Alperin weight conjecture. (Q2842795)

Alperin's Weight Conjecture (AWC) predicts that, for any prime \(p\) and any finite group \(G\), there exists a bijection between the set of irreducible \(p\)-Brauer characters of \(G\) and the set of conjugacy classes of \(p\)-weights of \(G\). Recall that a \(p\)-weight of \(G\) is a pair \((Q,\varphi)\) consisting of a \(p\)-subgroup \(Q\) of \(G\) and an irreducible character \(\varphi\) of \(N_G(Q)/Q\) of \(p\)-defect zero. Moreover, the bijection is expected to be compatible with \(p\)-blocks; this is the blockwise Alperin Weight Conjecture.NEWLINENEWLINE The main result of the paper under review shows that this blockwise AWC holds for any prime \(p\) and any finite group \(G\) provided that every nonabelian finite simple group \(S\) satisfies a strengthened version of AWC called the inductive BAW condition. The strengthening is concerned with central \(p'\)-extensions \(X\) of \(S\) and automorphisms of \(X\); the details are too involved to state here.NEWLINENEWLINE The paper under review is a refinement of results by \textit{G. Navarro} and \textit{P. H. Tiep} [Invent. Math. 184, No. 3, 529-565 (2011; Zbl 1234.20010)] who did not consider blocks. Somewhat different reductions of AWC to simple groups have been proposed by E. C. Dade (unpublished) and \textit{L. Puig} [J. Algebra 372, 211-236 (2012; Zbl 1285.20007)].NEWLINENEWLINE The author proves that simple groups of Lie type satisfy the inductive BAW condition for their defining prime, and that the blockwise AWC holds for arbitrary primes \(p\) in finite groups with Abelian Sylow 2-subgroups.