On a theorem of Barker-Puig that refines Alperin's weight conjecture for blocks in \(p\)-solvable finite groups via Clifford theory.
In this paper, the author gives a shorter Clifford theoretic proof of a theorem by Barker and Puig on a refinement of Alperin's weight conjecture for blocks of \(p\)-solvable groups. More precisely, let \(p\) be a prime number, and let \(k\) be an algebraically closed field of characteristic \(p\). Let \(G\) be a finite group, let \(P\) be a \(p\)-subgroup of \(G\), and let \(H=N_G(P)\). Suppose \(b\) is a block idempotent of \(kH\). Define \(\mathrm{Irr}((kH)b,P)\) to be a complete set of representatives of isomorphism classes of irreducible \((kH)b\)-modules with vertex \(P\). Define \(\mathcal{IG}(kG,P,b)\) to be a complete set of representatives of isomorphism classes of irreducible \(kG\)-modules \(X\) with vertex \(P\) such that \(b\,\mathcal{G}r_H^G(X)=\mathcal{G}r_H^G(X)\) for any Green correspondent \(\mathcal{G}r_H^G(X)\) of \(X\) in \(H\). The theorem of Barker and Puig [\textit{L. Barker}, Q. J. Math., Oxf. II. Ser. 48, No. 190, 133-160 (1997; Zbl 0882.20002)] that the author studies says that if \(G\) is a \(p\)-solvable group then \(|\mathcal{IG}(kG,P,b)|=|\mathrm{Irr}((kH)b,P)|\). The author's new proof uses a standard Clifford theoretic approach. He also demonstrates a relationship between the Green correspondence and Sibley's concept of vertex pairs [\textit{D. A. Sibley}, J. Algebra 132, No. 2, 501-507 (1990; Zbl 0707.20005)], which enables him to state the main theorem in terms of these vertex pairs.
scientific article; zbMATH DE number 5816636
scientific article