Vertices for irreducible characters of a class of blocks.
Let \(p\) be a prime, and let \(B\) be a \(p\)-block of a finite group \(G\) whose defect groups are contained in a \(p\)-solvable normal subgroup \(N\) of \(G\). Extending an idea of \textit{G. Navarro} [Trans. Am. Math. Soc. 354, No. 7, 2759-2773 (2002; Zbl 0998.20012)] the author associates with every irreducible character \(\chi\) in \(B\) pairs \((Q,\delta)\) consisting of a \(p\)-subgroup \(Q\) of \(G\) and an irreducible character \(\delta\) of \(Q\). These pairs are called the vertices of \(\chi\); they form a \(G\)-orbit under conjugation and are independent of the choice of the \(p\)-solvable normal subgroup \(N\). If \((Q,\delta)\) is a vertex of \(\chi\) then \(\chi\) is afforded by a relatively \(Q\)-projective module; in particular, \(\chi\) vanishes on elements whose \(p\)-factor is not conjugate to an element in \(Q\). Moreover, the defect of \(\chi\) coincides with the defect of \(\delta\), and \(Q=O_p(N_G(Q,\delta))\). Also, the irreducible characters in \(B\) with vertex \((Q,1_Q)\) are in canonical bijection with the irreducible Brauer characters of \(G\) with vertex \(Q\) (in Green's sense).
scientific article; zbMATH DE number 2164793
scientific article