On a minimal counterexample to Dade's projective conjecture
Dade's projective conjecture (DPC) can be formulated as: Let \(Z\) be a central \(p\)-subgroup of the finite group \(H\), let \(\lambda\) be a linear character of \(Z\) and let \(B\) be a \(p\)-block of \(H\) whose defect group strictly contains \(Z\), then \[ \sum_{\sigma\in{\mathcal N}(H,Z)/H}(-1)^{|\sigma|}k_d(B_\sigma,\lambda)=0.\tag{1} \] Here \({\mathcal N}(H,Z)\) consists of the strictly increasing chains of \(p\)-subgroups of \(H\) of the form \(\sigma=(Q_0=Z<Q_1<\cdots<Q_n)\) where each \(Q_i\trianglelefteq Q_n\) and \({\mathcal N}(H,Z)/H\) is any set of representatives of the \(H\)-conjugation orbits on \({\mathcal N}(H,Z)\). Also \(H_\sigma\) denotes the stabilizer of the chain \(\sigma\in{\mathcal N}(H,Z)\), \(B_\sigma\) denotes the sum of the Brauer correspondent blocks of \(B\) for the subgroup \(H_\sigma\) and \(k_d(B_\sigma,\lambda)\) denotes the number of irreducible complex characters of defect \(d\) in \(B_\sigma\) which lie over \(\lambda\). It is no loss of generality to assume that \(Z=O_p(H)\). Let the ring \(R\) be a complete discrete valuation ring of characteristic \(0\) whose residue field \(F=R/J(R)\) is algebraically closed of characteristic \(p\) that contains sufficiently many \(p\)-power roots of 1. The main result of this paper is: Theorem 1. Suppose that the formula (1) in DPC fails to hold for some defect \(d\) and some linear character \(\lambda\) of the central subgroup \(Z=O_p(G)\) for the block \(B\) of \(RG\) and that first \(|G:Z(G)|\) and then \(|G|\) have been minimalized subject to such a failure. Then for any subgroup \(M\) of \(G\) such that \(Z(G)<M\trianglelefteq G\), we have \(C_G(M)=Z(G)\) and \(F^*(M)>F(M)\). Remarks: 1. The conclusion of Theorem 1 is equivalent to asserting that \(F(G)=Z(G)\) and that \(G\) has a unique conjugacy class of components; and 2. The choice of \(G\) forces \(\lambda\) to be faithful.
numbers of irreducible complex characters
scientific article; zbMATH DE number 1750363