More cancellation theorems for conjectures of Alperin and Dade
Let \(G\) be a finite group, \(p\) a prime and \(B\) a \(p\)-block of \(G\). Alperin's weight conjecture (AWC) asserts that \[ \ell(B)=\sum_{Q/G}f^{(B)}_0(N_G(Q)/Q), \] where \(\ell(B)\) denotes the number of irreducible Brauer characters in \(B\), \(Q/G\) is a set of representatives for the \(G\)-conjugacy classes of \(p\)-subgroups of \(G\) and \(f^{(B)}_0(N_G(Q)/Q)\) denotes the number of \(p\)-blocks of defect \(0\) of \(N_G(Q)/Q\) contained in the \(B\)-Brauer correspondent blocks of \(N_G(Q)\). This conjecture was reformulated by R. Knörr and the author of this paper. This condition, equivalent to (AWC), was subsequently reformulated by E. C. Dade, Dade's projective conjecture (DPC), which is as follows: Let \(Z\) be a central \(p\)-subgroup of \(G\) and let \(\lambda\) be a linear character of \(Z\). Suppose that the defect group of \(B\) strictly contains \(Z\). Then \[ \sum_{\sigma\in{\mathcal N}(G,Z)/G}(-1)^{|\sigma|}k_d(B_\sigma,\lambda)=0\tag{1} \] for any integer \(d\geq 0\) and any such \(\lambda\) where \({\mathcal N}(G,Z)/G\) denotes a set of representatives for the \(G\)-conjugacy classes of strictly increasing normal chains of \(p\)-subgroups of \(G\) of the form \(\sigma=(Z=Q_0<Q_1<\cdots<Q_n)\) where all \(Q_i\triangleleft Q_n\), \(H_\sigma\) is the \(G\)-stabilizer of \(\sigma\), \(B_\sigma\) denotes the sum of \(B\)-Brauer correspondent blocks of \(H_\sigma\) and \(k_d(B_\sigma,\lambda)\) denotes the number of complex irreducible characters of defect \(d\) in the blocks of \(B_\sigma\) that cover \(\lambda\). To verify DPC, it is no loss to assume that \(Z=O_p(G)\). Moreover in a previous work, the author has shown that (1) is equivalent to \[ \sum_{\sigma\in{\mathcal N}(G,Z)/G}(-1)^{|\sigma|}k_d(B_\sigma)=0\tag{2} \] for all integers \(d\geq 0\) where \(k_d(B_\sigma)\) denotes the number of complex irreducible characters of defect \(d\) in the blocks of \(B_\sigma\). Note that \[ \sum_{\sigma\in{\mathcal N}(G,Z)}(-1)^{|\sigma|}k_d(B_\sigma)=k_d(B)-\sum_{U/G} \sum_{\sigma\in{\mathcal N}(G,U)/N_G(U)}(-1)^{|\sigma|}k_d(B_\sigma),\tag{3} \] where \(U/G\) denotes a set of representatives for the \(G\)-conjugacy orbits of \(p\)-subgroups of \(G\) that strictly contain \(Z\) and \({\mathcal N}(G,U)/N_G(U)\) denotes a set of representatives for the \(N_G(U)\)-conjugacy orbits of normal chains \(\sigma\), as above, whose initial subgroups is \(U\). The main results of this paper present some interesting conditions that ensure that chains in \({\mathcal N}(G,U)/N_G(U)\) make a zero contribution to (3) and some applications to a minimal counterexample to DPC.
numbers of irreducible Brauer characters
numbers of complex irreducible characters
scientific article; zbMATH DE number 1750364