Dade's conjecture for symmetric groups
Let \(G\) be a finite group and \(p\) be a prime number. A chain of \(p\)-subgroups \({\mathcal C}: P_0 \subset P_1 \subset\dots \subset P_m\) of \(G\) is called a radical chain if \(P_0= O_p (G)\) and \(P_i= O_p (\bigcap^i_{j=0} N_G (P_j))\) for \(i=1, 2, \dots, m\). The normalizer of the above chain is defined by \(N_G^{({\mathcal C})}= \bigcap^m_{j=0} N_G (P_j)\). Obviously \(G\) acts on the set of all the radical chains in \(G\) by conjugation. A set of representatives for the \(G\)-orbits is denoted by \({\mathcal R}/G\). For any subgroup \(H\) of \(G\), any \(p\)-block \(B\) of \(G\) and any non-negative integer \(d\) let \(k(H, B, d)\) denote the number of irreducible characters \(\varphi\) of \(H\) satisfying (i) \(\varphi\) belongs to a \(p\)-block \(b\) of \(H\) such that \(b^G =B\), (ii) The \(p\)-part of \(|H|/ \varphi(1)\) is equal to \(p^d\). A conjecture of Dade, called ordinary conjecture, states that if \(O_p (G)=1\) and \(B\) is a \(p\)-block of \(G\) with positive block, then for all \(d\geq 0\) the equality \(\sum_{c\in {\mathcal R}/G} (-1)^{|{cal C}|} k (N_G({\mathcal C}), B,d)=0\) holds. This conjecture has been verified for several simple groups. According to the authors' announcement the conjecture has been verified for any positive defect block of the group \(\text{GL}(n,q)\) in the defining characteristic. In the present paper the authors verify the conjecture of Dade for the symmetric groups in odd characteristic, i.e. when \(p\) is an odd prime.
scientific article; zbMATH DE number 815924