A canonical class of \(p'\)-degree characters of \(\mathfrak{S}_n\) (Q2404940)

Let \(p\) be an odd prime and let \(\Gamma =\mathrm{Irr}_{p'}(\mathfrak{S}_{p^k})\) denote the set of irreducible characters of \(p'\)-degree for the symmetric group of degree \(p^k\). Let \(H_k\) be the \(k\)-fold wreath product \(\mathfrak{S}_k\wr\mathfrak{S}_k\wr \dots \wr\mathfrak{S}_k\leq \mathfrak{S}_{p^k}\). The main theorems of this paper are as follows. (Theorem A): for each \(\chi \in \Gamma\) the restriction \(\chi \downarrow H_k\) has a unique constituent of \(p'\)-degree (denoted by \(\Phi_k(\chi)\)) and the corresponding mapping \(\Phi_k:\Gamma \rightarrow \mathrm{Irr}_{p'}(H_k)\) is a bijection. Furthermore, if \(T_k\) is the set of irreducible characters of \(H_k\) of \(p'\)-degree, then the set \(\Omega_{p^k} \subseteq \Gamma\) consisting of the characters \(\chi\) for which \(\Phi_k(\chi)\) is itself irreducible has size \(\left | \Omega_{p^k}\right | =(p -1)^k\). A direct characterization of the characters \(\chi\) in \(\Omega_{p^k}\) is given in terms of the shape of the partitions associated with them (Theorem B).