Non-linear Sylow branching coefficients for symmetric groups (Q2678957)

The object of this paper is to investigate the relationship between the characters of the symmetric group \(\mathfrak{S}_{n}\) and their restrictions to a \(p\)-Sylow subgroup. Let \(p\) be an odd prime and \(P_{n}\) be a \(p\)-Sylow subgroup of \(\mathfrak{S}_{n}\). The characters in \(\mathrm{Irr}(\mathfrak{S}_{n})\) are described by their Young diagrams and the characters in \(\mathrm{Irr}(P_{n})\) all have degrees a power of \(p\). In fact, for some \(\alpha _{n}\) it is shown the character degrees of \(\mathrm{Irr}(P_{n})\) are all the prime powers \(p^{k}\) for \( 0\leq k\leq \alpha _{n}\). We write \(\mathrm{Irr}_{k}(P_{n}):=\left\{ \varphi \in\mathrm{Irr}(P_{n})~|~\varphi (1)=p^{k}\right\} \) and use \(\Omega _{n}^{k}\) to denote the characters \(\chi \in \mathrm{Irr}(\mathfrak{S}_{n})\) whose restriction \(\chi _{P_{n}}\) to \(P_{n}\) has a constituent in \(\mathrm{Irr}_{k}(P_{n})\). In an earlier paper involving the first author [\textit{E. Giannelli} and \textit{G. Navarro}, Proc. Am. Math. Soc. 146, No. 5, 1963--1976 (2018; Zbl 1403.20011)] it was shown that the restriction of every character in \(\mathrm{Irr}(\mathfrak{S}_{n})\) has a constituent of degree \(1\) and so \(\Omega _{n}^{0}=\mathrm{Irr}(\mathfrak{S}_{n})\) for all \(n\). This paper describes the set \( \Omega _{n}^{k}\) where \(k>0\). Let \(\mathcal{B}_{n}(t)\) be the set of characters of \(\mathfrak{S}_{n}\) whose Young diagrams have at most \(t\) rows and \(t\) columns. Then there is an integer \(T_{n}^{k}\) such that \(\Omega _{n}^{k}=\mathcal{B}_{n}(T_{n}^{k})\) (Theorem 5.1). Further information is given about the value of \(T_{n}^{k}\), particularly when \(n\) is a power of \(p\), and computations give the precise value in various smaller cases. The case where \(p=2\) is not considered but it is pointed out that it more complicated than the case where \(p>2\).