Title:

On problems in the representation theory of symmetric groups

In this thesis, we study the representation theory of the symmetric groups $\mathfrak{S}_n$, their Sylow $p$subgroups $P_n$ and related algebras. For all primes $p$ and natural numbers $n$, we determine the maximum number of distinct irreducible constituents of degree coprime to $p$ of restrictions of irreducible characters of $\mathfrak{S}_n$ to $\mathfrak{S}_{n1}$, and show that every value between 1 and this maximum is attained. These results can be stated graphtheoretically in terms of the Young lattice, which describes branching for symmetric groups. We present new graph isomorphisms between certain subgraphs of the Young lattice and find selfsimilar structures. This generalises from $p=2$ to all $p$ work of Ayyer, Prasad and Spallone which was central in the construction of character correspondences for symmetric groups in the context of the McKay Conjecture, a fundamental open problem in the representation theory of finite groups. Linear characters of Sylow subgroups have also played a central role in character correspondences verifying the McKay Conjecture, becoming the focus of much current interest. For instance, a consequence of recent work of Giannelli and Navarro shows the existence of linear constituents in the restriction of every irreducible character of a symmetric group to its Sylow $p$subgroups. We now identify these linear constituents, using a mixture of algebraic and combinatorial techniques including Mackey theory and an analysis of LittlewoodRichardson coefficients. We determine precisely when the trivial character of $P_n$ appears as a constituent of the restriction of an irreducible character of $\mathfrak{S}_n$, for all $n$ and odd $p$. As a consequence, we determine the irreducible characters of the Hecke algebra corresponding to the induced permutation character. Analogous results are obtained for the alternating groups $\mathfrak{A}_n$. We then extend our scope to arbitrary linear characters of $P_n$, proving in particular that for all $p$, given linear characters $\phi$ and $\phi'$ of $P_n$, their inductions to $\mathfrak{S}_n$ are equal if and only if $\phi$ and $\phi'$ are $N_{\mathfrak{S}_n}(P_n)$conjugate. Finally, we consider the representation theory of Schur algebras in all characteristics. We classify the classical Schur algebras $S(n,r)$ which are Ringel selfdual, using decomposition numbers for symmetric groups, tilting module multiplicities and combinatorial methods.
