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On Problems in the Representation Theory of Symmetric Groups 10.17863/CAM.46174 Law, Stacey Wing Chee Martin, Stuart 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 $\mathfrak{S}_{n-1}$, and show that every value between 1 and this maximum is attained. These results can be stated graph-theoretically 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 self-similar 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 Littlewood--Richardson 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 self-dual, using decomposition numbers for symmetric groups, tilting module multiplicities and combinatorial methods. University of Cambridge 2020-01-25 Thesis Doctoral Doctor of Philosophy (PhD) en https://www.repository.cam.ac.uk/handle/1810/299110 https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34c36708-345d-466f-bed5-6120be931e92/download 7c22ceb47e90f7782cd2b495352960ff https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ab2511c7-5ec1-4a64-b759-ab92766e3f06/download 87eda9de84448d1f82354d60eee3eb5f https://www.rioxx.net/licenses/all-rights-reserved/ Representation theory symmetric groups linear characters of Sylow subgroups