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}_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.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2020

Author dc:creator

• Law, Stacey Wing Chee

Advisor dc:contributor.advisor

• Martin, Stuart

• Representation theory
• symmetric groups
• linear characters of Sylow subgroups

Licence

• https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ab2511c7-5ec1-4a64-b759-ab92766e3f06/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

en

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.46174

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/299110