Advanced search options

Advanced Search Options 🞨

Browse by author name (“Author name starts with…”).

Find ETDs with:

in
/  
in
/  
in
/  
in

Written in Published in Earliest date Latest date

Sorted by

Results per page:

You searched for id:"oai:ethos.bl.uk:793084". One record found.

Search Limiters

Last 2 Years | English Only

No search limiters apply to these results.

▼ Search Limiters


University of Cambridge

1. Law, Stacey Wing Chee. On problems in the representation theory of symmetric groups.

Degree: PhD, 2020, University of Cambridge

In this thesis, we study the representation theory of the symmetric groups \mathfrak{S}n, their Sylow p-subgroups Pn 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 Pn 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 Pn, proving in particular that for all p, given linear characters φ and φ' of Pn, their inductions to \mathfrak{S}n are equal if and only if φ and φ' are N\mathfrak{Sn}(Pn) – 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.

Subjects/Keywords: Representation theory; symmetric groups; linear characters of Sylow subgroups

Record DetailsSimilar RecordsGoogle PlusoneFacebookTwitterCiteULikeMendeleyreddit

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6th Edition):

Law, S. W. C. (2020). On problems in the representation theory of symmetric groups. (Doctoral Dissertation). University of Cambridge. Retrieved from https://www.repository.cam.ac.uk/handle/1810/299110 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.793084

Chicago Manual of Style (16th Edition):

Law, Stacey Wing Chee. “On problems in the representation theory of symmetric groups.” 2020. Doctoral Dissertation, University of Cambridge. Accessed February 27, 2020. https://www.repository.cam.ac.uk/handle/1810/299110 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.793084.

MLA Handbook (7th Edition):

Law, Stacey Wing Chee. “On problems in the representation theory of symmetric groups.” 2020. Web. 27 Feb 2020.

Vancouver:

Law SWC. On problems in the representation theory of symmetric groups. [Internet] [Doctoral dissertation]. University of Cambridge; 2020. [cited 2020 Feb 27]. Available from: https://www.repository.cam.ac.uk/handle/1810/299110 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.793084.

Council of Science Editors:

Law SWC. On problems in the representation theory of symmetric groups. [Doctoral Dissertation]. University of Cambridge; 2020. Available from: https://www.repository.cam.ac.uk/handle/1810/299110 ; https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.793084

.