{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/299110"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/299110","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"On Problems in the Representation Theory of Symmetric Groups","abstract":"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.","abstract_html":"In this thesis, we study the representation theory of the symmetric groups <span class=\"etd-inline-math\">\\mathfrak{S}<sub>n</sub></span>, their Sylow $p$-subgroups <span class=\"etd-inline-math\">P<sub>n</sub></span> 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 <span class=\"etd-inline-math\">\\mathfrak{S}<sub>n</sub></span> to <span class=\"etd-inline-math\">\\mathfrak{S}<sub>n-1</sub></span>, 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 <span class=\"etd-inline-math\">P<sub>n</sub></span> appears as a constituent of the restriction of an irreducible character of <span class=\"etd-inline-math\">\\mathfrak{S}<sub>n</sub></span>, 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 <span class=\"etd-inline-math\">\\mathfrak{A}<sub>n</sub></span>. We then extend our scope to arbitrary linear characters of <span class=\"etd-inline-math\">P<sub>n</sub></span>, proving in particular that for all $p$, given linear characters $\\phi$ and $\\phi&#x27;$ of <span class=\"etd-inline-math\">P<sub>n</sub></span>, their inductions to <span class=\"etd-inline-math\">\\mathfrak{S}<sub>n</sub></span> are equal if and only if $\\phi$ and $\\phi&#x27;$ are <span class=\"etd-inline-math\">N<sub>\\mathfrak{S}<sub>n</sub></sub>(P<sub>n</sub>)</span>--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.","abstract_has_math":true,"creators":["Law, Stacey Wing Chee"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Martin, Stuart"],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-01-25","date_published":"2020-01-25","updated_at":"2026-07-22T22:24:03Z","subjects":["Representation theory","symmetric groups","linear characters of Sylow subgroups"],"languages":["en"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ab2511c7-5ec1-4a64-b759-ab92766e3f06/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.46174","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Martin, Stuart"]},{"key":"dc:creator","label":"Author","values":["Law, Stacey Wing Chee"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2020-01-25"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/299110"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Representation theory","symmetric groups","linear characters of Sylow subgroups"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ab2511c7-5ec1-4a64-b759-ab92766e3f06/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.46174"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34c36708-345d-466f-bed5-6120be931e92/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["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."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["7c22ceb47e90f7782cd2b495352960ff","87eda9de84448d1f82354d60eee3eb5f"]},{"key":"dc:title","label":"Title","values":["On Problems in the Representation Theory of Symmetric Groups"]}]}],"canonical_facts":{"dc:contributor.advisor":["Martin, Stuart"],"dc:creator":["Law, Stacey Wing Chee"],"dc:date.issued":["2020-01-25"],"dc:description.abstract":["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."],"dc:format.checksum.md5":["7c22ceb47e90f7782cd2b495352960ff","87eda9de84448d1f82354d60eee3eb5f"],"dc:identifier.doi":["10.17863/CAM.46174"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/34c36708-345d-466f-bed5-6120be931e92/download"],"dc:language":["en"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/299110"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/ab2511c7-5ec1-4a64-b759-ab92766e3f06/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Representation theory","symmetric groups","linear characters of Sylow subgroups"],"dc:title":["On Problems in the Representation Theory of Symmetric Groups"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:03Z"}