{"id":{"repo_id":"glasgow","oai_identifier":"oai:theses.gla.ac.uk:300"},"canonical_url":"https://search.dev.ndltd.org/etd/glasgow/oai:theses.gla.ac.uk:300","repository":{"repo_id":"glasgow","name":"University of Glasgow","base_url":"https://theses.gla.ac.uk/cgi/oai2"},"display":{"title":"The ring of invariants of the orthogonal group over finite fields in odd characteristic","abstract":"Let $V$ be a non-zero finite dimensional vector space over a finite field $\\mathbb{F}_q$ of odd characteristic. Fixing a non-singular quadratic form $\\xi_0$ in $S^2(V^*)$, the symmetric square of the dual of V we are concerned with the Orthogonal group $O(\\xi_0)$, the subgroup of the General Linear Group $GL(V)$ that fixes $\\xi_0$ and with invariants of this group. We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of $O(\\xi_0)$. Considering the $O(\\xi_0)$ orbits of the dual vector space $\\vs$ we generate the Chern Orbit polynomials, the coefficients of which, the Chern Orbit Classes, are also invariants of the Orthogonal group. The invariants $\\xi_1, \\xi_2, \\dots $ are be generated from $\\xi_0$ by applying the action of the Steenrod Algebra to $S^2(V^*)$ which being natural takes invariants to invariants. Our aim is to discover further invariants from these known invariants with the intention of establishing a set of generators for the the Ring of invariants of the Orthogonal Group. In particular we calculate invariants of $O(\\xi_0)$ when the dimension of the vector space is $4$ the finite field is $\\mathbb{F}_3$ and the quadratic form is $\\xi_0=x_1^2+x_2^2+x_3^2+x_4^2$ and we are able to establish an explicit presentation of $O(\\xi_0)$ in this case.","abstract_html":"Let $V$ be a non-zero finite dimensional vector space over a finite field <span class=\"etd-inline-math\">\\mathbb{F}<sub>q</sub></span> of odd characteristic. Fixing a non-singular quadratic form <span class=\"etd-inline-math\">\\xi<sub>0</sub></span> in <span class=\"etd-inline-math\">S<sup>2</sup>(V<sup>*</sup>)</span>, the symmetric square of the dual of V we are concerned with the Orthogonal group <span class=\"etd-inline-math\">O(\\xi<sub>0</sub>)</span>, the subgroup of the General Linear Group $GL(V)$ that fixes <span class=\"etd-inline-math\">\\xi<sub>0</sub></span> and with invariants of this group. We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of <span class=\"etd-inline-math\">O(\\xi<sub>0</sub>)</span>. Considering the <span class=\"etd-inline-math\">O(\\xi<sub>0</sub>)</span> orbits of the dual vector space $\\vs$ we generate the Chern Orbit polynomials, the coefficients of which, the Chern Orbit Classes, are also invariants of the Orthogonal group. The invariants <span class=\"etd-inline-math\">\\xi<sub>1</sub>, \\xi<sub>2</sub>, \\dots </span> are be generated from <span class=\"etd-inline-math\">\\xi<sub>0</sub></span> by applying the action of the Steenrod Algebra to <span class=\"etd-inline-math\">S<sup>2</sup>(V<sup>*</sup>)</span> which being natural takes invariants to invariants. Our aim is to discover further invariants from these known invariants with the intention of establishing a set of generators for the the Ring of invariants of the Orthogonal Group. In particular we calculate invariants of <span class=\"etd-inline-math\">O(\\xi<sub>0</sub>)</span> when the dimension of the vector space is $4$ the finite field is <span class=\"etd-inline-math\">\\mathbb{F}<sub>3</sub></span> and the quadratic form is <span class=\"etd-inline-math\">\\xi<sub>0</sub>=x<sub>1</sub><sup>2</sup>+x<sub>2</sub><sup>2</sup>+x<sub>3</sub><sup>2</sup>+x<sub>4</sub><sup>2</sup></span> and we are able to establish an explicit presentation of <span class=\"etd-inline-math\">O(\\xi<sub>0</sub>)</span> in this case.","abstract_has_math":true,"creators":["Barnes, Sue"],"institution":"University of Glasgow","degree_name":null,"degree_level":"PhD","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2008,"date_issued":"2008","date_published":"2008","updated_at":"2026-07-24T02:23:18Z","subjects":["QA Mathematics"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Barnes, Sue"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2008"]},{"key":"dc:date.issued","label":"Date","values":["2008"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Glasgow"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://theses.gla.ac.uk/300/"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://gla.on.worldcat.org/oclc/248382877"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["PhD"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["QA Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://theses.gla.ac.uk/300/1/2008barnesphd.pdf.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let $V$ be a non-zero finite dimensional vector space over a finite field $\\mathbb{F}_q$ of odd characteristic. Fixing a non-singular quadratic form $\\xi_0$ in $S^2(V^*)$, the symmetric square of the dual of V we are concerned with the Orthogonal group $O(\\xi_0)$, the subgroup of the General Linear Group $GL(V)$ that fixes $\\xi_0$ and with invariants of this group. We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of $O(\\xi_0)$. Considering the $O(\\xi_0)$ orbits of the dual vector space $\\vs$ we generate the Chern Orbit polynomials, the coefficients of which, the Chern Orbit Classes, are also invariants of the Orthogonal group. The invariants $\\xi_1, \\xi_2, \\dots $ are be generated from $\\xi_0$ by applying the action of the Steenrod Algebra to $S^2(V^*)$ which being natural takes invariants to invariants. Our aim is to discover further invariants from these known invariants with the intention of establishing a set of generators for the the Ring of invariants of the Orthogonal Group. In particular we calculate invariants of $O(\\xi_0)$ when the dimension of the vector space is $4$ the finite field is $\\mathbb{F}_3$ and the quadratic form is $\\xi_0=x_1^2+x_2^2+x_3^2+x_4^2$ and we are able to establish an explicit presentation of $O(\\xi_0)$ in this case."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The ring of invariants of the orthogonal group over finite fields in odd characteristic"]}]}],"canonical_facts":{"dc:creator":["Barnes, Sue"],"dc:date":["2008"],"dc:date.issued":["2008"],"dc:description.abstract":["Let $V$ be a non-zero finite dimensional vector space over a finite field $\\mathbb{F}_q$ of odd characteristic. Fixing a non-singular quadratic form $\\xi_0$ in $S^2(V^*)$, the symmetric square of the dual of V we are concerned with the Orthogonal group $O(\\xi_0)$, the subgroup of the General Linear Group $GL(V)$ that fixes $\\xi_0$ and with invariants of this group. We have the Dickson Invariants which being invariants of the General Linear Group are then invariants of $O(\\xi_0)$. Considering the $O(\\xi_0)$ orbits of the dual vector space $\\vs$ we generate the Chern Orbit polynomials, the coefficients of which, the Chern Orbit Classes, are also invariants of the Orthogonal group. The invariants $\\xi_1, \\xi_2, \\dots $ are be generated from $\\xi_0$ by applying the action of the Steenrod Algebra to $S^2(V^*)$ which being natural takes invariants to invariants. Our aim is to discover further invariants from these known invariants with the intention of establishing a set of generators for the the Ring of invariants of the Orthogonal Group. In particular we calculate invariants of $O(\\xi_0)$ when the dimension of the vector space is $4$ the finite field is $\\mathbb{F}_3$ and the quadratic form is $\\xi_0=x_1^2+x_2^2+x_3^2+x_4^2$ and we are able to establish an explicit presentation of $O(\\xi_0)$ in this case."],"dc:format":["application/pdf"],"dc:identifier.uri":["https://theses.gla.ac.uk/300/1/2008barnesphd.pdf.pdf"],"dc:language":["en"],"dc:publisher.institution":["University of Glasgow"],"dc:relation.isreferencedby":["https://theses.gla.ac.uk/300/"],"dc:relation.isreferencedby.uri":["https://gla.on.worldcat.org/oclc/248382877"],"dc:subject":["QA Mathematics"],"dc:title":["The ring of invariants of the orthogonal group over finite fields in odd characteristic"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["PhD"]},"updated_at":"2026-07-24T02:23:18Z"}