{"id":{"repo_id":"wfu","oai_identifier":"oai:wakespace.lib.wfu.edu:10339/82199"},"canonical_url":"https://search.dev.ndltd.org/etd/wfu/oai:wakespace.lib.wfu.edu:10339/82199","repository":{"repo_id":"wfu","name":"Wake Forest University","base_url":"https://wakespace.lib.wfu.edu/oai/request"},"display":{"title":"ALGEBRA GENERATORS AND HILBERT SERIES OF Z(K_{-1}[X_1, ..., X_n]^{S_n})","abstract":"Given a field $k$ of characteristic zero, let $R = k_{-1}[x_1, ..., x_n]$ denote the $(-1)$ skew-polynomial ring; i.e., the ring generated from $n$ indeterminates $\\{x_i\\}_{i=1}^n$ over $k$ such that $x_i x_j = - x_j x_i$ for all $i \\ne j$. If $G$ is a subgroup of the symmetric group $S_n$ represented by permutation matrices, there is an induced $G$-action on $R$ given by permuting the indeterminates. We define $R^G$ as the subring of invariants under this $G$-action. In this thesis, we determine the algebra generators and the Hilbert series of the center of $R^{S_n}$, denoted $Z(R^{S_n})$. Moreover, we determine an algorithm for determining the ideal of relations on the generators.","abstract_html":"Given a field $k$ of characteristic zero, let <span class=\"etd-inline-math\">R = k<sub>-1</sub>[x<sub>1</sub>, ..., x<sub>n</sub>]</span> denote the $(-1)$ skew-polynomial ring; i.e., the ring generated from $n$ indeterminates <span class=\"etd-inline-math\">\\{x<sub>i</sub>\\}<sub>i=1</sub><sup>n</sup></span> over $k$ such that <span class=\"etd-inline-math\">x<sub>i</sub> x<sub>j</sub> = - x<sub>j</sub> x<sub>i</sub></span> for all $i \\ne j$. If $G$ is a subgroup of the symmetric group <span class=\"etd-inline-math\">S<sub>n</sub></span> represented by permutation matrices, there is an induced $G$-action on $R$ given by permuting the indeterminates. We define <span class=\"etd-inline-math\">R<sup>G</sup></span> as the subring of invariants under this $G$-action. In this thesis, we determine the algebra generators and the Hilbert series of the center of <span class=\"etd-inline-math\">R<sup>S<sub>n</sub></sup></span>, denoted <span class=\"etd-inline-math\">Z(R<sup>S<sub>n</sub></sup>)</span>. Moreover, we determine an algorithm for determining the ideal of relations on the generators.","abstract_has_math":true,"creators":["Hall, Bailey Thomas"],"institution":"Wake Forest University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2017,"date_issued":"2017","date_published":"2017","updated_at":"2026-07-27T22:02:11Z","subjects":["Algebra Generator"],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/10339/82199","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Hall, Bailey Thomas"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2017-06-15T08:35:57Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2017-06-15T08:35:57Z"]},{"key":"dc:date.issued","label":"Date","values":["2017"]},{"key":"dc:publisher","label":"Institution","values":["Wake Forest University"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Algebra Generator"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10339/82199"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Given a field $k$ of characteristic zero, let $R = k_{-1}[x_1, ..., x_n]$ denote the $(-1)$ skew-polynomial ring; i.e., the ring generated from $n$ indeterminates $\\{x_i\\}_{i=1}^n$ over $k$ such that $x_i x_j = - x_j x_i$ for all $i \\ne j$. If $G$ is a subgroup of the symmetric group $S_n$ represented by permutation matrices, there is an induced $G$-action on $R$ given by permuting the indeterminates. We define $R^G$ as the subring of invariants under this $G$-action. In this thesis, we determine the algebra generators and the Hilbert series of the center of $R^{S_n}$, denoted $Z(R^{S_n})$. Moreover, we determine an algorithm for determining the ideal of relations on the generators."]},{"key":"dc:title","label":"Title","values":["ALGEBRA GENERATORS AND HILBERT SERIES OF Z(K_{-1}[X_1, ..., X_n]^{S_n})"]}]}],"canonical_facts":{"dc:creator":["Hall, Bailey Thomas"],"dc:date.accessioned":["2017-06-15T08:35:57Z"],"dc:date.available":["2017-06-15T08:35:57Z"],"dc:date.issued":["2017"],"dc:description.abstract":["Given a field $k$ of characteristic zero, let $R = k_{-1}[x_1, ..., x_n]$ denote the $(-1)$ skew-polynomial ring; i.e., the ring generated from $n$ indeterminates $\\{x_i\\}_{i=1}^n$ over $k$ such that $x_i x_j = - x_j x_i$ for all $i \\ne j$. If $G$ is a subgroup of the symmetric group $S_n$ represented by permutation matrices, there is an induced $G$-action on $R$ given by permuting the indeterminates. We define $R^G$ as the subring of invariants under this $G$-action. In this thesis, we determine the algebra generators and the Hilbert series of the center of $R^{S_n}$, denoted $Z(R^{S_n})$. Moreover, we determine an algorithm for determining the ideal of relations on the generators."],"dc:identifier.uri":["http://hdl.handle.net/10339/82199"],"dc:language.iso":["en"],"dc:publisher":["Wake Forest University"],"dc:subject":["Algebra Generator"],"dc:title":["ALGEBRA GENERATORS AND HILBERT SERIES OF Z(K_{-1}[X_1, ..., X_n]^{S_n})"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T22:02:11Z"}