{"id":{"repo_id":"vt","oai_identifier":"oai:vtechworks.lib.vt.edu:10919/93039"},"canonical_url":"https://search.dev.ndltd.org/etd/vt/oai:vtechworks.lib.vt.edu:10919/93039","repository":{"repo_id":"vt","name":"Virginia Tech","base_url":"https://vtechworks.lib.vt.edu/oai/request"},"display":{"title":"The Combinatorial Curve Neighborhoods of Affine Flag Manifold in Type A<sub>n-1</sub><sup>(1)</sup>","abstract":"Let X be the affine flag manifold of Lie type A<sub>n-1</sub><sup>(1)</sup> where n ≥ 3 and let W<sub>aff</sub> be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W<sub>aff</sub> and the torus stable curves in X. Given a fixed point u ∈ W<sub>aff</sub> and a degree d = (d₀, d₁, ..., d<sub>n−1</sub>) ∈ ℤ<sub>≥0</sub><sup>n</sup>, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements.","abstract_html":"Let X be the affine flag manifold of Lie type A&lt;sub&gt;n-1&lt;/sub&gt;&lt;sup&gt;(1)&lt;/sup&gt; where n ≥ 3 and let W&lt;sub&gt;aff&lt;/sub&gt; be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W&lt;sub&gt;aff&lt;/sub&gt; and the torus stable curves in X. Given a fixed point u ∈ W&lt;sub&gt;aff&lt;/sub&gt; and a degree d = (d₀, d₁, ..., d&lt;sub&gt;n−1&lt;/sub&gt;) ∈ ℤ&lt;sub&gt;≥0&lt;/sub&gt;&lt;sup&gt;n&lt;/sup&gt;, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements.","abstract_has_math":false,"creators":["Aslan, Songul"],"institution":"Virginia Tech","degree_name":"Doctor of Philosophy","degree_level":"doctoral","degree_discipline":"Mathematics","degree_department":"Mathematics","school":null,"contributors":[],"advisors":[],"committee_chairs":["Mihalcea, Constantin Leonardo"],"committee_members":["Orr, Daniel D.","Shimozono, Mark M.","Floyd, William J."],"year":2019,"date_issued":"2019-08-12","date_published":"2019-08-12","updated_at":"2026-07-22T22:19:13Z","subjects":["Affine Flag Manifolds","Schubert Varieties","Curve Neighborhoods","Moment Graph","Combinatorial Curve Neighborhoods"],"languages":[],"rights":["In Copyright"],"rights_urls":["http://rightsstatements.org/vocab/InC/1.0/"],"identifier_entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:21133"],"render_values":[{"text":"vt_gsexam:21133","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/10919/93039","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.committeechair","label":"Committee Chair","values":["Mihalcea, Constantin Leonardo"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Orr, Daniel D.","Shimozono, Mark M.","Floyd, William J."]},{"key":"dc:contributor.department","label":"Department","values":["Mathematics"]},{"key":"dc:creator","label":"Author","values":["Aslan, Songul"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2019-08-13T08:00:28Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2019-08-13T08:00:28Z"]},{"key":"dc:date.issued","label":"Date","values":["2019-08-12"]},{"key":"dc:publisher","label":"Institution","values":["Virginia Tech"]},{"key":"dc:type","label":"Dc Type","values":["Dissertation"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["doctoral"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Doctor of Philosophy"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Virginia Polytechnic Institute and State University"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Affine Flag Manifolds","Schubert Varieties","Curve Neighborhoods","Moment Graph","Combinatorial Curve Neighborhoods"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]},{"key":"dc:rights.uri","label":"Rights URI","values":["http://rightsstatements.org/vocab/InC/1.0/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.other","label":"Dc Identifier Other","values":["vt_gsexam:21133"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/10919/93039"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let X be the affine flag manifold of Lie type A<sub>n-1</sub><sup>(1)</sup> where n ≥ 3 and let W<sub>aff</sub> be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W<sub>aff</sub> and the torus stable curves in X. Given a fixed point u ∈ W<sub>aff</sub> and a degree d = (d₀, d₁, ..., d<sub>n−1</sub>) ∈ ℤ<sub>≥0</sub><sup>n</sup>, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements."]},{"key":"dc:description.abstractgeneral","label":"General Abstract","values":["The study of curves on flag manifolds is motivated by questions in enumerative geometry and physics. To a space of curves and incidence conditions one can associate a combinatorial object called the ‘combinatorial curve neighborhood’. For a fixed degree d and a (Schubert) cycle, the curve neighborhood consists of the set of all elements in the Weyl group which can be reached from the given cycle by a path of fixed degree in the moment graph of the flag manifold, and are also Bruhat maximal with respect to this property. For finite dimensional flag manifolds, a description of the curve neighborhoods helped answer questions related to the quantum cohomology and quantum K theory rings, and ultimately about enumerative geometry of the flag manifolds. In this thesis we study the situation of the affine flag manifolds, which are infinite dimensional. We obtain explicit combinatorial formulas and recursions which characterize the curve neighborhoods for flag manifolds of affine Lie type A. Among the conclusions, we mention that, unlike in the finite dimensional case, the curve neighborhoods have more than one component, although all components have the same length. In general, calculations reflect a close connection between the curve neighborhoods and the Lie combinatorics of the affine root system, especially the imaginary roots."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Doctor of Philosophy"]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["ETD"]},{"key":"dc:title","label":"Title","values":["The Combinatorial Curve Neighborhoods of Affine Flag Manifold in Type A<sub>n-1</sub><sup>(1)</sup>"]}]}],"canonical_facts":{"dc:contributor.committeechair":["Mihalcea, Constantin Leonardo"],"dc:contributor.committeemember":["Orr, Daniel D.","Shimozono, Mark M.","Floyd, William J."],"dc:contributor.department":["Mathematics"],"dc:creator":["Aslan, Songul"],"dc:date.accessioned":["2019-08-13T08:00:28Z"],"dc:date.available":["2019-08-13T08:00:28Z"],"dc:date.issued":["2019-08-12"],"dc:description.abstract":["Let X be the affine flag manifold of Lie type A<sub>n-1</sub><sup>(1)</sup> where n ≥ 3 and let W<sub>aff</sub> be the associated affine Weyl group. The moment graph for X encodes the torus fixed points (which are elements of the affine Weyl group W<sub>aff</sub> and the torus stable curves in X. Given a fixed point u ∈ W<sub>aff</sub> and a degree d = (d₀, d₁, ..., d<sub>n−1</sub>) ∈ ℤ<sub>≥0</sub><sup>n</sup>, the combinatorial curve neighborhood is the set of maximal elements in the moment graph of X which can be reached from u′ ≤ u by a chain of curves of total degree ≤ d. In this thesis we give combinatorial formulas and algorithms for calculating these elements."],"dc:description.abstractgeneral":["The study of curves on flag manifolds is motivated by questions in enumerative geometry and physics. To a space of curves and incidence conditions one can associate a combinatorial object called the ‘combinatorial curve neighborhood’. For a fixed degree d and a (Schubert) cycle, the curve neighborhood consists of the set of all elements in the Weyl group which can be reached from the given cycle by a path of fixed degree in the moment graph of the flag manifold, and are also Bruhat maximal with respect to this property. For finite dimensional flag manifolds, a description of the curve neighborhoods helped answer questions related to the quantum cohomology and quantum K theory rings, and ultimately about enumerative geometry of the flag manifolds. In this thesis we study the situation of the affine flag manifolds, which are infinite dimensional. We obtain explicit combinatorial formulas and recursions which characterize the curve neighborhoods for flag manifolds of affine Lie type A. Among the conclusions, we mention that, unlike in the finite dimensional case, the curve neighborhoods have more than one component, although all components have the same length. In general, calculations reflect a close connection between the curve neighborhoods and the Lie combinatorics of the affine root system, especially the imaginary roots."],"dc:description.degree":["Doctor of Philosophy"],"dc:format.medium":["ETD"],"dc:identifier.other":["vt_gsexam:21133"],"dc:identifier.uri":["http://hdl.handle.net/10919/93039"],"dc:publisher":["Virginia Tech"],"dc:rights":["In Copyright"],"dc:rights.uri":["http://rightsstatements.org/vocab/InC/1.0/"],"dc:subject":["Affine Flag Manifolds","Schubert Varieties","Curve Neighborhoods","Moment Graph","Combinatorial Curve Neighborhoods"],"dc:title":["The Combinatorial Curve Neighborhoods of Affine Flag Manifold in Type A<sub>n-1</sub><sup>(1)</sup>"],"dc:type":["Dissertation"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["doctoral"],"thesis:degree_name":["Doctor of Philosophy"],"thesis:institution_name":["Virginia Polytechnic Institute and State University"]},"updated_at":"2026-07-22T22:19:13Z"}