{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/24090"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/24090","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Group-invariant CR mappings","abstract":"We consider group-invariant CR mappings from spheres to hyperquadrics. Given a finite subgroup $\\Gamma \\subset U(n)$, a construction of D'Angelo and Lichtblau yields a target hyperquadric $Q(\\Gamma)$ and a canonical non-constant CR map $h_{\\Gamma} : S^{2n-1}/\\Gamma \\to Q(\\Gamma)$. For every $\\Gamma \\subset SU(2)$, we determine this hyperquadric $Q(\\Gamma)$, that is, the numbers of positive and negative eigenvalues in its defining equation. For families of cyclic and dihedral subgroups of $U(2)$, we study these numbers asymptotically as the order of the group tends to infinity. Next we study number-theoretic and combinatorial aspects of $h_{\\Gamma}$ for cyclic $\\Gamma \\subset U(2)$. In particular, we show that the mappings $h_{\\Gamma}$ associated to the lens spaces $L(p,q)$ satisfy a linear recurrence relation of order $2^q-1$ and no smaller. We also give explicit but complicated formulas for the coefficients. Finally, we explore connections with representation theory and invariant theory.","abstract_html":"We consider group-invariant CR mappings from spheres to hyperquadrics. Given a finite subgroup $\\Gamma \\subset U(n)$, a construction of D&#x27;Angelo and Lichtblau yields a target hyperquadric $Q(\\Gamma)$ and a canonical non-constant CR map <span class=\"etd-inline-math\">h<sub>\\Gamma</sub> : S<sup>2n-1</sup>/\\Gamma \\to Q(\\Gamma)</span>. For every $\\Gamma \\subset SU(2)$, we determine this hyperquadric $Q(\\Gamma)$, that is, the numbers of positive and negative eigenvalues in its defining equation. For families of cyclic and dihedral subgroups of $U(2)$, we study these numbers asymptotically as the order of the group tends to infinity. Next we study number-theoretic and combinatorial aspects of <span class=\"etd-inline-math\">h<sub>\\Gamma</sub></span> for cyclic $\\Gamma \\subset U(2)$. In particular, we show that the mappings <span class=\"etd-inline-math\">h<sub>\\Gamma</sub></span> associated to the lens spaces $L(p,q)$ satisfy a linear recurrence relation of order <span class=\"etd-inline-math\">2<sup>q</sup>-1</span> and no smaller. We also give explicit but complicated formulas for the coefficients. Finally, we explore connections with representation theory and invariant theory.","abstract_has_math":true,"creators":["Grundmeier, Dusty E."],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["D'Angelo, John P.","Tyson, Jeremy T.","Leininger, Christopher J.","Lebl, Jiri"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-25T15:05:03Z","date_published":"2011-05-25T15:05:03Z","updated_at":"2026-07-22T22:25:23Z","subjects":["Group-Invariant CR Mappings","Hermitian Polynomials","mappings to hyperquadrics"],"languages":["en"],"rights":["Copyright 2011 Dusty E. Grundmeier"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/24090","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["D'Angelo, John P.","Tyson, Jeremy T.","Leininger, Christopher J.","Lebl, Jiri"]},{"key":"dc:creator","label":"Author","values":["Grundmeier, Dusty E."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-25T15:05:03Z","2011-05"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Group-Invariant CR Mappings","Hermitian Polynomials","mappings to hyperquadrics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2011 Dusty E. 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In particular, we show that the mappings $h_{\\Gamma}$ associated to the lens spaces $L(p,q)$ satisfy a linear recurrence relation of order $2^q-1$ and no smaller. We also give explicit but complicated formulas for the coefficients. Finally, we explore connections with representation theory and invariant theory.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2011-04-12T21:03:19Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 4 sourcecode.tex: 1236 bytes, checksum: 00af13ea4f8f71a4b923751e87006cbd (MD5) thesisrefs.bib: 5413 bytes, checksum: ca7da7aa0bf510f1e2b9fc4fe5838572 (MD5) grundmeierthesis.tex: 146006 bytes, checksum: 8308d934a3623a103d06f35b530a617b (MD5) Grundmeier_Dusty.pdf: 642581 bytes, checksum: bad7c48d0d176247b47c2368e05775d6 (MD5)","Made available in DSpace on 2011-05-25T15:05:03Z (GMT). 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Given a finite subgroup $\\Gamma \\subset U(n)$, a construction of D'Angelo and Lichtblau yields a target hyperquadric $Q(\\Gamma)$ and a canonical non-constant CR map $h_{\\Gamma} : S^{2n-1}/\\Gamma \\to Q(\\Gamma)$. For every $\\Gamma \\subset SU(2)$, we determine this hyperquadric $Q(\\Gamma)$, that is, the numbers of positive and negative eigenvalues in its defining equation. For families of cyclic and dihedral subgroups of $U(2)$, we study these numbers asymptotically as the order of the group tends to infinity. Next we study number-theoretic and combinatorial aspects of $h_{\\Gamma}$ for cyclic $\\Gamma \\subset U(2)$. In particular, we show that the mappings $h_{\\Gamma}$ associated to the lens spaces $L(p,q)$ satisfy a linear recurrence relation of order $2^q-1$ and no smaller. We also give explicit but complicated formulas for the coefficients. 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