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 Title: Group-invariant CR mappings Author(s): Grundmeier, Dusty E. Director of Research: D'Angelo, John P. Doctoral Committee Chair(s): Tyson, Jeremy T. Doctoral Committee Member(s): D'Angelo, John P.; Leininger, Christopher J.; Lebl, Jiri Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Group-Invariant CR Mappings Hermitian Polynomials mappings to hyperquadrics 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. Issue Date: 2011-05-25 URI: http://hdl.handle.net/2142/24090 Rights Information: Copyright 2011 Dusty E. Grundmeier Date Available in IDEALS: 2011-05-25 Date Deposited: 2011-05
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