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Group-invariant CR mappings

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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
Doctoral Committee Member(s): D'Angelo, John P.; Leininger, Christopher; 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
Rights Information: Copyright 2011 Dusty E. Grundmeier
Date Available in IDEALS: 2011-05-25
Date Deposited: 2011-05

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