University of Illinois Urbana-Champaign

Group-invariant CR mappings

Grundmeier, Dusty E.

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Grundmeier_Dusty.pdf

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https://hdl.handle.net/2142/24090

Description

Title
Group-invariant CR mappings
Author(s)
Grundmeier, Dusty E.
Issue Date
2011-05-25T15:05:03Z
Director of Research (if dissertation) or Advisor (if thesis)
D'Angelo, John P.
Doctoral Committee Chair(s)
Tyson, Jeremy T.
Committee Member(s)
  • D'Angelo, John P.
  • Leininger, Christopher J.
  • Lebl, Jiri
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2011-05-25T15:05:03Z
Keyword(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.
Graduation Semester
2011-05
Permalink
http://hdl.handle.net/2142/24090
Copyright and License Information
Copyright 2011 Dusty E. Grundmeier

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