Group-invariant CR mappings

Grundmeier, Dusty E.

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

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

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

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

Copyright and License Information

Copyright 2011 Dusty E. Grundmeier

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