Group-invariant CR mappings

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2011

Author dc:creator

• Grundmeier, Dusty E.

Contributors dc:contributor

• D'Angelo, John P.
• Tyson, Jeremy T.
• Leininger, Christopher J.
• Lebl, Jiri

• Group-Invariant CR Mappings
• Hermitian Polynomials
• mappings to hyperquadrics

Statement dc:rights

• Copyright 2011 Dusty E. Grundmeier

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/24090