IJPAM: Volume 28, No. 2 (2006)


Ken Dykema$^1$, Nate Strawn$^2$
$^{1,2}$Department of Mathematics
Texas A&M University
College Station, TX 77843-3368, USA
$^1$e-mail: Ken.Dykema@math.tamu.edu
$^2$e-mail: natestrawn@neo.tamu.edu

Abstract.We consider the space $\Fc^\Eb_{k,n}$ of all spherical tight frames of $k$ vectors in the $n$-dimensional Hilbert space $\Eb^n$ ($k>n$), for $\Eb=\Reals$ or $\Eb=\Cpx$, and its orbit space $\Gc^\Eb_{k,n}=\Fc^\Eb_{k,n}/\Oc^\Eb_n$ under the obvious action of the group $\Oc^\Eb_n$ of structure preserving transformations of $\Eb^n$. We show that the quotient map $\Fc^\Eb_{k,n}\to\Gc^\Eb_{k,n}$ is a locally trivial fiber bundle (also in the more general case of ellipsoidal tight frames) and that there is a homeomorphism $\Gc^\Eb_{k,n}\to\Gc^\Eb_{k,k-n}$. We show that $\Gc^\Eb_{k,n}$ and $\Fc^\Eb_{k,n}$ are real manifolds whenever $k$ and $n$ are relatively prime, and we describe them as a disjoint union of finitely many manifolds (of various dimensions) when when $k$ and $n$ have a common divisor. We also prove that $\Fc^\Reals_{k,2}$ is connected ($k\ge4$) and $\Fc^\Reals_{n+2,n}$ is connected ($n\ge2$). The spaces $\Gc^\Reals_{4,2}$ and $\Gc^\Reals_{5,2}$ are investigated in detail. The former is found to be a graph and the latter is the orientable surface of genus $25$.

Received: March 31, 2006

AMS Subject Classification: 54C50, 18F15, 55R05

Key Words and Phrases: spherical tight frames, group of structure preserving transformations, fiber bundle

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 28
Issue: 2