Tight frames generated by finite nonabelian groups

Richard Vale and Shayne Waldron

Abstract:

Let $\cH$ be a Hilbert space of finite dimension $d$, such as the finite signals $\Cd$ or a space of multivariate orthogonal polynomials, and $n\ge d$. There is a finite number of tight frames of $n$ vectors for $\cH$ which can be obtained as the orbit of a single vector under the unitary action of an abelian group $G$ (of symmetries of the frame). Each of these so called {\it harmonic frames} or {\it geometrically uniform frames} can be obtained from the character table of $G$ in a simple way. These frames are used in signal processing and information theory.

For a nonabelian group $G$ there are in general uncountably many inequivalent tight frames of $n$ vectors for $\cH$ which can be obtained as such a $G$--orbit. However, by adding an additional natural symmetry condition (which automatically holds if $G$ is abelian), we obtain a finite class of such frames which can be constructed from the character table of $G$ in a similar fashion to the harmonic frames. This is done by identifying each $G$--orbit with an element of the group algebra $\CC G$ (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.

Keywords: signal processing, information theory, finite nonabelian groups, representation theory, group matrices, tight frames, harmonic frames, geometrically uniform frames, Gramian matrix, central tight $G$--frames

Math Review Classification: Primary 20C15, 42C15, 41A63, 41A65; Secondary 65T60, 94A11, 94A12, 94A15

Length: 15 pages

Last Updated: 18 July 2007

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