Tight frames generated by finite nonabelian groups

Richard Vale and Shayne Waldron


Let $cH$ be a Hilbert space of finite dimension $d$, such as the finite signals $Cd$ or a space of multivariate orthogonal polynomials, and $nge d$. 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.

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

Last Updated
18 July 2007

15 Pages

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