Tight frames generated by finite nonabelian groups

Abstract

Let $\cal H$ be a Hilbert space of finite dimension d, such as the finite signals ℓ ₂(d) or a space of multivariate orthogonal polynomials, and n ≥ d. There is a finite number of tight frames of n vectors for $\cal H$ 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 harmonic frames or 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 $\cal H$ 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 ℂG (via its Gramian), imposing the condition in the group algebra, and then describing the corresponding class of tight frames.