Abstract:

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NOTE: Background in frames will not be needed! If you are curious about the subject, this is an opportunity to take an accessible bite.

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The Mercedes Benz frame, which is the typical first example of a frame which is not an orthonormal basis, is very symmetric in the following sense: it is the orbit of one vector under the group of rotations of \(\mathbb{R}^2\) by angles which are multiples of \( 2\pi/3\). More generally, one may ask: given a finite group \(G\) acting by unitaries on a finite-dimensional Hilbert space \(H\), for which vectors does the corresponding orbit form a tight frame? We will prove a characterization due to Vale and Waldron: the mass of the vector has to be correctly distributed among the different components of the decomposition of \(H\) into irreducibles (namely, proportionally to the dimension of each component), plus an extra compatibility condition relating the pieces of the vector which belong to different components.