What's a Tight Group Frame?

We can lift each node from a Schreier coset graph (at left) to the full permutahedron based on different permutations of candidates in a tableau. At right, there are three unique liftings, which give us 3 vectors to span a 2 dimensional space. Each of the Schreier graphs corresponds to a clustering of certain candidates into particular positions. Thus, when the eigenvectors are lifted onto the permutahedron and the inner product is taken with the signal, it reveals how similar the data is to the given frame vector, which is interpretable.

(One minute is not long enough to capture the nice-ness of this, so please check out the full explanation on the mathematical walkthrough!)