Suppose we look at the Heisenberg group Hd as a matrix group of upper triangular matrices over the ring ℤ/dℤ. You can even choose d to be prime if you want. A natural irrep of Hd acting on ℂd maps the group elements into the "shift" and "phase" operators, plus roots of unity. More specifically, the two natural generators map the orthonormal basis vectors from j \to j+1 mod d, and the Fourier transform of that operation, plus overall phases by roots of unity. The question is this:
Can you find a unit vector v such that |(v,Ug v)| = c for all g not in the center of Hd? One can solve for the constant: c=1/sqrt(d+1).
Numerics suggests that these vectors exist in all the dimensions < 67, hence they may exist in every dimension, but the form of the vectors contains no (obvious) hint as to how to prove this.
This problem seems extremely truculent and any help is greatly appreciated!