Tight frames over the quaternions and equiangular lines

Shayne Waldron


We show that much of the theory of finite tight frames can be generalised to vector spaces over the quaternions. This includes the variational characterisation, group frames, and the characterisations of projective and unitary equivalence. We are particularly interested in sets of equiangular lines (equi-isoclinic subspaces) and the groups associated with them, and how to move them between the spaces $\R^d$, $\C^d$ and $\H^d$. We present and discuss the analogue of Zauner's conjecture for equiangular lines in $\H^d$.

Keywords: finite tight frames, quaternionic equiangular lines, equi-isoclinic subspaces, equichordal subspaces, projective unitary equivalence over the quaternions, group frames, quaternionic reflection groups, double cover of $A_6$.

Math Review Classification: Primary 05B30, 15B33, 42C15, 51M20; Secondary 15B57, 51M15, 65D30, 94A12.

Length: 34 pages

Last Updated: 11 June 2020