Quaternionic MUBs in $\mathbb{H}^2$ and their reflection symmetries

Abstract

We consider the primitive quaternionic reflection groups of type P for $\mathbb{H}^2$ that are obtained from Blichfeldt’s collineation groups for $\mathbb{C}^4$ . These are seen to be intimately related to the maximal set of five quaternionic mutually unbiased bases (MUBs) in $\mathbb{H}^2$ , for which they are symmetries. From these groups, we construct other interesting sets of lines that they fix, including a new quaternionic spherical 3-design of 16 lines in $\mathbb{H}^2$ with angles {1/5,3/5}, which meets the special bound. Some interesting consequences of this investigation include finding imprimitive quaternionic reflection groups with several systems of imprimitivity, and finding a nontrivial reducible subgroup which has a continuous family of eigenvectors.