Quaternionic MUBs in H^2 and their reflection symmetries
Zachary Buckley and Shayne Waldron
Abstract:
We consider the primitive quaternionic reflection groups of type P for H^2
that are obtained from Blichfeldt’s collineation groups for C^4 .
These are seen to be intimately related to the maximal set of five quaternionic
mutually unbiased bases (MUBs) in 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 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.
Keywords:
finite tight frames, quaternionic MUBs (mutually unbiased bases), quater-
nionic reflection groups, representations over the quaternions, Frobenius-Schur indicator,
projective spherical t-designs, special and absolute bounds on lines
Math Review Classification:
Primary 05B30, 15B33, 20C25, 20F55, 20G20, 51F15;
Secondary 51M20, 65D30.
Length: 23 Pages
Last Updated: 2 September 2025
Availability: