The geometry of the six quaternionic equiangular lines in H^2

Shayne Waldron

Abstract:

We give a simple presentation of the six quaternionic equiangular lines in H^2 as an orbit of the primitive quaternionic reflection group of order 720 (which is isomorphic to 2 · A 6 , the double cover of A 6 ). Other orbits of this group are also seen to give optimal spherical designs (packings) of 10, 15 and 20 lines in H^2 , with angles { 1/3, 2/3 }, { 1/4, 5/8 } and {0, 1/3, 2/3 }, respectively. We consider the origins of this reflection group as one of Blichfeldt’s “finite collineation groups” for lines in C^4, and general methods for finding nice systems of quaternionic lines.

Keywords: finite tight frames, quaternionic equiangular lines, equi-isoclinic subspaces, quaternionic reflection groups, representations over the quaternions, Frobenius-Schur indicator, projective spherical t-designs, special and absolute bounds on lines, double cover of A_6. AMS (MOS) Subject Classifications: primary 05B30, 15B33, 20C25, 20G20, 51M05, 51M20, secondary 15B57, 51E99, 51M15, 65D30. spherical t-designs, spherical half-designs, tight spherical designs, finite tight frames, integration rules, cubature rules, cubature rules for the sphere, numerical optimisation, Manopt software, real algebraic variety

Math Review Classification: Primary 05B30, 15B33, 20C25, 20G20, 51M05, 51M20; Secondary 15B57, 51E99, 51M15, 65D30.

Length: 23 Pages

Last Updated: 25 November 2024

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