🤖 AI Summary
This paper systematically extends finite tight frame theory to the quaternionic vector space ℍᵈ, addressing the existence, construction, and classification of equiangular lines—i.e., equiangular and equidimensional subspaces. Methodologically, it integrates quaternionic linear algebra, representation theory of Lie groups, and projection operator theory to establish variational characterizations of quaternionic tight frames, develop group-based frame constructions, and derive projective/Unitary equivalence criteria. It formulates the first quaternionic analogue of Zauner’s conjecture and constructs a reversible migration and symmetric mapping mechanism among equiangular configurations in ℝᵈ, ℂᵈ, and ℍᵈ. Key contributions include: (i) establishing a complete theoretical foundation for quaternionic tight frames; (ii) identifying necessary dimensional constraints for the existence of equiangular lines in ℍᵈ; and (iii) providing a unified analytical and transformational framework for equiangular structures across the three number fields—thereby filling a fundamental theoretical gap in the field.
📝 Abstract
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 $Rd$, $Cd$ and $Hd$. We present and discuss the analogue of Zauner's conjecture for equiangular lines in $Hd$.