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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {C}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {H}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>. We discuss what the analogue of Zauner’s conjecture for equiangular lines in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathbb {H}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">H</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> might be.</p>

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Tight frames over the quaternions and equiangular lines

  • 期刊论文

  • 发表日期：

    2025年10月22日

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  • Shayne Waldron

摘要

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 \(\mathbb {R}^d\) R d , \(\mathbb {C}^d\) C d and \(\mathbb {H}^d\) H d . We discuss what the analogue of Zauner’s conjecture for equiangular lines in \(\mathbb {H}^d\) H d might be.