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$k$-Homogeneous Equiangular Tight Frames

Emily J. King

Abstract

We consider geometric and combinatorial characterizations of equiangular tight frames (ETFs), with the former concerning homogeneity of the vector and line symmetry groups and the latter the matroid structure. We introduce the concept of the bender of a frame, which is the collection of short circuits, which in turn are the dependent subsets of frame vectors of minimum size. We also show that ETFs with $k$-homogeneous line symmetry groups have benders which are $k$-designs. Paley ETFs are a known class of ETFs constructed using number theory. We determine the line and vector symmetry groups of all Paley ETFs and show that they are $2$-homogeneous. We additionally characterize all $k$-homogeneous ETFs for $k\geq 3$. Finally, we revisit David Larson's AMS Memoirs \emph{Frames, Bases, and Group Representations} coauthored with Deguang Han and \emph{Wandering Vectors for Unitary Systems and Orthogonal Wavelets} coauthored with Xingde Dai with a modern eye and focus on finite-dimensional Hilbert spaces.

$k$-Homogeneous Equiangular Tight Frames

Abstract

We consider geometric and combinatorial characterizations of equiangular tight frames (ETFs), with the former concerning homogeneity of the vector and line symmetry groups and the latter the matroid structure. We introduce the concept of the bender of a frame, which is the collection of short circuits, which in turn are the dependent subsets of frame vectors of minimum size. We also show that ETFs with -homogeneous line symmetry groups have benders which are -designs. Paley ETFs are a known class of ETFs constructed using number theory. We determine the line and vector symmetry groups of all Paley ETFs and show that they are -homogeneous. We additionally characterize all -homogeneous ETFs for . Finally, we revisit David Larson's AMS Memoirs \emph{Frames, Bases, and Group Representations} coauthored with Deguang Han and \emph{Wandering Vectors for Unitary Systems and Orthogonal Wavelets} coauthored with Xingde Dai with a modern eye and focus on finite-dimensional Hilbert spaces.
Paper Structure (12 sections, 24 theorems, 70 equations)

This paper contains 12 sections, 24 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Equiangular Tight Frames
  3. The Geometric Structure of ETFs
  4. The Combinatorial Structure of ETFs
  5. Paley ETFs
  6. Finite Fields
  7. Paley ETFs over Fields of Prime Order
  8. Alternate Proof of Theorem \ref{['thm:Paleydbhm']}
  9. Paley ETFs over Fields of Prime Power Order
  10. $k$-Homogeneous ETFs
  11. Dave Larson's AMS Memoirs
  12. Problems

Key Result

Proposition 1.5

Let $\Phi = (\varphi_j)_{j=1}^n,\Psi = (\psi_j)_{j=1}^n \in \mathbb{F}^{d\times n}$ be ETFs. They are switching equivalent if and only if for $j \neq k \neq \ell \neq j$

Theorems & Definitions (67)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Proposition 1.5
  • Proposition 1.6
  • Definition 1.7
  • Definition 1.8
  • Example 1.9
  • Theorem 1.10
  • ...and 57 more