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Totally symmetric Grassmannian codes

Matthew Fickus, Joseph W. Iverson, John Jasper, Dustin G. Mixon

Abstract

We introduce a general technique to construct tight fusion frames with prescribed symmetries. Applying this technique with a prescription for "all the symmetries", we construct a new family of equi-isoclinic tight fusion frames (EITFFs), which consequently form optimal Grassmannian codes. By virtue of their construction, our EITFFs have the remarkable property of total symmetry: any permutation of subspaces can be achieved by an appropriate unitary.

Totally symmetric Grassmannian codes

Abstract

We introduce a general technique to construct tight fusion frames with prescribed symmetries. Applying this technique with a prescription for "all the symmetries", we construct a new family of equi-isoclinic tight fusion frames (EITFFs), which consequently form optimal Grassmannian codes. By virtue of their construction, our EITFFs have the remarkable property of total symmetry: any permutation of subspaces can be achieved by an appropriate unitary.
Paper Structure (14 sections, 10 theorems, 88 equations, 3 figures, 2 tables)

This paper contains 14 sections, 10 theorems, 88 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Group actions
  4. Subspace ensembles
  5. Representation theory
  6. Tight fusion frames from symmetry
  7. One layer
  8. Examples
  9. Multiple layers
  10. Review of representation theory of $S_n$
  11. Total symmetry: One layer
  12. Total symmetry: Multiple layers
  13. Isoclinic partitions
  14. Examples

Key Result

Theorem 3.1

Let $G$ be a finite group acting transitively on a set $X$ via $\sigma \colon G \to S_X$. Fix a base point $x_0 \in X$ with stabilizer $H \leq G$, and select a transversal $\{ t_x \}_{x\in X}$ of elements in $G$ such that $\sigma(t_x) x_0 = x$ for each $x \in X$. Next, let $\rho \colon G \to \operat

Figures (3)

  • Figure 1: (a) The Young diagram with shape $\mu := (4,2^2)$, where removable boxes are colored blue. (b) A standard tableau $T \in \operatorname{Tab}(\mu)$. (c) The action of a transposition on $T$. (d) The embedding of $T$ into $\operatorname{Tab}(\lambda)$ for $\lambda := (4,3,2) \in \mu^\uparrow$, where the $\lambda-\mu$ box is colored red. (e) Hook lengths for $\mu$, where one hook is colored green.
  • Figure 2: Illustration of the Young diagrams for cases (i)--(iii) of Theorem \ref{['thm: single layer']}, where the red box belongs to $\lambda$ but not $\mu$, and where the removable boxes of $\mu$ are colored blue.
  • Figure 3: Let $\mu$ be a partition with $m$ distinct parts. Then we label $\lfloor \mu \rfloor = \{ (k_1,l_1),\dotsc,(k_m,l_m) \}$ by descending superdiagonal order, and $\mu^\uparrow = \{ \lambda_1,\dotsc,\lambda_{m+1}\}$ by descending superdiagonal order of differences with $\mu$.

Theorems & Definitions (39)

  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm: one layer general group']}
  • Remark 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Example 3.8
  • Example 3.9
  • ...and 29 more