Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Automorphisms of Kimura Hadamard Matrices

Santiago Barrera Acevedo, Melissa Lee

Abstract

We investigate the structure of the automorphism groups of Kimura Hadamard matrices (KHMs) constructed from dihedral groups. We identify several different types of automorphisms, and show that the automorphism group of a KHM always has a subgroup isomorphic to $D_{2k}\times Q_8$, or $C_2\times D_{2k}\times Q_8$ if it is $y$-invariant. We exhibit additional automorphisms arising from the holomorph of the dihedral group under suitable structural conditions. A comparison with known examples, including those of Kimura, Niwasaki, and matrices arising from the Shinoda--Yamada construction, reveals counterexamples to a conjecture of Ó Cathaín and suggests that no further automorphisms occur beyond those predicted by our framework.

Automorphisms of Kimura Hadamard Matrices

Abstract

We investigate the structure of the automorphism groups of Kimura Hadamard matrices (KHMs) constructed from dihedral groups. We identify several different types of automorphisms, and show that the automorphism group of a KHM always has a subgroup isomorphic to , or if it is -invariant. We exhibit additional automorphisms arising from the holomorph of the dihedral group under suitable structural conditions. A comparison with known examples, including those of Kimura, Niwasaki, and matrices arising from the Shinoda--Yamada construction, reveals counterexamples to a conjecture of Ó Cathaín and suggests that no further automorphisms occur beyond those predicted by our framework.
Paper Structure (12 sections, 14 theorems, 23 equations, 2 tables)

This paper contains 12 sections, 14 theorems, 23 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Primitive and imprimitive actions
  4. A permutation representation of $\text{Aut}(H)$
  5. Kimura HMs
  6. Automorphisms of Kimura Hadamard matrices
  7. General automorphisms of KHMs
  8. Automorphisms of KHMs arising from $\mathrm{Hol}(D_{2k})$
  9. Induced action of $\text{Aut}(H)$ on the rows of $H$
  10. Automorphisms of the known examples of KHMs
  11. Further properties of automorphisms of KHMs
  12. Open questions

Key Result

Lemma 2.1

Let $a,b,c,d\in \mathbb{Z}D_{2k}$. Then the $\pm 1$--matrices associated to $\rho(a),\rho(b),\rho(c),\rho(d)$ satisfy the conditions in Equation eq1 if and only if $\pm 1$-matrices associated to $\lambda(a),\lambda(b),\lambda(c),\lambda(d)$ satisfy the conditions in Equation eq1. Moreover, if $H_1$

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2: Kimura, Proposition 4
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • ...and 19 more