Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Maximal determinants of matrices over the roots of unity

Guillermo Nuñez Ponasso

TL;DR

The paper extends Hadamard's maximal determinant problem to matrices with entries among the $\ell$-th roots of unity, introducing the central quantity $\gamma_{\ell}(n)$ and connecting it to Butson-type Hadamard matrices $\mathrm{BH}(n,\ell)$. It develops a generalized Barba bound via the minimal sums $\sigma_{\ell}(n)$, provides equality characterizations, and offers broad determinantal lower bounds through bordered/Hadamard-type and Bush-type constructions, yielding fixed fractions of classical bounds. It gives concrete results for $\ell=3$ (Paley-core based lower bounds, small-order maximal determinants, and computational certificates) and for $\ell=4$ (Turyn morphism lifting from $\mu_2$, record matrices, and open-order data), including infinite families and explicit examples. Finally, it introduces algorithmic Gram-matrix certificate methods to certify maximal determinants at orders $n \equiv 2\pmod{3}$, demonstrating practical verification tools alongside new theoretical bounds and constructions with significant implications for design theory and combinatorial matrix theory.

Abstract

We study the maximum absolute value of the determinant of matrices with entries in the set of $\ell$-th roots of unity; this is a generalization of $D$-optimal designs and Hadamard's maximal determinant problem, which involves $\pm 1$ matrices. For general values of $\ell$, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases $\ell = 3$, $\ell = 4$ is similar to the classical Hadamard maximal determinant problem for matrices with entries $\pm 1$, and many techniques can be generalized. For $\ell = 3$ we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant over $μ_3$ for all orders $n < 14$. Additionally, we survey the case $\ell = 4$ and exhibit an infinite family of maximal determinant matrices over the fourth roots of unity.

Maximal determinants of matrices over the roots of unity

TL;DR

The paper extends Hadamard's maximal determinant problem to matrices with entries among the -th roots of unity, introducing the central quantity and connecting it to Butson-type Hadamard matrices . It develops a generalized Barba bound via the minimal sums , provides equality characterizations, and offers broad determinantal lower bounds through bordered/Hadamard-type and Bush-type constructions, yielding fixed fractions of classical bounds. It gives concrete results for (Paley-core based lower bounds, small-order maximal determinants, and computational certificates) and for (Turyn morphism lifting from , record matrices, and open-order data), including infinite families and explicit examples. Finally, it introduces algorithmic Gram-matrix certificate methods to certify maximal determinants at orders , demonstrating practical verification tools alongside new theoretical bounds and constructions with significant implications for design theory and combinatorial matrix theory.

Abstract

We study the maximum absolute value of the determinant of matrices with entries in the set of -th roots of unity; this is a generalization of -optimal designs and Hadamard's maximal determinant problem, which involves matrices. For general values of , we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases , is similar to the classical Hadamard maximal determinant problem for matrices with entries , and many techniques can be generalized. For we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant over for all orders . Additionally, we survey the case and exhibit an infinite family of maximal determinant matrices over the fourth roots of unity.

Paper Structure

This paper contains 10 sections, 35 theorems, 146 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. General upper and lower bounds
  3. A generalization of Barba's bound
  4. A general determinantal lower bound
  5. Lower bounds for matrices over the third roots
  6. Maximal determinant matrices of small order over the third roots
  7. Butson matrices over the third roots
  8. Barba matrices over the third roots
  9. Maximal determinant matrices at orders n ≡ 2 (mod 3)
  10. Maximal determinants of matrices over the fourth roots

Key Result

Theorem 1

If $M$ is a matrix of order $n$ with entries belonging to the complex unit disk $D=\{z\in\mathop{\mathrm{\mathbb{C}}}\nolimits:|z|\leq 1\}$, then Furthermore, equality holds above if and only if $MM^*=nI_n$.

Figures (1)

  • Figure 1: An example of a graph $\mathcal{P}(G)$ obtained from a matrix $G\in \mathcal{G}$, where $\mathcal{G}$ is a set of $2\times 2$ Hermitian matrices with entries in $\Phi=\{n,-1,-1-3\omega,2+3\omega\}$. Notice that in the rightmost column of nodes we have included the node labeled $-1$, even if this entry does not appear in the matrix $G$.

Theorems & Definitions (79)

  • Theorem 1: Hadamard's determinantal bound Hadamard-Determinants
  • Definition 1
  • Theorem 2: Barba, 1931 Barba-DetBound
  • Theorem 3: Muir-Kelvin bound; cf., Theorem 7.8.1. Horn-Johnson
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 4: Generalized Barba bound. Theorem 5.2.3 Ponasso-Thesis
  • proof
  • ...and 69 more