Functional van Lint-Seidel Relative and Gerzon Universal Bounds

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Functional van Lint-Seidel Relative and Gerzon Universal Bounds

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K. Mahesh Krishna^*

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Submitted:

12 August 2024

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12 August 2024

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Abstract

We introduce the notion of functional equiangular lines in finite rank modules over subrings of $\mathbb{R}$. We show that van Lint-Seidel relative and Gerzon universal bounds hold for this most general lines.

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 52A40; 05C50

1. Introduction

Let

d \in N

and

γ \in [0, 1]

. Recall that a collection

{τ_{j}}_{j = 1}^{n}

of unit vectors in

R^{d}

is said to be $γ$ -equiangular lines [1,2] if

\begin{matrix} | 〈 τ_{j}, τ_{k} 〉 | = γ, \forall 1 \leq j, k \leq n, j \neq k . \end{matrix}

A fundamental problem associated with equiangular lines is the following.

Problem 1.1.

Given

d \in N

and

γ \in [0, 1]

, what is the upper bound on n such that there exists a collection

{τ_{j}}_{j = 1}^{n}

of γ-equiangular lines in

R^{d}

Two answers to Problem (1.1) which are driving forces in the study of equiangular lines is the following relative bound of van Lint and Seidel [2,3] and universal bound of Gerzon [4].

Theorem 1.2.

[2,3] (van Lint-Seidel Relative Bound) Let

{τ_{j}}_{j = 1}^{n}

be γ-equiangular lines in

R^{d}

. Then

\begin{matrix} n (1 - d γ^{2}) \leq d (1 - γ^{2}) . \end{matrix}

In particular, if

\begin{matrix} γ < \frac{1}{\sqrt{d}}, \end{matrix}

then

\begin{matrix} n \leq \frac{d (1 - γ^{2})}{1 - d γ^{2}} . \end{matrix}

Theorem 1.3.

[4] (Gerzon Universal Bound) Let

{τ_{j}}_{j = 1}^{n}

be γ-equiangular lines in

R^{d}

. Then

\begin{matrix} n \leq \frac{d (d + 1)}{2} . \end{matrix}

The notion of functional equiangular lines is hinted in [5]. In this paper, we define it in most general form and derive functional forms of Theorems 1.2 and 1.3.

2. Functional Equiangular Lines

In the paper,

R

denotes a subring of

R

and

M

is a rank d module over

R

. By

M^{*}

we mean the module of all homomorphisms from

M

R

. Module of all homomorphisms from

M

M

is denoted by

Mor (M)

Definition 2.1.

Let

{τ_{j}}_{j = 1}^{n}

be a collection in a module

M

of rank d over

R

and

{f_{j}}_{j = 1}^{n}

be a collection in

M^{*} .

Let

γ \geq 0

. The pair

({f_{j}}_{j = 1}^{n}, {τ_{j}}_{j = 1}^{n})

is said to befunctional $γ$ -equiangularif following conditions hold.

(i): $f_{j} (τ_{j}) = 1$ for all $1 \leq j \leq n$ .
(ii): $| f_{j} (τ_{k}) f_{k} (τ_{j}) | = γ^{2}$ for all $1 \leq j, k \leq n, j \neq k .$
(iii): The operator

$\begin{matrix} S_{f, τ} : M ∋ x \mapsto \sum_{j = 1}^{n} f_{j} (x) τ_{j} \in M \end{matrix}$

is similar (through invertible operator) to a diagonal operator.

Theorem 2.2.

(Functional van Lint-Seidel Relative Bound) If

({f_{j}}_{j = 1}^{n}, {τ_{j}}_{j = 1}^{n})

is functional γ-equiangular for rank d module

M

, then

\begin{matrix} n (1 - d γ^{2}) \leq d (1 - γ^{2}) . \end{matrix}

In particular, if

\begin{matrix} γ < \frac{1}{\sqrt{d}}, \end{matrix}

then

\begin{matrix} n \leq \frac{d (1 - γ^{2})}{1 - d γ^{2}} . \end{matrix}

Proof.

Define

\begin{matrix} S_{f, τ} : M ∋ x \mapsto S_{f, τ} x \sum_{j = 1}^{n} f_{j} (x) τ_{j} \in M . \end{matrix}

Since

S_{f, τ}

is diagonalizable,

\begin{matrix} n^{2} & = {(\sum_{j = 1}^{n} f_{j} (τ_{j}))}^{2} = {(Tra (S_{f, τ}))}^{2} = {(\sum_{k = 1}^{d} λ_{k})}^{2} \\ \leq d \sum_{k = 1}^{d} λ_{k}^{2} = d Tra (S_{f, τ}^{2}) = d \sum_{j = 1}^{n} \sum_{k = 1}^{n} f_{j} (τ_{k}) f_{k} (τ_{j}) \\ = d \sum_{j = 1}^{n} f_{j} {(τ_{j})}^{2} + d \sum_{j, k = 1, j \neq k}^{n} f_{j} (τ_{k}) f_{k} (τ_{j}) = d n^{2} + d \sum_{j, k = 1, j \neq k}^{n} f_{j} (τ_{k}) f_{k} (τ_{j}) \\ \leq d n^{2} + d \sum_{j, k = 1, j \neq k}^{n} | f_{j} (τ_{k}) f_{k} (τ_{j}) | = d n^{2} + d (n^{2} - n) γ . \end{matrix}

Therefore

\begin{matrix} n \leq d n + d (n - 1) γ . \end{matrix}

□

We are unable to derive Gerzon bound for functional equiangular lines. However, we derive following Gerzon bound for functionals.

Theorem 2.3.

(Functional Gerzon Universal Bound) Let

{τ_{j}}_{j = 1}^{n}

be a collection in a module

M

of rank d over

R

and

{f_{j}}_{j = 1}^{n}

be a collection in

M^{*} .

Assume the following.

(i): $f_{j} (τ_{j}) = 1$ for all $1 \leq j \leq n$ .
(ii): There is a $γ \neq 1$ such that $f_{j} (τ_{k}) f_{k} (τ_{j}) = γ^{2}$ for all $1 \leq j, k \leq n, j \neq k .$

Then

\begin{matrix} n \leq d^{2} . \end{matrix}

Proof.

For

1 \leq j \leq n

, define

\begin{matrix} τ_{j} \otimes f_{j} : M ∋ x \mapsto (τ_{j} \otimes f_{j}) f_{j} (x) τ_{j} \in M \end{matrix}

We show that the collection

{τ_{j} \otimes f_{j}}_{j = 1}^{n}

Mor (M)

is linearly independent over

R

. Let

c_{1}, \dots, c_{n} \in R

be such that

\begin{matrix} \sum_{j = 1}^{n} c_{j} (τ_{j} \otimes f_{j}) = 0 . \end{matrix}

Let

1 \leq k \leq n

be fixed. Then previous equation gives

\begin{matrix} \sum_{j = 1}^{n} c_{j} (τ_{j} \otimes f_{j}) (t a u_{k} \otimes f_{k}) = 0 . \end{matrix}

By taking trace we get

\begin{matrix} 0 & = \sum_{j = 1}^{n} c_{j} Tra ((τ_{j} \otimes f_{j}) (τ_{k} \otimes f_{k})) = \sum_{j = 1}^{n} c_{j} f_{j} (τ_{k}) f_{k} (τ_{j}) \\ = \sum_{j = 1, j \neq k}^{n} c_{j} f_{j} (τ_{k}) f_{k} (τ_{j}) + c_{k} f_{k} {(τ_{k})}^{2} = \sum_{j = 1, j \neq k}^{n} c_{j} γ^{2} + c_{k} \\ = γ^{2} (\sum_{j = 1}^{n} c_{j} - c_{k}) + c_{k} = γ^{2} (\sum_{j = 1}^{n} c_{j}) + (1 - γ^{2}) c_{k} . \end{matrix}

Therefore

\begin{matrix} c_{k} = \frac{γ^{2}}{γ^{2} - 1} \sum_{j = 1}^{n} c_{j} = : c, \forall 1 \leq k \leq n . \end{matrix}

Now

\begin{matrix} 0 & = Tra (\sum_{j = 1}^{n} c_{j} (τ_{j} \otimes f_{j})) = Tra (\sum_{j = 1}^{n} c (τ_{j} \otimes f_{j})) \\ = \sum_{j = 1}^{n} c Tra (τ_{j} \otimes f_{j}) = \sum_{j = 1}^{n} c f_{j} (τ_{j}) = c n . \end{matrix}

Hence

c = 0

. Therefore

{τ_{j} \otimes f_{j}}_{j = 1}^{n}

is linearly independent. Since the rank of

Mor (M)

d^{2}

we must have

n \leq d^{2}

. □

References

Haantjes, J. Equilateral point-sets in elliptic two- and three-dimensional spaces. Nieuw Arch. Wiskunde (2) 1948, 22, 355–362. [Google Scholar]
Lemmens, P.W.H.; Seidel, J.J. Equiangular lines. J. Algebra 1973, 24, 494–512. [Google Scholar] [CrossRef]
van Lint, J.H.; Seidel, J.J. Equilateral point sets in elliptic geometry. Indag. Math. 1966, 28, 335–348. [Google Scholar] [CrossRef]
Waldron, S.F.D. An introduction to finite tight frames; Applied and Numerical Harmonic Analysis, Birkhäuser/ Springer, New York, 2018; pp. xx+587. [CrossRef]
Krishna, K.M. Discrete and continuous Welch bounds for Banach spaces with applications. J. Class. Anal. 2023, 22, 81–111. [Google Scholar] [CrossRef]

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Functional van Lint-Seidel Relative and Gerzon Universal Bounds

Abstract

1. Introduction

2. Functional Equiangular Lines

References

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