Week 1: Preliminaries and Motivation

Reading:

• I. Dolgachev: Lectures on Invariant Theory : Chapter 1

Topics:

• What is ETF?

Motivation

Let \(V\) be a \(K\)-vector space where \(K\) is algebraically closed.

Frame

Let \(\mathrm{Fr}_n(V) := \left\{ (v_i)_{i=1}^{n} \in V^n \;\middle|\; v_i \neq \lambda v_j \text{ for } i \neq j,\; \operatorname{span}\{v_i\} = V \right\}\) be called set of frames.

We define actions of different groups on \(\mathrm{Fr}_n(V) \times \mathrm{Fr}_n(V^*)\) to define equivalence between frames:

Group actions

• Left action of \(\mathrm{GL}(V)\): \(M\big((v_i),(\phi_i)\big) = \big((Mv_i),\; (v \mapsto \phi_i(M^{-1}v))\big)\)

• Right action of \((K^\times)^n \rtimes S_n\)
  (scaling and permuting frames):

  • Scaling: \(((v_i),(\phi_i))\lambda_i = \big((\lambda_i v_i),\; (v \mapsto \phi_i(\lambda_i^{-1}v))\big)\)

  • Permutation: \(((v_i),(\phi_i))\sigma = \big((v_{\sigma(i)}),(\phi_{\sigma(i)})\big)\)

Definition ETB

Define \(\mathrm{ETB}_n(V) \subset \mathrm{GL}(V)\backslash \big(\mathrm{Fr}_n(V)\times \mathrm{Fr}_n(V^*)\big) /\big((K^\times)^n \rtimes S_n\big)\) to be the subspace consisting of points satisfying:

1. \(\phi_i(v_i) = 1\) (similar to unit norm)
2. \(\phi_i(v_j)\phi_j(v_i) = \frac{n-d}{d(n-1)}\) for \(i \neq j\)
3. \(v_j = \frac{d}{n} \sum_\limits{i=1}^{n} \phi_i(v_j)\, v_i\) (tightness)

Big Questions

• What is \(\mathrm{ETB}_n(V)\)? Is it a variety?
• When is \(\dim \mathrm{ETB}_n(V) = 0\)?
  In that case, what is the number of points \(\#\mathrm{ETB}_n(V)\)?

Special Case

Let \(K = \mathbb{C}\) and \(\phi_i(v) = \langle v_i, v \rangle .\) Then \((v_i)\) is called an ETF (Equiangular Tight Frame).

The defining conditions become:

1. Unit norm \(\langle v_i, v_i \rangle = 1\)

2. Equiangular \(\big|\langle v_i, v_j \rangle\big|^2 = \frac{n-d}{d(n-1)}\)

3. Tight \(v = \frac{d}{n} \sum\limits_{i=1}^{n} \langle v_i, v \rangle v_i\)

Comment Thread

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