Parseval’s Theorem

If \(f_1, \dots , f_d \in \mathfrak{H}\) is an orthonormal basis,

\(\|v\|^2 = \sum |\langle f_i, v \rangle |^2 = \|F^* v\|^2\)

for any \(v \in \mathfrak{H}\).

Even better,

\(v = \sum \langle f_i, v \rangle f_i = FF^* v \).

If \(F = [f_1 \, f_2 \dots f_N]\), the measurement vector is \(F^*v\).

\(\Leftrightarrow FF^* = \mathrm{Id}_{d \times d}\)

Parseval frame

Fact: For any \(n \geq d\), there exist Parseval frames \(f_1, \dots , f_N \in \mathfrak{H}\) so that \(\|f_i\|=\|f_j\|\) for all \( i,j\); these are the equal-norm Parseval frames (ENPs).

Unitary equivalence classes of frames have the same Gram matrix: if \(U \in O(d)\) or \(U(d)\), then

\((UF)^\ast(UF) = F^\ast U^\ast U F = F^\ast F\).

For \(\boldsymbol{\lambda} = (\lambda_1, \dots , \lambda_d, 0, \dots 0)\),

\(\mathcal{O}_{\boldsymbol{\lambda}} := \{U \operatorname{diag}(\boldsymbol{\lambda}) U^\ast | U \in U(N) \text{ or } O(N)\}\subset \mathscr{H}(N)\)

is a flag manifold.

Examples.

Examples.

The Lie bracket \([\cdot , \cdot ]\) is the matrix commutator on matrix groups: \([A,B] = AB-BA\).

For matrix groups, the adjoint action is the conjugation action.

Example. \(\mathfrak{so}(3) \simeq \left\{\begin{pmatrix} 0 & -z & y \\ z & 0 & -x \\ -y & x & 0\end{pmatrix}:x,y,z \in \mathbb{R}\right\}\simeq (\mathbb{R}^3, \times)\)

Example: \((\mathbb{R}^2,dx \wedge dy) = (\mathbb{C},\frac{i}{2}dz \wedge d\bar{z})\)

Non-Example: \((\mathbb{R}^3, dx \wedge dy)\)

For any vector \(v\), \(dx \wedge dy\left(\frac{\partial}{\partial z}, v\right) = 0\), so \(dx \wedge dy\) is degenerate.

Non-Example: \((\mathbb{R}^4, dx_1 \wedge dy_1 + y_1 dx_2 \wedge dy_2)\)

Because of the coefficient \(y_1\), this is not closed (“not divergence-free”).

\((S^2,d\theta\wedge dz)\)

\((S^2,\omega)\), where \(\omega_p(u,v) = (u \times v) \cdot p\)

\((\mathbb{R}^2,\omega)\) where \(\omega(u,v) = \langle i u, v \rangle \)

\((\mathbb{C}^n, \frac{i}{2} \sum dz_k \wedge d\overline{z}_k)\)

\((\mathbb{C}^{m \times n}, \omega)\) with \(\omega(X_1,X_2) = -\operatorname{Im} \operatorname{trace}(X_1^* X_2)\).

\((T^* \mathbb{R}^n,\sum dq_i \wedge dp_i)\)

phase space

position

momentum

Proposition. The map recording diagonal entries is the momentum map of a Hamiltonian torus action.

Theorem [Atiyah and Guillemin–Sternberg]

Momentum maps of Hamiltonian torus actions have convex image and connected level sets.

Theorem [with Needham, 2021]

The space \(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{C}^d,N}(\boldsymbol{r})\subset \mathbb{C}^{d \times N}\) of frames with \(\operatorname{spec}(FF^\ast) = (\lambda_1, \dots , \lambda_d)\) and \(\|f_i\|^2 = r_i\) for \(i=1, \dots , N\) is path-connected.

This solves the general frame homotopy problem for complex frames.

Theorem [with Needham]

Fix the spectrum of \(FF^\ast\) and fix \(\|f_1\|,\dots,\|f_N\|\). There are three possibilities:

• It is impossible for a frame to have these data
• It is impossible for a frame with these data to have full spark
• A random frame with these data will have full spark with probability 1.

Proposition/Example. If \(\boldsymbol{\lambda}\) is generic (i.e., \(\lambda_i \neq \lambda_j\) for \(i \neq j\)), then \(\mathcal{O}_{\boldsymbol{\lambda}}\subset \mathscr{H}(N)\) is isoparametric.

If \(\boldsymbol{\lambda}\) is not generic, then \(\mathcal{O}_{\boldsymbol{\lambda}}\) is parallel to an isoparametric submanifold.

If \(\xi\) is a parallel section of \(\nu(M)\), then \(M_\xi :=\{p+\xi(p):p \in M\}\) is parallel to \(M\)

Theorem [Atiyah and Guillemin–Sternberg]

Momentum maps of Hamiltonian torus actions have convex image and connected level sets.

Example. \(\theta: \mathfrak{sl}_n(\mathbb{R}) \to \mathfrak{sl}_n(\mathbb{R})\) defined by \(\theta(X) = -X^T\). This is the derivative of the involution on \(SL_n(\mathbb{R})\) given by \(U \mapsto (U^{-1})^T\).

\(E_{+1} =: \mathfrak{k}\) consists of skew-symmetric matrices

\(E_{-1} =: \mathfrak{p}\) consists of (traceless) symmetric matrices

\(\mathfrak{sl}_n(\mathbb{R}) = \mathfrak{k} \oplus \mathfrak{p}\) is an orthogonal decomposition — the Cartan decomposition

\([\mathfrak{k},\mathfrak{p}]\subseteq \mathfrak{p}\), so the adjoint action of \(SO(n) = \exp(\mathfrak{k})\) restricts to an action on \(\mathfrak{p}\)

\([\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}\), so any Lie subalgebra of \(\mathfrak{p}\) is abelian.

Theorem. \(G\) non-compact, semisimple, and \(\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}\) a Cartan decomposition with \(K = \exp(\mathfrak{k})\) and \(P = \exp(\mathfrak{p})\), then \(G = KP\) and \(T_{[I]} G/K \simeq \mathfrak{p}\).

The adjoint action of \(K\) on \(\mathfrak{p}\) is often called the isotropy representation.

Example.

\(G = SL_n(\mathbb{C})\), \(K=SU(N)\), \(\mathfrak{k} = \mathfrak{su}(N)\), \(\mathfrak{p} = \mathscr{H}_0(N)\), \(a = \operatorname{diag}(a_1, \dots , a_N)\),

\(\mathcal{O}_a = \{UaU^\ast | U \in SU(N)\}\)

\(P(\mathcal{O}_a) = \operatorname{conv}(S_N \cdot a)\) is essentially Schur–Horn.

This gives a non-symplectic proof of frame homotopy for complex frame spaces!

In particular, Terng and Mare’s theorems imply the projection of \(\mathcal{O}_a\) onto \(\mathfrak{a}\) has convex image and connected level sets.

\(z+w \boldsymbol{j} \mapsto \begin{pmatrix} z & w \\ -\overline{w} & \overline{z}\end{pmatrix}\)

\(\mathbb{H}^d\) is a right vector space over the skew-field \(\mathbb{H}\).

\(\Psi(\operatorname{Sp}(N)) \subset SU(2N)\) is the fixed point set of the involution

\(U \mapsto \Omega^\ast \overline{U} \Omega\),

where \(\Omega = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\).

Corresponding Cartan decomposition of the Lie algebra:

\(\mathfrak{su}(2N) = \mathfrak{k} \oplus \mathfrak{p} \simeq \Psi(\mathfrak{sp}(N)) \oplus \Psi(\mathscr{H}_0(N))\).

Admissibility for Quaternionic Frames [with Needham]

\(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r})\neq \emptyset\) if and only if \(\boldsymbol{r} \in \operatorname{conv}(S_N \cdot (\lambda_1, \dots , \lambda_d, 0, \dots , 0))\) (cf. Casazza–Leon).

Key Idea. Terng/Kostant convexity.

Generalized Frame Homotopy for Quaternionic Frames [with Needham]

\(\mathscr{F}_{\boldsymbol{\lambda}}^{\mathbb{H}^d,N}(\boldsymbol{r})\) is always path-connected.

Key Idea. Mare’s connectedness theorem.