Hamiltonian Group Actions on Frame Spaces

Clayton Shonkwiler

Colorado State University

/jmm20

This talk!

Take Home Message

Symplectic geometry is a powerful set of tools which is useful for the study of frames

Frames

A frame in \(\mathbb{C}^d\) is an ordered spanning set \(\{\phi_1,\dots ,\phi_N\}\), which we often represent as a short, fat matrix

\Phi = \begin{bmatrix} \phi_1 | \phi_2 | \dots | \phi_N \end{bmatrix} \in \mathbb{C}^{d \times N}

Symplectic Geometry

Definition. A symplectic manifold is a smooth manifold \(M\) together with a closed, non-degenerate 2-form \({\omega \in \Omega^2(M)}\).

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

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

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

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

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

dx \wedge dy \left( a \frac{\partial}{\partial x} + b \frac{\partial}{\partial y}, c \frac{\partial }{\partial x} + d \frac{\partial}{\partial y} \right) = ad - bc

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

Maps

If \(H: M \to \mathbb{R}\) is smooth, then there exists a unique vector field \(X_H\) so that \({dH = \iota_{X_H}\omega}\), i.e.,

dH(\cdot) = \omega(X_H, \cdot)

(\(X_H\) is called the Hamiltonian vector field for \(H\), or sometimes the symplectic gradient of \(H\))

Example. \(H: (S^2, d\theta\wedge dz) \to \mathbb{R}\) given by \(H(\theta,z) = z\).

\(dH = dz = \iota_{\frac{\partial}{\partial \theta}}(d\theta\wedge dz)\), so \(X_H = \frac{\partial}{\partial \theta}\).

Lie Group Actions

Let \(G\) be a Lie group, and let \(\mathfrak{g}\) be its Lie algebra. If \(G\) acts on \((M,\omega)\), then each \(V \in \mathfrak{g}\) determines a vector field \(X_V\) on \(M\) by

X_V(p) = \left.\frac{d}{dt}\right|_{t=0}\exp(t V) \cdot p

\(S^1=U(1)\) acts on \((S^2,d\theta \wedge dz)\) by

e^{it} \cdot(\theta, z) = (\theta + t, z)

For \(r \in \mathbb{R} \simeq \mathfrak{u}(1)\), \(X_r = r \frac{\partial}{\partial \theta}\).

Momentum Maps

Definition. An action of \(U(1)\) on \((M,\omega)\) is Hamiltonian if there exists a map

\mu: M \to \mathfrak{u}(1)^* \simeq \mathbb{R}

so that \(d\mu = \iota_{X}\omega\), where \(X\) is the vector field generated by the circle action.

\(X = \frac{\partial}{\partial \theta}\)

\(\iota_X\omega = \iota_{\frac{\partial}{\partial \theta}} d\theta \wedge dz = dz \)

\(\mu(\theta,z) = z\)

Momentum Maps

Definition. An action of \(G\) on \((M,\omega)\) is Hamiltonian if each one-parameter subgroup action is Hamiltonian. Equivalently, there exists a map

\mu: M \to \mathfrak{g}^*

so that \(\omega_p(X_V, X) = D_p \mu(X)(V)\) for each \(p \in M\), \(X \in T_pM\), and \(V \in \mathfrak{g}\).

\(X_{V_{(a,b,c)}}(x,y,z) = (a,b,c) \times (x,y,z)\)

\((\iota_{X_{V_{(a,b,c)}}}\omega)_{(x,y,z)} = a dx + b dy + c dz \)

\(\mu(x,y,z)(V_{(a,b,c)}) = (x,y,z)\cdot(a,b,c)\)

Group Actions on Frames

What compact Lie groups act nicely on \(\mathbb{C}^{d \times N}\)?

\(U(d)\) acts on the left
\(U(N)\) acts on the right
\(U(1)^d\) acts on the left
\(U(1)^N\) acts on the right

\mu_{U(d)}(\Phi) = \Phi \Phi^*

\mu_{U(N)}(\Phi) = -\Phi^*\Phi

\mu_{U(1)^d}\left(\begin{bmatrix} \rule[.8mm]{4mm}{.5px}\, \phi^1 \rule[.8mm]{4mm}{.5px}\\ \vdots \\ \rule[.8mm]{4mm}{.5px}\, \phi^d \rule[.8mm]{4mm}{.5px} \end{bmatrix}\right) = \left(\frac{1}{2}\|\phi^1\|^2 , \dots , \frac{1}{2}\|\phi^d\|^2\right)

\mu_{U(1)^N} \left(\begin{bmatrix} \phi_1 | \cdots | \phi_N \end{bmatrix}\right) = \left(-\frac{1}{2}\|\phi_1\|^2 , \dots , -\frac{1}{2}\|\phi_N\|^2\right)

Parseval frames

\(\mu_{U(d)}^{-1}(I_d)\)

unit-norm frames

\(\mu_{U(1)^N}^{-1}\left(-\frac{1}{2},\dots , -\frac{1}{2}\right)\)

FUNTFs

The (finite) unit-norm tight frames (FUNTFs) are the level set

Let \(\mu\) be the momentum map of the product group

U(d) \times U(1)^N

\mu^{-1}\left(\frac{N}{d} I_d, \left(-\frac{1}{2},\dots , -\frac{1}{2}\right)\right)= \mu^{-1}\left(\frac{N}{d}I_d, -\vec{\frac{1}{2}}\right)

The Generalized Frame Homotopy Conjecture

Theorem [Cahill–Mixon–Strawn ’17]

The space of length-\(N\) FUNTFs in \(\mathbb{C}^d\) is path-connected for all \(N \geq d\geq 1\).

is either empty or path-connected.

and

\|\phi_i\|^2 = r_i

\Phi \Phi^* = S

Theorem [with Needham]

For any invertible, Hermitian matrix \(S\) and any \(r_1, \dots , r_N \geq 0\), the space of frames \(\Phi = [\phi_1 | \cdots | \phi_N]\) with

Eigensteps

\begin{array}{ccccccccccc} \frac{6}{3}&{}&\frac{6}{3}&{}&\frac{6}{3}&{}&0&{}&0&{}&0\\ {}&\lambda_{51}&{}&\lambda_{52}&{}&\lambda_{53}&{}&0&{}&0&{}\\ {}&{}&\lambda_{41}&{}&\lambda_{42}&{}&\lambda_{43}&{}&0&{}&{}\\ {}&{}&{}&\lambda_{31}&{}&\lambda_{32}&{}&\lambda_{33}&{}&{}&{}\\ {}&{}&{}&{}&\lambda_{21}&{}&\lambda_{22}&{}&{}&{}&{}\\ {}&{}&{}&{}&{}&1&{}&{}&{}&{}&{} \end{array}

Example for FUNTFs with \(N=6, d=3\):

\begin{array}{c} \sum_j \lambda_{6j} = 6\\ \sum_j \lambda_{5j} = 5\\ \sum_j \lambda_{4j} = 4\\ \sum_j \lambda_{3j} = 3\\ \sum_j \lambda_{2j} = 2\\ {} \end{array}

The eigensteps of a frame \(\Phi = [\phi_1 | \dots | \phi_N]\) are the eigenvalues \(\lambda_{ij}\) of the partial frame operators

\Phi_i := \phi_1\phi_1^* + \dots + \phi_i \phi_i^*.

Must satisfy \(\ldots \geq \lambda_{i,j} \geq \lambda_{i-1,j} \geq \lambda_{i,j+1} \geq \dots\)

Proposition [with Needham; cf. Flaschka–Millson]

The eigensteps of a FUNTF are coordinates of the moment map associated to a maximal torus action on (an open, dense subset of a quotient of) FUNTF space.

Corollary [using Duistermaat–Heckman]

The eigensteps of uniform random FUNTFs are distributed according to Lebesgue measure on the eigenstep polytope.