Grassmannians...

As in Erik’s talk, points in the Grassmannian \(\mathrm{Gr}_2(\mathbb{C}^n)\) are conformations of closed, framed polygons in \(\mathbb{R}^3\).

Think of this as a discrete, 3-dimensional version of Younes–Michor–Shah–Mumford’s shape space. Tom has developed the smooth version.

Torus Action

Scalar matrices act trivially on \(\mathrm{Gr}_2(\mathbb{C}^n)\), so there is an effective \(U(1)^{n-1}\) action, and \(\mathrm{Gr}_2(\mathbb{C}^n)/U(1)^{n-1}\) is the space of closed polygons in \(\mathbb{R}^3\).

Fixing Edgelengths

Geometrically, this choice is equivalent to choosing the lengths \(r_1,\ldots , r_n\) of the edges in the polygon.

Definition. \(\mathrm{Pol}(n,r) := \mathrm{Gr}_2(\mathbb{C}^n)/\!/\!_r U(1)^{n-1}\) is the conformation space of closed polygons in \(\mathbb{R}^3\) with edgelength vector \(r=(r_1,\ldots , r_n)\).

In polymer physics, primarily interested in equilateral polygons, i.e., the space \(\mathrm{ePol}(n) := \mathrm{Pol}(n,(1,\ldots , 1))\).

Equilateral Polygons

Symmetries

The Gelfand–Tsetlin integrable system on \(\mathrm{Gr}_2(\mathbb{C}^n)\) descends to \(n-3\) commuting symmetries on \(\mathrm{Pol}(n,r)\).

A polytope

The \((n-3)\)-dimensional  moment polytope \(\mathcal{P}_n \subset \mathbb{R}^{n-3}\) is defined by the triangle inequalities

Sampling

Theorem (with Cantarella, 2016)

The joint distribution of \(d_1,\ldots , d_{n-3}\) is uniform on \(\mathcal{P}_n\) and \(\theta_1, \ldots , \theta_{n-3}\) are each uniform on \([0,2\pi]\).

Corollary

Any algorithm for sampling the convex polytope \(\mathcal{P}_n\subseteq \mathbb{R}^{n-3}\) gives an algorithm for sampling \(\mathrm{ePol}(n)\).

A Sampling Algorithm

Introduce fake chordlengths \(d_0=1=d_{n-2}\) and make the linear change of variables

\(s_i = d_i - d_{i-1} \text{ for } 1 \leq i \leq n-2\).

Then \(\sum s_i = d_{n-2} - d_0 = 0\), so \(s_{n-2}\) is determined by \(s_1, \ldots , s_{n-3}\)

The polytope is surprisingly large!

Let \(\mathcal{C}_n \subset \mathbb{R}^{n-3}\) be determined by

\(-1 \leq s_i \leq 1, -1 \leq \sum_{i=1}^{n-3} s_i \leq 1\)

\(\sum_{j=1}^i s_j + \sum_{j=1}^{i+1}s_j \geq -1\)

Direct sampling

Action-Angle Method

• Generate \((s_1,\ldots , s_{n-3})\) uniformly on \([-1,1]^{n-3}\)
• Test whether \((s_1,\ldots , s_{n-3})\in \mathcal{C}_n\)
• Change to \((d_1, \ldots , d_{n-3})\) coordinates
• Generate dihedral angles from \(T^{n-3}\)
• Build corresponding polygon

Diagonal sampling in 3 lines of code

A random 10,000-gon

Frames

Definition. A frame in a Hilbert space \(\mathcal{H}\) is a collection \(\{f_i\}\) of elements of \(\mathcal{H}\) so that

for all \(x \in \mathcal{H}\). If \(A=B\), the frame is tight. If \(\mathcal{H} = \mathbb{R}^d\) or \(\mathbb{C}^d\), the frame is finite. If \(\|f_i\|=1\) for all \(i\), the frame is unit norm. Finite unit norm tight frames are called FUNTFs.

FUNTFs

\(\mathcal{F}^n_{\frac{n}{d} \mathbb{I}_d} =\{F \subseteq \mathbb{C}^d : T_F^*T_F=\frac{n}{d}\mathbb{I}_d\} \simeq \mathrm{St}_d(\mathbb{C}^n) \), the Stiefel manifold of orthonormal \(d\)-frames in \(\mathbb{C}^n\), and hence \(U(d)\backslash \mathcal{F}^n_{\frac{n}{d}\mathbb{I}_d} \simeq \mathrm{Gr}_d(\mathbb{C}^n)\).

If \(S\) is an invertible, positive definite, Hermitian \(d \times d\) matrix, let \(\mathcal{F}^n_S := \{F = \{f_i\}_{i=1}^n \subseteq \mathbb{C}^d : T_F^*T_F=S\}\)

FUNTFs in \(\mathbb{C}^2\)

Equilateral polygons in \(\mathbb{R}^3\) lift to FUNTFs in \(\mathbb{C}^2\)!

More generally...

Similar symplectic reasoning leads to:

Theorem (with Needham, 2018)

If \(S\) is an invertible, positive-definite, \(d \times d\) matrix, the space \(\mathcal{F}^n_S(r)\) (of length-\(n\) frames in \(\mathbb{C}^d\) with frame operator \(S\) and \(\|f_i\|=r_i\)) is path-connected.

This generalizes Cahill, Mixon, and Strawn’s (2017) resolution of the frame homotopy conjecture, which showed that the space of FUNTFs in \(\mathbb{C}^d\) is path-connected.