# Admissibility and Frame Homotopy for Quaternionic Frames

Tom Needham  Department of Mathematics, Florida State University, Tallahassee,
FL  Clayton Shonkwiler

###### Abstract

We consider the following questions: when do there exist quaternionic frames
with given frame spectrum and given frame vector norms? When such frames
exist, is it always possible to interpolate between any two while fixing their
spectra and norms? In other words, the first question is the admissibility
question for quaternionic frames and the second is a generalization of the
frame homotopy conjecture. We give complete answers to both questions. For the
first question, the existence criterion is exactly the same as in the real and
complex cases. For the second, the non-empty spaces of quaternionic frames
with specified frame spectrum and frame vector norms are always path-
connected, just as in the complex case. Our strategy for proving these results
is based on interpreting equivalence classes of frames with given frame
spectrum as adjoint orbits, which is an approach that is also well-suited to
the study of real and complex frames.

##  1 Introduction

In a finite-dimensional real or complex Hilbert space $\mathfrak{H}$, an
(ordered) _frame_ is simply a collection $f_{1},\dots,f_{N}\in\mathfrak{H}$
that spans $\mathfrak{H}$. When $N=d:=\dim\mathfrak{H}$, this is just a basis
for $\mathfrak{H}$, but when $N>d$ a frame gives a redundant representation of
a signal $v\in\mathfrak{H}$ by

$(\langle v,f_{1}\rangle,\dots,\langle v,f_{N}\rangle)$ (1)

which can be more robust to erasures and other corruption of the data than a
basis representation [14, 22, 27].

Identifying the frame with the matrix $F=[f_{1}|\dots|f_{N}]$ whose columns
are the frame vectors, the redundant representation (1) corresponds to
evaluation of the _analysis operator_ $v\mapsto F^{\ast}v$. Composition of the
analysis operator with its adjoint _synthesis operator_ $w\mapsto Fw$ gives
the _frame operator_

$v\mapsto FF^{\ast}v.$

For orthonormal bases—or more generally, and by definition, _Parseval frames_
—the frame operator is the identity and the synthesis operator provides a
simple method for reconstructing the signal $v$ from the data $F^{\ast}v$.

As in the case of Parseval frames, it is often desirable to choose frames with
a fixed spectrum of the frame operator, for example to provide optimal
reconstruction in a given noise model [22, 13, 56]. Likewise, the squared
norms $\|f_{1}\|^{2},\dots,\|f_{N}\|^{2}$ of the frame vectors are often fixed
for both practical and theoretical reasons [36, 55, 48, 12].

In other words, given vectors $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{d})$
and $\bm{r}=(r_{1},\dots,r_{N})$ of positive numbers, we are often interested
in selecting frames from the space
$\mathcal{F}^{\mathfrak{H},N}_{\bm{\lambda}}(\bm{r})$ of frames
$f_{1},\dots,f_{N}\in\mathfrak{H}$ so that $\bm{\lambda}$ is the spectrum of
the frame operator $FF^{\ast}$ and $\|f_{i}\|^{2}=r_{i}$ for $i=1,\dots,N$. In
particular, two natural questions immediately present themselves:

  1. 1.

Does there exist a frame for $\mathfrak{H}$ with prescribed data
$\bm{\lambda}$ and $\bm{r}$? In other words, is
$\mathcal{F}^{\mathfrak{H},N}_{\bm{\lambda}}(\bm{r})$ non-empty?

  2. 2.

Can we interpolate between arbitrary elements of
$\mathcal{F}^{\mathfrak{H},N}_{\bm{\lambda}}(\bm{r})$? In other words, is
$\mathcal{F}^{\mathfrak{H},N}_{\bm{\lambda}}(\bm{r})$ path-connected?

Question 1 has been completely answered by Casazza and Leon [15], who give a
simple compatibility criterion for $\bm{\lambda}$ and $\bm{r}$ which
determines whether or not
$\mathcal{F}^{\mathfrak{H},N}_{\bm{\lambda}}(\bm{r})$ is empty. Moreover, the
same criterion applies in both the real and the complex cases.

Question 2 is a generalization of the well-known _frame homotopy conjecture_ ,
which was posed by Larson in a 2002 REU and first appeared in the literature
in Dykema and Strawn’s 2006 paper [17]. This conjecture says that when
$\bm{\lambda}$ and $\bm{r}$ are constant vectors, the space
$\mathcal{F}^{\mathfrak{H},N}_{\bm{\lambda}}(\bm{r})$ is path-connected (when
considered with the natural subspace topology). Up to scale one can assume
that the constant vector $\bm{r}=(1,\dots,1)$, and hence the frames under
consideration are _unit-norm tight frames_ , so the frame homotopy conjecture
says that the space of unit-norm tight frames in $\mathfrak{H}$ is path-
connected.

The frame homotopy conjecture was proved for both real and complex frames by
Cahill, Mixon, and Strawn in a 2017 paper [6]. In previous work [44], we gave
a complete answer to Question 2 for complex frames, showing that the space
$\mathcal{F}^{\mathbb{C}^{d},N}_{\bm{\lambda}}(\bm{r})$ is always path-
connected. For real frames with nonconstant $\bm{\lambda}$ or $\bm{r}$,
Question 2 remains open.

There has been a recent flourishing of interest in _quaternionic frames_ [31,
57, 16, 19, 58, 33, 50, 54]; that is, frames in $\mathbb{H}^{d}$ for $d\geq 1$
and $\mathbb{H}$ being the 4-dimensional skew field of quaternions. Since the
group $\operatorname{Sp}(1)$ of unit quaternions is isomorphic to
$\operatorname{Spin}(3)$ (the universal cover of the rotation group
$\operatorname{SO}(3)$), the quaternions can be interpreted as a cone over
$\operatorname{Spin}(3)$, so they are well-suited to parameterizing rotations
in $\mathbb{R}^{3}$ [24]. Just as complex numbers consist of a magnitude and a
phase, quaternions consist of a magnitude and a versor, which determines a
particular 3D rotation. Consequently, vectors $v\in\mathbb{H}^{d}$ can be used
to record both magnitude and orientation information, for example in framed
curves like polymers or inflatable elastic rods [7, 25, 29, 43]. Moreover,
quaternions have become an increasingly popular tool for representing signals
in data-driven applications. For example, quaternion-valued signals have
recently been used to encode RGB images [18, 21], measurements in industrial
machinery [59], and multicomponent seismic measurements [61]; see also the
special issue of _Signal Processing_ on Hypercomplex Signal Processing [4].
This motivates the extension of signal processing techniques for classical
(i.e., real- or complex-valued) signals to quaternionic signals.

Despite the difficulties introduced by the non-commutativity of the
quaternions, it is still possible to define frames in $\mathbb{H}^{d}$, the
norms of the frame vectors, and the spectrum of the frame operator (see
Section 2.2). In other words, the space
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ of length-$N$ frames
in $\mathbb{H}^{d}$ with fixed frame spectrum $\bm{\lambda}$ and fixed frame
vector norms $\|f_{i}\|^{2}=r_{i}$ exists, and the goal of this paper is to
give complete answers to Questions 1 and 2 in this case. Specifically, we
show:

######  Theorem 1.1.

Let $N$, $d$, $\bm{\lambda}$, and $\bm{r}$ be as above, and additionally
assume that $\bm{\lambda}$ and $\bm{r}$ are sorted in non-increasing order:
$\lambda_{1}\geq\dots\geq\lambda_{d}>0$ and $r_{1}\geq\dots\geq r_{N}>0$. Then
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is non-empty if and
only if

$\sum_{i=1}^{k}r_{i}\leq\sum_{i=1}^{k}\lambda_{i}\text{ for all
}k=1,\dots,d\qquad\text{and}\qquad\sum_{i=1}^{N}r_{i}=\sum_{i=1}^{d}\lambda_{i}.$

######  Theorem 1.2.

For any $\bm{r}$ and $\bm{\lambda}$, the space
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is path-connected.

Since the empty set is trivially path-connected, the substance of Theorem 1.2
is that all of the non-empty
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ spaces are path-
connected.

Note that the condition in Theorem 1.1 is exactly the same as that given by
Casazza and Leon in the real and complex cases [15]. In other words, for given
$\bm{\lambda}$ and $\bm{r}$ either all three of
$\mathcal{F}^{\mathbb{R}^{d},N}_{\bm{\lambda}}(\bm{r})$,
$\mathcal{F}^{\mathbb{C}^{d},N}_{\bm{\lambda}}(\bm{r})$, and
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ are empty, or all
three are non-empty. Likewise, the result in Theorem 1.2 is the same as in the
complex case [44], so that
$\mathcal{F}^{\mathbb{C}^{d},N}_{\bm{\lambda}}(\bm{r})$ and
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ are both path-
connected. In particular, Theorems 1.1 and 1.2 imply that all spaces of
quaternionic unit-norm tight frames are non-empty and path-connected, so the
standard frame homotopy conjecture is true in the quaternionic setting.

Our strategy for proving the complex version of Theorem 1.2 in [44] was based
on symplectic geometry, which is extremely well-suited to the complex setting,
but not to either the real or quaternionic settings. In this paper, we instead
identify (equivalence classes of) frames with fixed frame spectrum
$\bm{\lambda}$ with an adjoint orbit of the action of the symplectic
group111The (compact) symplectic group is the quaternionic analog of the
unitary group; see Section 2.1 for the definition. $\operatorname{Sp}(N)$ on
the space of $N\times N$ Hermitian matrices, which is the natural home of the
Gram matrix $F^{\ast}F$ associated to a frame $F$. This is not special to
quaternionic frames: one can make an analogous identification for real and
complex frames, so we view this as a unifying perspective on all three classes
of frames.

The advantage is that such adjoint orbits are extremely nice geometrically. In
particular, they are motivating special cases of both Kostant’s convexity
theorem [35] and of the theory of isoparametric submanifolds [46, Chapter 6],
and the basic strategy for proving Theorems 1.1 and 1.2 is to apply those very
general tools to these specific problems.

Specifically, since $\bm{\lambda}$ and $\bm{r}$ are essentially the spectrum
and diagonal entries, respectively, of the Gram matrix $F^{\ast}F$, the real
and complex versions of Theorem 1.1 can be proved using the Schur–Horn theorem
[1, 53]. Likewise, we will prove Theorem 1.1 using a quaternionic Schur–Horn
theorem. This quaternionic version of Schur–Horn is an easy consequence of
Kostant’s convexity theorem, but its statement does not seem to be readily
accessible in the literature (though see [34, Example 8.6]). Since it may be
of some independent interest, we give a statement and proof in Theorem 3.2. On
the other hand, adjoint orbits (and, more generally, isoparametric
submanifolds) share many of the nice features of Hamiltonian manifolds [52,
39]. In particular, in this setting we have access to a version of Atiyah’s
connectedness theorem [2] (see Theorem 3.9), and with that tool in hand the
proof of Theorem 1.2 goes much as it did in the complex case.

##  2 Quaternions and Quaternionic Frames

###  2.1 Quaternions

In thinking about $\mathbb{H}^{m}$ as a vector space over $\mathbb{H}$, some
care is required. First, recall that $\mathbb{H}$ is a skew field whose
elements can be written as

$a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$

with multiplication given by the identities

$\mathbf{i}^{2}=\mathbf{j}^{2}=\mathbf{k}^{2}=\mathbf{i}\mathbf{j}\mathbf{k}=-1,$

which in particular implies $\mathbf{i}\mathbf{j}=\mathbf{k}$,
$\mathbf{j}\mathbf{i}=-\mathbf{k}$, etc. The products of distinct elements of
$\\{\mathbf{i},\mathbf{j},\mathbf{k}\\}$ are encoded in the diagram

$\mathbf{i}$$\mathbf{j}$$\mathbf{k}$

which for example says that $\mathbf{k}\mathbf{i}=\mathbf{j}$ since the
product follows arrows with their correct orientation and
$\mathbf{k}\mathbf{j}=-\mathbf{i}$ since this product reverses directions of
arrows.

Instead of thinking in terms of quadruples of real numbers, it is also often
convenient to interpret quaternions as pairs of complex numbers:

$q=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}=(a+b\mathbf{i})+(c+d\mathbf{i})\mathbf{j}.$

This makes it easy to identify an element of $\mathbb{H}$ with a $2\times 2$
complex matrix:

$z+w\mathbf{j}\mapsto\begin{bmatrix}z&w\\\
-\overline{w}&\overline{z}\end{bmatrix}.$ (2)

This map is an algebra isomorphism between $\mathbb{H}$ and the collection of
$2\times 2$ complex matrices of the given form.

The _conjugate_ of a quaternion
$q=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}\in\mathbb{H}$ is defined by

$\overline{q}=a-b\mathbf{i}-c\mathbf{j}-d\mathbf{k},$

and the _modulus_ by

$|q|=\sqrt{q\overline{q}}=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}}.$

Since quaternionic multiplication is non-commutative, we have to distinguish
between left and right vector spaces over $\mathbb{H}$. In this paper we will
follow Waldron’s conventions [58] and consider $\mathbb{H}^{m}$ as a right
vector space over $\mathbb{H}$, meaning that scalar multiplication happens on
the right. This is done so that left matrix multiplication is linear: for
$v_{1},\dots,v_{N}\in\mathbb{H}^{d}$ thought of as column vectors,
$\alpha_{1},\dots,\alpha_{N}\in\mathbb{H}$, and a matrix
$A\in\mathbb{H}^{m\times d}$, we have

$A(v_{1}\alpha_{1}+\dots+v_{N}\alpha_{N})=(Av_{1})\alpha_{1}+\dots+(Av_{N})\alpha_{N}.$

Much of linear algebra over $\mathbb{H}$ goes through as over $\mathbb{R}$ or
$\mathbb{C}$; see [60, 47] for more, but in particular recall that
$\mathbb{H}^{m}$ has a standard $\mathbb{H}$-valued Hermitian inner product
given by

$\langle v,w\rangle=\sum_{i}\overline{w}_{j}v_{j}\in\mathbb{H}$ (3)

for all $v,w\in\mathbb{H}^{m}$. The _Frobenius inner product_ on matrices
$A,B\in\mathbb{H}^{m\times k}$ is given by

$\langle A,B\rangle_{F}:=\operatorname{tr}(B^{\ast}A),$

where, as usual, $B^{\ast}$ is the conjugate transpose of the matrix $B$. This
agrees with the Hermitian inner product (3) on vectorized versions of the
matrices.

The quaternionic analog of the unitary group is usually called the _symplectic
group_ (sometimes the _compact symplectic group_) and denoted
$\operatorname{Sp}(m)$. That is, a $m\times m$ quaternionic matrix
$U\in\mathbb{H}^{m\times m}$ is symplectic if

$\langle Uv,Uw\rangle=\langle v,w\rangle$

for all $v,w\in\mathbb{H}^{m}$; equivalently, $U^{\ast}U=\mathbb{I}_{m}$, the
$m\times m$ identity matrix. $\operatorname{Sp}(m)$ is a compact, semisimple
Lie group with type-$C$ Lie algebra $\mathfrak{sp}(m)$ consisting of the skew-
Hermitian $m\times m$ quaternionic matrices; that is, those matrices $A$ so
that $A^{\ast}=-A$. A simple parameter count shows that

$\dim_{\mathbb{R}}\operatorname{Sp}(m)=\dim_{\mathbb{R}}\mathfrak{sp}(m)=3m+4\frac{m(m-1)}{2}=2m^{2}+m.$

Finally, we introduce the notation

$\mathcal{H}(m)=\\{A\in\mathbb{H}^{m\times m}:A^{\ast}=A\\}$

for the $(2m^{2}-m)$-dimensional real vector space of $m\times m$ Hermitian
quaternionic matrices, and let $\mathcal{H}_{0}(m)$ be the subspace of
traceless Hermitian matrices.

###  2.2 Quaternionic Frames

Frames in $\mathbb{H}^{d}$ are defined in the usual way [58, 33, 50, 54]: a
sequence $(f_{i})$ of vectors in $\mathbb{H}^{d}$ is a _frame_ if there exist
$0<A\leq B<\infty$ so that

$A\|v\|^{2}\leq\sum_{i}|\langle v,f_{i}\rangle|^{2}\leq B\|v\|^{2}.$ (4)

For finite collections of vectors the upper bound is automatically satisfied
with $B=\sum_{i}\|f_{i}\|^{2}$ and the lower bound is satisfied when the
$f_{i}$ span $\mathbb{H}^{d}$. In other words, a finite collection of vectors
$f_{1},\dots,f_{N}\in\mathbb{H}^{d}$ is a frame for $\mathbb{H}^{d}$ if and
only if $\\{f_{1},\dots,f_{N}\\}$ is a spanning set for $\mathbb{H}^{d}$. We
will use $\mathcal{F}^{\mathbb{H}^{d},N}$ to denote the collection of all
frames consisting of $N$ vectors in $\mathbb{H}^{d}$.

A frame is called _tight_ if we can choose $A=B$ in (4), and _Parseval_ if
$A=B=1$. We get a nice alternative characterization of these frames by
introducing some operators. To do so, we will usually identify a frame
$f_{1},\dots,f_{N}$ with the $d\times N$ matrix
$F=[f_{1}|\dots|f_{N}]\in\mathbb{H}^{d\times N}$ whose columns are the frame
vectors, and we will often just write $F\in\mathcal{F}^{\mathbb{H}^{d},N}$.
The _analysis operator_ associated to the frame is the map
$\mathbb{H}^{d}\to\mathbb{H}^{N}$ given by

$v\mapsto(\langle v,f_{1}\rangle,\dots,\langle v,f_{N}\rangle)=F^{\ast}v,$

and the _synthesis operator_ is the map $\mathbb{H}^{N}\to\mathbb{H}^{d}$
given by

$w\mapsto\sum_{i=1}^{N}f_{i}w_{i}=Fw.$

Composing these two operators one way gives the _frame operator_
$S:\mathbb{H}^{d}\to\mathbb{H}^{d}$ given by

$S(v)=\sum_{i=1}^{N}f_{i}\langle v,f_{i}\rangle=FF^{\ast}v.$

In this interpretation, a tight frame has frame operator
$S=FF^{\ast}=A\mathbb{I}_{d}$, so in particular a Parseval frame satisfies
$FF^{\ast}=\mathbb{I}_{d}$. Composing in the other way gives the _Gram matrix_
$F^{\ast}F$ whose entries are the pairwise inner products of the frame
vectors. For real and complex frames, cyclic invariance of the trace
immediately implies that
$\operatorname{tr}(FF^{\ast})=\operatorname{tr}(F^{\ast}F)$, which is an
important and frequently-used identity. Over the quaternions, the trace of a
product of matrices is not generally invariant under cyclic permutations of
terms, but the real part is:
$\operatorname{Re}(\operatorname{tr}(AB))=\operatorname{Re}(\operatorname{tr}(BA))$
for quaternionic matrices $A$ and $B$ of dimensions for which these products
make sense. Since the frame operator and Gram matrix are both Hermitian, and
hence have real eigenvalues and real trace, it follows that

$\operatorname{tr}(FF^{\ast})=\operatorname{Re}(\operatorname{tr}(FF^{\ast}))=\operatorname{Re}(\operatorname{tr}(F^{\ast}F))=\operatorname{tr}(F^{\ast}F)=\sum_{i=1}^{N}\|f_{i}\|^{2}.$
(5)

In fact, since there is a singular value decomposition for quaternionic
matrices [60, 47], the usual argument from the real and complex cases shows
that the frame operator $FF^{\ast}\in\mathcal{H}(d)$ and the Gram matrix
$F^{\ast}F\in\mathcal{H}(N)$ have the same nonzero (right) eigenvalues. In
particular:

######  Proposition 2.1.

If $F\in\mathcal{F}^{\mathbb{H}^{d},N}$ has frame operator $FF^{\ast}$ with
spectrum
$\bm{\lambda}=(\lambda_{1},\dots,\lambda_{d})\in\mathbb{R}_{+}^{d}$,222Since
the columns of $F$ are a spanning set for $\mathbb{H}^{d}$, the frame operator
$FF^{\ast}$ must be nonsingular, so we know $\lambda_{i}>0$ for all
$i=1,\dots,d$. then the Gram matrix $F^{\ast}F$ has spectrum
$\widetilde{\bm{\lambda}}:=(\lambda_{1},\dots,\lambda_{d},0,\dots,0)$.

Since we are interested in frames with specified spectrum of the frame
operator, we introduce some terminology and notation:

######  Definition 2.2.

If $F\in\mathcal{F}^{\mathbb{H}^{d},N}$, call the spectrum of $FF^{\ast}$ the
_frame spectrum_ of $F$. For given
$\bm{\lambda}=(\lambda_{1},\dots,\lambda_{d})\in\mathbb{R}_{+}^{d}$, let
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}$ be the set of all frames in
$\mathcal{F}^{\mathbb{H}^{d},N}$ with frame spectrum equal to $\bm{\lambda}$.

Since Gram matrices are positive semidefinite and the spectral decomposition
of Hermitian quaternionic matrices works as expected [60, 47], the same proof
as in the real or complex case shows that every positive semidefinite
Hermitian quaternionic matrix is the Gram matrix of a collection of vectors
which is a frame for its span, and that this collection is uniquely determined
up to the action of the symplectic group. In other words, if
$M\in\mathcal{H}(N)$ is positive semidefinite of rank $d$, then there exists
$F=[f_{1}|\dots|f_{N}]\in\mathcal{F}^{\mathbb{H}^{d},N}\subset\mathbb{H}^{d\times
N}$ so that $M=F^{\ast}F$. Notice that, for $U\in\operatorname{Sp}(d)$, the
frame $UF$ has the same Gram matrix:
$(UF)^{\ast}(UF)=F^{\ast}U^{\ast}UF=F^{\ast}F=M$. Conversely, if
$G\in\mathbb{H}^{d\times N}$ so that $G^{\ast}G=M$, then $G=UF$ for some
$U\in\operatorname{Sp}(d)$.

We summarize the above discussion in the following proposition:

######  Proposition 2.3.

$\operatorname{Sp}(d)$-equivalence classes of frames in
$\mathcal{F}^{\mathbb{H}^{d},N}$ are uniquely determined by their Gram
matrices, which consist of all rank-$d$ positive semidefinite elements of
$\mathcal{H}(N)$.

##  3 Adjoint Orbits, Convexity, and Isoparametric Submanifolds

Our next goal is to connect the story of Gram matrices to adjoint orbits and
isoparametric submanifolds. We will largely follow Mare’s discussion [39,
Example 5.4]; see [26, §III.7] for a more general exposition of Cartan
decompositions.

Thinking of a square quaternionic matrix $A\in\mathbb{H}^{m\times m}$ as

$A=Z+W\mathbf{j}$

for $Z,W\in\mathbb{C}^{m\times m}$, we can define a map
$\Psi_{m}:\mathbb{H}^{m\times m}\to\mathbb{C}^{2m\times 2m}$ analogously to
(2):

$\Psi_{m}(Z+W\mathbf{j}):=\begin{bmatrix}Z&W\\\
-\overline{W}&\overline{Z}\end{bmatrix}.$

If $A\in\mathbb{H}^{m\times m}$ is invertible, then $\Psi_{m}(A)$ is as well.
Moreover, the image $\Psi_{m}(\operatorname{Sp}(m))$ of $\operatorname{Sp}(m)$
under this map is a subgroup of $\operatorname{SU}(2m)$, the group of
$2m\times 2m$ unitary matrices with determinant 1, and in fact
$\Psi_{m}(\operatorname{Sp}(m))$ is precisely the fixed point set inside
$\operatorname{SU}(2m)$ of the involution $\sigma:\mathbb{C}^{2m\times
2m}\to\mathbb{C}^{2m\times 2m}$ given by

$\sigma(M)=\Omega^{\ast}\overline{M}\Omega,$

where $\Omega=\begin{bmatrix}0&\mathbb{I}_{m}\\\
-\mathbb{I}_{m}&0\end{bmatrix}$.

Passing to the Lie algebra gives the Cartan decomposition

$\mathfrak{su}(2m)=\mathfrak{k}\oplus\mathfrak{p},$

where $\mathfrak{k}$ is the $(+1)$-eigenspace of the linearization
$d_{\mathbb{I}_{m}}\sigma:\mathfrak{su}(2m)\to\mathfrak{su}(2m)$ and
$\mathfrak{p}$ is the $(-1)$-eigenspace.333Since
$d_{\mathbb{I}_{m}}\sigma(A)=-A^{\ast}$ is an involution, its only eigenvalues
are $\pm 1$. The subalgebra $\mathfrak{k}$ is just the Lie algebra of
$\Psi_{m}(\operatorname{Sp}(m))\simeq\operatorname{Sp}(m)$, so
$\mathfrak{k}\simeq\mathfrak{sp}(m)$, with an explicit isomorphism given by
$\Psi_{m}|_{\mathfrak{sp}(m)}:\mathfrak{sp}(m)\to\mathfrak{k}$.

Notice that $[\mathfrak{k},\mathfrak{p}]\subseteq\mathfrak{p}$, and so the
adjoint action of $\Psi_{m}(\operatorname{Sp}(m))$ on $\mathfrak{su}(2m)$
restricts to an action on $\mathfrak{p}$. At first glance this appears
slightly esoteric, but we can use $\Psi_{m}$ to relate $\mathfrak{p}$ to a
more familiar collection of matrices. Specifically,
$\Psi_{m}|_{\mathcal{H}_{0}(m)}$ gives an $\operatorname{Sp}(m)$-equivariant
linear isomorphism between the traceless Hermitian matrices
$\mathcal{H}_{0}(m)$ and $\mathfrak{p}$, and the adjoint action of
$\Psi_{m}(\operatorname{Sp}(m))$ on $\mathfrak{p}$ corresponds to the
conjugation action of $\operatorname{Sp}(m)$ on $\mathcal{H}_{0}(m)$.

Under this isomorphism, we can identify the standard maximal abelian subspace
of $\mathfrak{p}$ with the set $\mathfrak{a}\subset\mathcal{H}_{0}(m)$ of real
diagonal $m\times m$ matrices with trace $0$. The orbit
$\mathcal{O}_{\Lambda}:=\operatorname{Sp}(m)\cdot\Lambda$ of a point
$\Lambda=\operatorname{diag}(\lambda_{1},\dots,\lambda_{m})\in\mathfrak{a}$ is
simply

$\mathcal{O}_{\Lambda}=\\{U\Lambda U^{\ast}:U\in\operatorname{Sp}(m)\\},$

which is precisely the collection of Hermitian $m\times m$ quaternionic
matrices with spectrum $(\lambda_{1},\dots,\lambda_{m})$. At this point, it is
hopefully clear that we intend to use Propositions 2.1 and 2.3 to relate
$\operatorname{Sp}(d)$-equivalence classes of frames to orbits of this form.
Before doing so, let’s see what desirable features these orbits have.

Needless to say, we could have introduced these orbits without recourse to
Cartan decompositions or adjoint actions. The point of approaching things in
this slightly roundabout way, though, is that we can now easily see that
$\mathcal{O}_{\Lambda}$ fits into both _Kostant’s convexity theorem_ and the
story of so-called _isoparametric submanifolds_.

###  3.1 Kostant’s Convexity Theorem

Consider the Cartan decomposition

$\mathfrak{G}=\mathfrak{K}\oplus\mathfrak{P}$

of the Lie algebra $\mathfrak{G}$ of a semisimple Lie group $G$.444In defiance
of the usual convention, we are using capital fraktur letters here for the Lie
algebra and its subspaces, so as not to confuse the general $\mathfrak{K}$ and
$\mathfrak{P}$ discussed here with the specific $\mathfrak{k}$ and
$\mathfrak{p}$ defined above. Then, as above, $\mathfrak{K}$ is the Lie
algebra of a compact Lie subgroup $K\subset G$, and the adjoint action of $K$
on $\mathfrak{G}$ restricts to an action on $\mathfrak{P}$. Let
$\mathfrak{A}\subset\mathfrak{P}$ be a maximal abelian subspace and let
$P:\mathfrak{P}\to\mathfrak{A}$ be the orthogonal (with respect to the Killing
form) projection. The _Weyl group_ $W$ associated with the pair
$(\mathfrak{A},\mathfrak{G})$ is the finite group
$N_{K}(\mathfrak{A})/Z_{K}(\mathfrak{A})$, where $N_{K}(\mathfrak{A})$ is the
normalizer of $\mathfrak{A}$ in $K$, and $Z_{K}(\mathfrak{A})$ is its
centralizer.

Now, let $a\in\mathfrak{A}\subset\mathfrak{P}$ and let $\mathcal{O}_{a}$ be
the orbit of $a$ under the adjoint action of $K$ on $\mathfrak{P}$. Then
Kostant’s convexity theorem characterizes the image of $\mathcal{O}_{a}$ under
$P$:

######  Theorem 3.1 (Kostant [35]).

$P(\mathcal{O}_{a})=\operatorname{conv}(W\cdot a)$, the convex hull of the
Weyl orbit of $a\in\mathfrak{A}$.

In the case where $G=\operatorname{SL}_{n}(\mathbb{C})$ and
$K=\operatorname{SU}(n)$, this is essentially the Schur–Horn theorem [49, 28],
which says that the diagonal entries of Hermitian matrices with fixed spectrum
fill out the convex hull of the collection of vectors given by all possible
re-orderings of the spectrum.

With $G=\operatorname{SU}(2m)$, $K=\operatorname{Sp}(m)$, and the Cartan
decomposition $\mathfrak{su}(2m)=\mathfrak{k}\oplus\mathfrak{p}$ as above,
Theorem 3.1 implies the following quaternionic analog of the Schur–Horn
theorem (compare to [34, Example 8.6]):

######  Theorem 3.2.

Let $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{m})\in\mathbb{R}^{m}$. Let
$\mathcal{H}_{\bm{\lambda}}(m)$ be the collection of quaternionic Hermitian
$m\times m$ matrices with spectrum $\bm{\lambda}$. Let
$\Delta:\mathcal{H}(m)\to\mathbb{R}^{m}$ record the diagonal entries of a
matrix. Then

$\Delta(\mathcal{H}_{\bm{\lambda}}(m))=\operatorname{conv}(S_{m}\cdot\bm{\lambda}),$

the convex hull of the permutation orbit of $\bm{\lambda}$.

###### Proof.

First of all, if $\lambda_{1}+\dots+\lambda_{m}=0$, then
$\mathcal{H}_{\bm{\lambda}}(m)\subset\mathcal{H}_{0}(m)$ corresponds to the
orbit of $\Lambda=\operatorname{diag}(\lambda_{1},\dots,\lambda_{m})$ under
the adjoint action of $\operatorname{Sp}(m)$ on
$\mathfrak{p}\simeq\mathcal{H}_{0}(m)$. Moreover, the Weyl group associated to
$(\mathfrak{a},\mathfrak{su}(2m))$ is the symmetric group $S_{m}$ and the
orthogonal projection $\mathfrak{p}\to\mathfrak{a}$ corresponds to the
diagonal entry map $\Delta$, so Theorem 3.1 says exactly that

$\Delta(\mathcal{H}_{\bm{\lambda}}(m))=\operatorname{conv}(S_{m}\cdot\bm{\lambda}).$

If $\lambda_{1}+\dots+\lambda_{m}\neq 0$, then we just need to re-center,
which we do using the map
$\tau:\mathcal{H}_{\bm{\lambda}}(m)\to\mathcal{O}_{\Lambda}$ defined in 3.3
below. In the notation of that lemma, Theorem 3.1 implies that

$(\Delta\circ\tau)(\mathcal{H}_{\bm{\lambda}}(m))=\operatorname{conv}(t(S_{m}\cdot\bm{\lambda})).$

Since taking the convex hull is $t$-equivariant and since
$\Delta\circ\tau=t\circ\Delta$, it follows that
$\Delta(\mathcal{H}_{\bm{\lambda}}(m))=\operatorname{conv}(S_{m}\cdot\bm{\lambda})$,
as desired. ∎

######  Lemma 3.3.

Let $\bm{\lambda}=(\lambda_{1},\dots,\lambda_{m})\in\mathbb{R}^{m}$, let
$\sigma=\lambda_{1}+\dots+\lambda_{m}$, and define
$\Lambda:=\operatorname{diag}(\lambda_{1},\dots,\lambda_{m})-\frac{\sigma}{m}\mathbb{I}_{m}$.
Then the map $B\mapsto B-\frac{\sigma}{m}\mathbb{I}_{m}$ defines a
diffeomorphism $\tau:\mathcal{H}_{\bm{\lambda}}(m)\to\mathcal{O}_{\Lambda}$
which is equivariant with respect to the $\operatorname{Sp}(m)$ actions on
domain and range and makes the following diagram commute:

${{\mathcal{H}_{\bm{\lambda}}(m)}}$${{\mathcal{O}_{\Lambda}}}$${{\mathbb{R}^{m}}}$${{\mathbb{R}^{m}}}$$\scriptstyle{\tau}$$\scriptstyle{\Delta}$$\scriptstyle{t}$$\scriptstyle{\Delta}$

Here $t:\mathbb{R}^{m}\to\mathbb{R}^{m}$ is the translation map
$t(\bm{x}):=\bm{x}-\left(\frac{\sigma}{m},\dots,\frac{\sigma}{m}\right)$.

###### Proof.

First, $\tau$ is certainly well-defined and smooth with smooth inverse, so it
is a diffeomorphism.

If $B\in\mathcal{H}_{\bm{\lambda}}(m)$ and $U\in\operatorname{Sp}(d)$, then

$U\tau(B)U^{\ast}=U\left(B-\frac{\sigma}{m}\mathbb{I}_{m}\right)U^{\ast}=UBU^{\ast}-\frac{\sigma}{m}\mathbb{I}_{m}=\tau(UBU^{\ast}),$

so $\tau$ is $\operatorname{Sp}(m)$-equivariant.

Finally, for $B\in\mathcal{H}_{\bm{\lambda}}(m)$,

$(\Delta\circ\tau)(B)=\Delta\left(B-\frac{\sigma}{m}\mathbb{I}_{m}\right)=\Delta(B)-\left(\frac{\sigma}{m},\dots,\frac{\sigma}{m}\right)=(t\circ\Delta)(B),$

so the diagram commutes. ∎

###  3.2 Isoparametric Submanifolds

Isoparametric hypersurfaces were first studied by Cartan [8, 9, 10, 11, 45];
for a more modern and general introduction see [46, Chapter 6] or [51]. First,
we give the definition, the details of which will not concern us greatly:

######  Definition 3.4.

A submanifold $M$ of a Riemannian manifold $N$ is _isoparametric_ if its
normal bundle $\nu(M)$ is flat and the principal curvatures along any parallel
normal field of $M$ are constant.

Principal orbits of isotropy representations are classic examples of
isoparametric submanifolds. In the context of our story, we have:

######  Proposition 3.5 (see, e.g., [46, Example 6.5.6]).

If $\Lambda\in\mathfrak{a}$ is generic (i.e., $\lambda_{i}\neq\lambda_{j}$ for
all $i\neq j$), then $\mathcal{O}_{\Lambda}$ is an isoparametric submanifold
of $\mathcal{H}_{0}(N)$, thought of as a Riemannian manifold by taking the
Frobenius inner product on each tangent space. Moreover, the normal space to
$\mathcal{O}_{\Lambda}$ at $\Lambda$ is just $\mathfrak{a}$.

If $\Lambda$ is not generic, then $\mathcal{O}_{\Lambda}$ is parallel to an
isoparametric submanifold of $\mathcal{H}_{0}(N)$.

To explain the terminology in the previous sentence, if
$M\subset\mathbb{R}^{n}$ is isoparametric and $\xi$ is a parallel section of
the normal bundle $\nu(M)$, then $M_{\xi}:=\\{p+\xi(p):p\in M\\}$ is a
_parallel submanifold_ to $M$, and these parallel submanifolds give a singular
foliation of the ambient space. In the special case of isotropy orbits, the
orbit foliation and the parallel foliation coincide.

Isoparametric submanifolds and their parallels have some of the nice features
of symplectic manifolds admitting Hamiltonian torus actions without
necessarily being symplectic. For example, the following fundamental result in
symplectic geometry has an analog in the isoparametric setting.

######  Theorem 3.6 (Atiyah [2] and Guillemin–Sternberg [23]).

Let $(M,\omega)$ be a compact symplectic manifold admitting a Hamiltonian
action of a torus $T\simeq\operatorname{U}(1)^{n}$. Let
$\mu:M\to\mathbb{R}^{n}$ be the associated momentum map.

  * •

For any $v\in\mathbb{R}^{n}$, $\mu^{-1}(v)$ is either empty or connected.

  * •

$\mu(M)$ is the convex hull of the images of the fixed points of the
$T$-action.

######  Example 3.7.

Let $K$ be a compact Lie group of rank $n$ with Lie algebra $\mathfrak{K}$.
Let $\mathcal{O}\subset\mathfrak{K}^{\ast}$ be an orbit of the coadjoint
action of $K$ on $\mathfrak{K}^{\ast}$. This action of $K$ on $\mathcal{O}$ is
Hamiltonian with momentum map being the inclusion
$\mathcal{O}\hookrightarrow\mathfrak{K}^{\ast}$ [42, Example 5.3.11].

If $T\subset K$ is a maximal torus, then $T\simeq U(1)^{n}$ and the coadjoint
action of $T$ on $\mathcal{O}$ is also Hamiltonian, with moment map given by
the restriction of the projection
$\mathfrak{K}^{\ast}\to\mathfrak{T}^{\ast}\simeq\mathbb{R}^{n}$ induced by the
inclusion $T\hookrightarrow K$ [3, Proposition II.1.10]. Then Theorem 3.6
implies that the image of this map is convex and its non-empty level sets are
connected.

In this case, convexity actually follows from Theorem 3.1: let
$G=K_{\mathbb{C}}$ be the complexification of $K$, with Lie algebra
$\mathfrak{G}=\mathfrak{K}\oplus\mathbf{i}\mathfrak{K}$. This is a Cartan
decomposition and the dual Lie algebra $\mathfrak{K}^{\ast}$ can be identified
with the complementary subspace $\mathbf{i}\mathfrak{K}$ so that the coadjoint
action of $K$ on $\mathfrak{K}^{\ast}$ corresponds to the adjoint action of
$K$ on $\mathbf{i}\mathfrak{K}$ and
$\mathfrak{T}^{\ast}\subset\mathfrak{K}^{\ast}$ corresponds to a maximal
abelian subalgebra $\mathfrak{A}\subset\mathbf{i}\mathfrak{K}$. Hence, the
projection map that Theorem 3.1 says has convex image is just the momentum map
of the torus action. Of course, Kostant’s convexity theorem was a major
inspiration for Atiyah and Guillemin–Sternberg, whose result can be
interpreted as generalizing this case of Kostant’s theorem to arbitrary
Hamiltonian torus actions.

On the other hand, if $\mathcal{O}$ is a principal orbit, then $\mathcal{O}$
is an isoparametric submanifold of $\mathfrak{K}^{\ast}$ and the normal space
at a point can be identified with $\mathfrak{A}\simeq\mathfrak{T}^{\ast}$.
This gives yet another interpretation of the momentum map as the orthogonal
projection onto the normal space at a point. Terng showed that, under this
interpretation, the convexity part of Theorem 3.6 generalizes to arbitrary
isoparametric submanifolds:

######  Theorem 3.8 (Terng [52]).

Let $M_{\xi}\subset\mathbb{R}^{n}$ be parallel to an isoparametric submanifold
$M$. Let $p\in M$ so that $p+\xi(p)\in M_{\xi}$, let $\nu_{p}(M)$ be the
normal space to $M$ at $p$, and let $P:M_{\xi}\to\nu_{p}(M)$ be orthogonal
projection. Then $P(M_{\xi})$ is a convex polytope.

Given this, the projection map $P$ is like a momentum map for isoparametric
submanifolds and their parallels, even though these submanifolds need not be
symplectic. It is then reasonable to ask whether, as in the first part of
Theorem 3.6, the level sets of $P$ are connected. They needn’t be in general,
but Mare gave a sufficient condition for all non-empty level sets of $P$ to be
connected:

######  Theorem 3.9 (Mare [39, Theorem 1.2 and Remark 1.3(a)]).

Let $M\subset\mathbb{R}^{n}$ be an isoparametric submanifold with all
multiplicities $\geq 2$ and let $M_{\xi}$ be parallel to $M$. If $p\in M$ and
$b\in\nu_{p}(M)$ is in the image of the projection $P:M_{\xi}\to\nu_{p}(M)$,
then $P^{-1}(b)$ is connected.

See Mare’s paper for a general definition of multiplicities; in the setting of
Theorem 3.1, where we are considering orbits of the adjoint action of
$K\subset G$ on the complementary subspace $\mathfrak{P}\subset\mathfrak{G}$,
the multiplicities are the differences in dimension between a principal orbit
and subprincipal orbits [46, 30]. Recall from Proposition 3.5 that the normal
space can be identified with the maximal abelian subspace $\mathfrak{A}$.

##  4 Existence

Let $N\geq d\geq 1$ be integers and let
$\bm{\lambda}=(\lambda_{1},\dots,\lambda_{d})$ and
$\bm{r}=(r_{1},\dots,r_{N})$ be lists of positive numbers. Our main object of
interest is the collection
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ of frames
$F=[f_{1}|\dots|f_{N}]\in\mathcal{F}^{\mathbb{H}^{d},N}\subset\mathbb{H}^{d\times
N}$ so that $FF^{\ast}$ has spectrum $\bm{\lambda}$ and $\|f_{i}\|^{2}=r_{i}$
for $i=1,\dots,N$. Our first goal is to answer the question: are there any
such frames?

The analogous question for real and complex frames was answered by Casazza and
Leon [15], and the answer in the quaternionic case is essentially the same:

######  Theorem 4.1.

Let $N$, $d$, $\bm{\lambda}$, and $\bm{r}$ be as above. The space
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is non-empty if and
only if $\bm{r}\in\operatorname{conv}(S_{N}\cdot\widetilde{\bm{\lambda}})$,
the convex hull of the collection of vectors in $\mathbb{R}^{N}$ given by
permuting the entries of
$\widetilde{\bm{\lambda}}:=(\lambda_{1},\dots,\lambda_{d},0,\dots,0)$.

###### Proof.

Theorem 3.2 will be the key to the proof, so the goal is to relate
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ to a space of the form
$\mathcal{H}_{\bm{\eta}}(m)$ for some choice of $\bm{\eta}$ and $m$.

Recall from Proposition 2.3 that $\operatorname{Sp}(d)$-equivalence classes of
frames in $\mathcal{F}^{\mathbb{H}^{d},N}$ are determined by their Gram
matrices. Specifically, the space
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}/\operatorname{Sp}(d)$ of
$\operatorname{Sp}(d)$-equivalence classes of frames with frame spectrum
$\bm{\lambda}$ is diffeomorphic to the space of all possible Gram matrices of
frames in $\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}$.

Now, if $F\in\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}$, then
definitionally $FF^{\ast}$ has spectrum $\bm{\lambda}$, and hence the Gram
matrix $F^{\ast}F$ has spectrum
$\widetilde{\bm{\lambda}}=(\lambda_{1},\dots,\lambda_{d},0,\dots,0)\in\mathbb{R}^{N}$.
Conversely, any matrix in $\mathcal{H}_{\widetilde{\bm{\lambda}}}(N)$ can be
realized as the Gram matrix of a frame in
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}$, so we see that

$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}/\operatorname{Sp}(d)\simeq\mathcal{H}_{\widetilde{\bm{\lambda}}}(N).$

In turn, if
$\Delta:\mathcal{H}_{\widetilde{\bm{\lambda}}}(N)\to\mathbb{R}^{N}$ is the map
which records diagonal entries, then the level set $\Delta^{-1}(\bm{r})$ is
the collection of $\operatorname{Sp}(d)$-equivalence classes of frames with
frame spectrum $\bm{\lambda}$ and squared frame norms $\bm{r}$; that is,

$\Delta^{-1}(\bm{r})\simeq\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})/\operatorname{Sp}(d).$

This means that $\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is
non-empty if and only if $\bm{r}$ is in the image of $\Delta$. But now Theorem
3.2 tells us that the image of $\Delta$ is exactly the convex hull of
$S_{N}\cdot\widetilde{\bm{\lambda}}$, as desired. ∎

Since permuting entries doesn’t change their sum,
$\operatorname{conv}(S_{N}\cdot\widetilde{\bm{\lambda}})$ lies in the affine
hyperplane
$\\{(x_{1},\dots,x_{N}):\sum_{i=1}^{N}x_{i}=\sum_{i=1}^{d}\lambda_{i}\\}$.
Hence, $\bm{r}\in\operatorname{conv}(S_{N}\cdot\widetilde{\bm{\lambda}})$ only
if

$\sum_{i=1}^{N}r_{i}=\sum_{i=1}^{d}\lambda_{i}.$ (6)

This is just a restatement of (5), which said the frame operator and the Gram
matrix have the same trace.

In practice, the order of the frame vectors is no more than a bookkeeping
convenience, so it is no problem to permute the frame vectors. Likewise, we
can freely permute the numbers comprising the frame spectrum. In particular,
we can get a more straightforward criterion for non-emptiness of
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ by sorting
$\bm{\lambda}$ and $\bm{r}$ in non-increasing order. Specifically, if we make
the additional assumption that $\lambda_{1}\geq\dots\geq\lambda_{d}>0$ and
$r_{1}\geq\dots\geq r_{N}>0$, then we see that
$\bm{r}\in\operatorname{conv}(S_{N}\cdot\widetilde{\bm{\lambda}})$ if and only
if it satisfies (6) and

$\sum_{i=1}^{k}r_{i}\leq\sum_{i=1}^{k}\lambda_{i}\quad\text{for all
}k=1,\dots,d.$ (7)

We call the (sorted) $\bm{r}$ satisfying (6) and (7) _$\bm{\lambda}$
-admissible_. Notice that the admissibility criterion is exactly the same as
that given by Casazza and Leon in the real and complex cases [15]. Thus,
Theorem 4.1 is equivalent to Theorem 1.1, which we restate in a more compact
form:

######  Theorem 1.1.

Let $N$, $d$, $\bm{\lambda}$, and $\bm{r}$ be as above, so that $\bm{\lambda}$
and $\bm{r}$ are sorted in non-increasing order. Then
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is non-empty if and
only if $\bm{r}$ is $\bm{\lambda}$-admissible.

##  5 Connectedness

Next, we turn to the question of when
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is connected. As in
the previous section, we will focus on
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})/\operatorname{Sp}(d)$,
since the following lemma combined with the fact that $\operatorname{Sp}(d)$
is connected implies the quotient is connected if and only if
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is connected.

######  Lemma 5.1.

Let $X$ be a topological space and let $G$ be a connected topological group
acting continuously on $X$. If $X/G$ is connected, then $X$ is connected.

This lemma follows from a standard point-set topology argument (see, e.g.,
[38, Exercise 5.5]) since connectedness of $G$ implies the fibers of the
quotient map $X\to X/G$ are connected.

The strategy is to prove connectedness using Theorem 3.9, which applies to
adjoint orbits of a compact group acting on the complementary subspace of the
Lie algebra of some larger group $G$. Following the setup in Section 3, let
$G=\operatorname{SU}(2N)$, so that $K=\operatorname{Sp}(N)$ and the
complementary subspace $\mathfrak{p}\subset\mathfrak{su}(2N)$ can be
identified with $\mathcal{H}_{0}(N)$, the space of traceless Hermitian
quaternionic $N\times N$ matrices, and the standard maximal abelian subspace
of $\mathfrak{p}$ corresponds to the subset
$\mathfrak{a}\subset\mathcal{H}_{0}(N)$ of real diagonal $N\times N$ matrices
with trace 0.

We saw in the proof of Theorem 4.1 that
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})/\operatorname{Sp}(d)\simeq\mathcal{H}_{\widetilde{\bm{\lambda}}}(N)$,
the space of quaternionic Hermitian $N\times N$ matrices with spectrum
$\widetilde{\bm{\lambda}}=(\lambda_{1},\dots,\lambda_{d},0,\dots 0)$. Since
$\lambda_{1}+\dots+\lambda_{d}\neq 0$, this is not quite an orbit of the
adjoint action of $\operatorname{Sp}(N)$ on $\mathcal{H}_{0}(N)$, but it is a
simple translation of such an orbit.

Letting $\sigma:=\lambda_{1}+\dots+\lambda_{d}$ and

$\widetilde{\Lambda}:=\operatorname{diag}\left(\lambda_{1},\dots,\lambda_{d},0,\dots,0\right)-\frac{\sigma}{N}\mathbb{I}_{N}\in\mathcal{H}_{0}(N),$

define
$\tau:\mathcal{H}_{\widetilde{\bm{\lambda}}}(N)\to\mathcal{O}_{\widetilde{\Lambda}}$
by $\tau(B)=B-\frac{\sigma}{N}\mathbb{I}_{N}$.

If $\bm{r}$ is $\bm{\lambda}$-admissible, then
$\Delta^{-1}(\bm{r})\subset\mathcal{H}_{\bm{\lambda}}(N)$ is non-empty, and
hence so is

$\tau(\Delta^{-1}(\bm{r}))=\Delta^{-1}(t(\bm{r}))\subset\mathcal{O}_{\widetilde{\Lambda}},$

where $t:\mathbb{R}^{N}\to\mathbb{R}^{N}$ is defined by
$t(\bm{x}):=\bm{x}-\left(\frac{\sigma}{N},\dots,\frac{\sigma}{N}\right)$ and
the above equality follows from the fact that the diagram in 3.3 commutes. But
now

$\Delta^{-1}(t(\bm{r}))=P^{-1}(\operatorname{diag}(t(\bm{r}))),$

where $P:\mathcal{O}_{\widetilde{\Lambda}}\to\mathfrak{a}$ is the orthogonal
projection. Since $\mathcal{O}_{\widetilde{\Lambda}}$ is an isotropy
orbit555In fact, it is an example of a quaternionic flag manifold, and much is
known about its cohomology [41, 30, 40]. and hence also a parallel submanifold
to some principal orbit, and since all multiplicities in this setting are
equal to 4 [30, §3.3], connectedness of $\Delta^{-1}(t(\bm{r}))$ follows from
Theorem 3.9. Since $\tau$ is a diffeomorphism, we conclude that

$\Delta^{-1}(\bm{r})\simeq\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})/\operatorname{Sp}(d)$

is connected.

Finally, applying 5.1 shows that
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is connected whenever
it is non-empty; i.e., whenever $\bm{r}$ is $\bm{\lambda}$-admissible. In
turn, since $\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is a real
algebraic set in $\mathbb{H}^{d\times N}\simeq\mathbb{R}^{4dN}$, it is locally
path-connected (in fact, triangulable by Łojasiewicz’s triangulation theorem
[37]) so that connectivity implies path-connectivity.

Thus, we have proved Theorem 1.2, which we now restate:

######  Theorem 1.2.

For any $\bm{r}$ and $\bm{\lambda}$, the space
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is path-connected.

If we prefer to focus on the space

$\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r}):=\\{F\in\mathcal{F}^{\mathbb{H}^{d},N}:FF^{\ast}=S\text{
and }\|f_{i}\|^{2}=r_{i}\text{ for all }i=1,\dots,N\\}$

of frames with fixed frame operator (rather than fixed frame spectrum) and
fixed frame vector norms, connectivity still holds:

######  Corollary 5.2.

If $S\in\mathcal{H}(N)$ is positive definite with spectrum $\bm{\lambda}$ and
$\bm{r}\in\mathbb{R}_{+}^{N}$, the space
$\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})$ is (i) non-empty if and only if
$\bm{r}$ is $\bm{\lambda}$-admissible and (ii) path-connected.

###### Proof.

If
$\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})\subset\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$
is non-empty, then certainly $\bm{r}$ is $\bm{\lambda}$-admissible by Theorem
4.1. Conversely, if $\bm{r}$ is $\bm{\lambda}$-admissible, then
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is non-empty, so that
there exists $F\in\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$, and
$FF^{\ast}=VSV^{\ast}$ for some $V\in\operatorname{Sp}(d)$. But then
$V^{\ast}F\in\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})$, so we see that
$\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})$ is non-empty as well.

Since the empty set is trivially path-connected, the only thing to prove is
that $\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})$ is path-connected whenever
it is non-empty. In this case, we know from Theorem 1.2 that
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$ is path-connected. So
if
$F_{0},F_{1}\in\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})\subset\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$,
then there exists a path
$\widetilde{\gamma}:[0,1]\to\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$
so that $\widetilde{\gamma}(0)=F_{0}$ and $\widetilde{\gamma}(1)=F_{1}$.

Of course, the path $\widetilde{\gamma}$ need not stay in
$\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})$, but we can easily fix it up to
do so. Indeed, the frame operator

$\widetilde{\gamma}(t)\widetilde{\gamma}(t)^{\ast}=U_{t}SU_{t}^{\ast}$

determines a continuous path
$\\{U_{t}:t\in[0,1]\\}\subset\operatorname{Sp}(d)$, and hence

$\gamma(t):=U_{t}^{\ast}\widetilde{\gamma}(t)$

is a path in $\mathcal{F}^{\mathbb{H}^{d},N}_{S}(\bm{r})$ connecting $F_{0}$
and $F_{1}$. ∎

##  6 Discussion

Corollary 5.2 is the exact analog of the main theorem in our previous paper
[44] on complex frames. While our argument in the complex case was based on
symplectic geometry, the strategy employed here for proving Theorem 1.2 and
Corollary 5.2 can be adapted to the complex setting, so we view this approach
as somewhat more general.

Indeed, much of the setup goes through in the real case as well: orthogonal
equivalence classes of real frames with fixed frame spectrum correspond to
adjoint orbits by way of their Gram matrices, and hence are parallel to
isoparametric submanifolds of the space of symmetric matrices. Unfortunately,
all multiplicities in this case are equal to 1 [30, §3.2], so Theorem 3.9 does
not apply. Indeed, while some of the real frame spaces
$\mathcal{F}^{\mathbb{R}^{d},N}_{\bm{\lambda}}(\bm{r})$ are connected [6],
others are not [32] (compare with [22, Theorem 2.7]), and it seems challenging
(but interesting!) to characterize which $\bm{\lambda}$ and $\bm{r}$ lead to
which outcome. Since the language of isoparametric submanifolds provides a
common framework for understanding real, complex, and quaternionic frames,
this perspective seems ripe for further exploration.

Theorems 1.1 and 1.2 have the same statements as the corresponding results in
the complex case, which gives some reason to hope that direct translations of
other results about complex frames might give true statements about
quaternionic frames. For example, it seems likely that there is a quaternionic
extension of the eigenstep method [5] which would construct all elements of
$\mathcal{F}^{\mathbb{H}^{d},N}_{\bm{\lambda}}(\bm{r})$, and more generally
give a constructive proof of Theorem 3.2 (cf. [20]).

### Acknowledgments

We are very grateful to Augustin-Liviu Mare, Emily King, and Colin Roberts for
providing inspiration and sharing their knowledge and insight. This work was
partially supported by grants from the National Science Foundation
(DMS–2107808, Tom Needham; DMS–2107700, Clayton Shonkwiler).

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