# Phase retrieval for nilpotent groups

Hartmut Führ , Vignon Oussa  0000-0001-5718-6449 0000-0003-2686-6470 H. Führ
Lehrstuhl für Geometrie und Analysis
RWTH Aachen University
D-52056 Aachen
Germany
email<EMAIL_ADDRESS>V. Oussa
Department of Mathematics
Bridgewater State University
MA 02234
USA
email<EMAIL_ADDRESS>
###### Abstract.

We study the phase retrieval property for orbits of general irreducible
representations of nilpotent groups, for the classes of simply connected
connected Lie groups, and for finite groups. We prove by induction that in the
Lie group case, all irreducible representations do phase retrieval.

For the finite group case, we mostly focus on $p$-groups. Here our main result
states that every irreducible representation of an arbitrary $p$-group with
exponent $p$ and size $\leq p^{2+p/2}$ does phase retrieval.

Despite the fundamental differences between the two settings, our inductive
proof methods are remarkably similar.

Keywords: phase retrieval; nilpotent Lie group; finite nilpotent group;
$p$-group; group frame;

AMS Subject Classification: 42C15; 42A38; 65T50; 94A12

##  1\. Introduction

Phase retrieval originated in the Fourier-analytic treatment of questions
arising in crystallography. The phase retrieval problem for (finite) frames as
a problem in mathematical signal processing was introduced in [3], with
motivation stemming from applications such as speech recognition. Since its
introduction phase retrieval has developed into a legitimate branch of applied
mathematics, with links to diverse subjects such as harmonic analysis,
algebraic geometry, optimization, numerical analysis, etc. We refer to the
review papers [10, 8] and their extensive lists of references.

Despite the general interest in the problem, sources containing explicit
constructions of frames with guaranteed phase retrieval properties are still
relatively scarce. In this paper, we study the construction of such systems
using strongly continuous, unitary representations of locally compact groups.
Precisely, let $\pi$ be a continuous unitary representation of a locally
compact group $G$ acting in some Hilbert space $\mathcal{H}_{\pi}.$ Fixing a
vector $\eta\in\mathcal{H}_{\pi},$ we define a continuous linear map
$V_{\eta}:\mathcal{H}_{\pi}\to C(G)$ as
$V_{\eta}(\psi)(g)=\langle\psi,\pi(g)\eta\rangle.$ Next, let $\mathbb{T}$
denote the group of complex numbers of modulus one, and let
$\mathcal{H}_{\pi}/\mathbb{T}$ denote the orbit space of $\mathcal{H}$ modulo
the canonical scalar action of $\mathbb{T}$. We say that $\eta$ does phase
retrieval if the map

$\mathcal{A}_{\eta}:\mathcal{H}_{\pi}/\mathbb{T}\to
C(G)~{},~{}\mathcal{A}_{\eta}(\mathbb{T}\cdot f)(x)=|V_{\eta}f(x)|~{},x\in G$

is injective. Similarly, we say that $\pi$ does phase retrieval if there
exists a vector $\eta\in\mathcal{H}_{\pi}$ that does phase retrieval. It is
worth noting that if $\eta\in\mathcal{H}_{\pi}$ does phase retrieval for $\pi$
then the linear map $V_{\eta}$ is necessarily injective. When this is the
case, $V_{\eta}$ defines a linear bijection onto a subspace
$\mathcal{C}_{\eta}\subset C(G)$, and the phase retrieval problem is
equivalent to the statement that each $F\in\mathcal{C}_{\eta}$ can be
recovered from $|F|$ up to a scalar. This observation serves as motivation for
the name ‘phase retrieval’; the phase of $F\in\mathcal{C}_{\eta}$ can be
recovered from its modulus.

Even though the phase retrieval problem for finite frames has been studied for
over a decade, the bulk of the literature addressing this problem for group
frames has only been devoted to some specific classes of groups, such as the
finite Heisenberg groups [5, 2, 10]. The literature also contains a treatment
of the affine Lie group (also known as the ax+b group) over the reals [15, 1]
as well as the finite ax+b group defined over prime fields [4]. Recently, the
results of [5] were extended to the case of projective unitary representations
of finite abelian groups [12]. To the best of our knowledge, this is the
largest class of general group representations for which the phase retrieval
property has been studied. The present work aims to significantly expand the
class of available representations. Precisely, we show that the class of
groups admitting representations for which phase retrieval is possible
includes all simply connected, connected nilpotent Lie groups, as well as all
sufficiently small $p$-groups with exponent $p$, for an arbitrary prime $p$.

The following examples describe the group representations that are probably
understood best so far, namely the Schrödinger representations of (finite or
Lie) Heisenberg groups. These will turn out to be the cornerstones of our
general inductive approach. Throughout the paper, we let
$\mathbb{Z}_{n}=\mathbb{Z}/n\mathbb{Z}$ be the finite cyclic group of order
$n$, and denote elements of $\mathbb{Z}_{n}$ (by slight abuse of notation) by
$k=0,\ldots,n-1$.

######  Example 1.

(The finite Heisenberg group) Fix $n\in\mathbb{N}$, $n\geq 2$, and define the
associated finite Heisenberg group by

$\mathbb{H}_{n}=\mathbb{Z}_{n}^{3}$ (1)

with group law

$(k,l,m)\cdot(k^{\prime},l^{\prime},m^{\prime})=(k+k^{\prime},l+l^{\prime},m+m^{\prime}-l^{\prime}k)~{},$
(2)

and arithmetic operations taken modulo $n$. The center of this group is given
by $Z(\mathbb{H})=\\{0\\}\times\\{0\\}\times\mathbb{Z}_{n}$, and
$\mathbb{H}/Z(\mathbb{H})\cong\mathbb{Z}_{n}\times\mathbb{Z}_{n}.$ Next,
define the Schrödinger representation of $\mathbb{H}_{n}$ acting on
$\mathbb{C}^{n}$ by first introducing the translation and modulation operators
$T_{k},M_{l}$ respectively as

$(T_{k}f)(y)=f(y-k)~{},~{}(M_{l}f)(y)=e^{2\pi ily/n}f(y)~{}.$

Let $\pi(k,l,m)=e^{2\pi im/n}M_{l}T_{k}~{}.$ Explicitly,

$(\pi(k,l,m)f)(y)=e^{2\pi im/n}e^{2\pi
ily/n}f(y-k)~{},k,l,m,y\in\\{0,\ldots,n-1\\}~{}.$

This representation is known to be irreducible and also does phase retrieval
[5]. To construct a vector $\eta$ that does phase retrieval for the
representation $\pi,$ define the ambiguity function of $\eta,$ as follows:

$A_{\eta}(k,l)=\langle\eta,M_{l}T_{k}\eta\rangle~{}.$

It is shown in [5, 12] that any vector $\eta$ for which the zero set of the
matrix coefficient $A\eta$ is empty does phase retrieval.

Note that the existence of vectors with nonvanishing ambiguity function is
established in Proposition 2.1 of [5].

######  Example 2.

(The Heisenberg Lie group) Define the three-dimensional Heisenberg Lie group
$\mathbb{H}$ as $\mathbb{H}=\mathbb{R}^{3}$, with group law

$(p,q,r)(p^{\prime},q^{\prime},r^{\prime})=(p+p^{\prime},q+q^{\prime},r+r^{\prime}+pq^{\prime})~{}.$
(3)

$\mathbb{H}$ is a simply connected, connected Lie group with center
$\\{0\\}\times\\{0\\}\times\mathbb{R}\subset\mathbb{H}$. If we define the
continuous-domain translation and modulation operators as
$(T_{p}f)(y)=f(y-p)$, $(M_{q}f)(y)=e^{2\pi iqy}f(y)$, acting on $f\in
L^{2}(\mathbb{R})$, then $\pi(p,q,r)=e^{2\pi ir}M_{q}T_{p}$ defines an
irreducible, unitary representation $\pi$ of $\mathbb{H}$ on
$L^{2}(\mathbb{R}).$ The representation $\pi$ is often referred to as the
Schrödinger representation, and as its finite counterpart, it does phase
retrieval, with an analogous sufficient criterion for vectors guaranteeing
that property: Defining the ambiguity function of $\eta\in L^{2}(\mathbb{R})$
as

$A_{\eta}(p,q)=\langle\eta,\pi(p,q,0)\eta\rangle~{},$

we have that $\eta$ does phase retrieval if $A_{\eta}$ vanishes only on a set
of Lebesgue measure zero. A prominent example of a window $g$ with that
property is provided by the Gaussian.

###  1.1. Main contributions of our paper

In this paper, we intend to address the problem of phase retrieval for general
irreducible representations of nilpotent groups. Our main results in this
direction are the following.

  * •

Any irreducible representation of an arbitrary simply connected, connected
nilpotent Lie group does phase retrieval (Theorem 12).

  * •

Explicitly checkable sufficient criteria for the phase retrieval property of
irreducible representations of $p$-groups (Corollary 33).

  * •

As a corollary to the previous result: Any irreducible representation of an
arbitrary $p$-group with exponent $p$ and size $\leq p^{2+p/2}$ does phase
retrieval (Corollary 34).

The techniques we use to establish these results seem of potentially
independent interest. In particular, the treatment of the $p$-group case is
surprisingly analogous to the Lie group case, due to a $p$-group version of
Kirillov’s lemma that appears to be new (Lemma 22), and an associated explicit
construction method for irreducible representations (Lemma 26). Under suitable
assumptions on the underlying groups, this approach establishes a close
relationship between irreducible representations of $p$-groups and finite
Schrödinger representations that we systematically exploit for the study of
phase retrieval.

As a second technical novelty of potential independent interest, we introduce
the quantity $p_{0}(\pi)$ (see Definition 16), which is an upper bound on the
proportion of zeros in a nonzero matrix coefficient associated to $\pi$. Our
results estimating $p_{0}(\pi)$, specifically Lemma 32, serve as further
illustration of the usefulness of the $p$-group analog of the Kirillov lemma
and its representation-theoretic ramifications.

##  2\. Nilpotent Lie groups

This section establishes phase retrieval for irreducible representations of
simply connected, connected nilpotent Lie groups. Before we turn to groups and
representations, we first state a few basic observations and ideas regarding
phase retrieval, which will be repeatedly useful. We expect them to be well-
known, but include short arguments for lack of a handy reference.

######  Definition 3.

Let $\mathcal{H}$ denote a Hilbert space. We say that $(x_{i})_{i\in
I}\subset\mathcal{H}$ does phase retrieval if

$\forall f,g\in\mathcal{H}~{}:~{}\left(\forall i\in I:|\langle
f,x_{i}\rangle|=|\langle g,x_{i}\rangle|\right)\Leftrightarrow
f\in\mathbb{T}\cdot g~{}.$

With this definition, a unitary representation $\pi$ of a locally compact
group does phase retrieval iff there exists a vector
$\eta\in\mathcal{H}_{\pi}$ such that $(\pi(x)\eta)_{x\in G}$ does phase
retrieval. Part (b) of the following lemma has been observed e.g. in [6].

######  Lemma 4.

Let $\mathcal{H}$ denote a Hilbert space and $(x_{i})_{i\in
I}\subset\mathcal{H}$.

  1. (a)

If $(x_{i})_{i\in I}$ does phase retrieval, then it has dense span in
$\mathcal{H}$.

  2. (b)

If $(x_{i})_{i\in I}$ does phase retrieval, and
$f,g\in\mathcal{H}\setminus\\{0\\}$ are arbitrary, there exists $i\in I$ such
that

$\langle f,x_{i}\rangle\langle g,x_{i}\rangle\not=0~{}.$

  3. (c)

Let $(y_{j})_{j\in J}$ be related to $(x_{i})_{i\in I}$ by the following
property: There exists a map $\kappa:J\to I$ such that for all $j\in J$

$y_{j}\in\mathbb{T}x_{\kappa(j)}$

and in addition,

$\forall i\in I\exists j\in J~{}:~{}x_{i}\in\mathbb{T}x_{\kappa(j)}~{}.$

Then $(x_{i})_{i\in I}$ does phase retrieval iff $(y_{j})_{j\in J}$ does.

###### Proof.

Part (a) follows from the fact that any $f$ orthogonal to all $x_{i}$ must be
in $\mathbb{T}\cdot 0=\\{0\\}$.

For part (b), assuming the existence of $f,g\in\mathcal{H}\setminus\\{0\\}$
with

$\forall i\in I~{}:~{}\langle f,x_{i}\rangle\langle g,x_{i}\rangle=0~{},$

we first note that by part (a), the fact that $f,g$ are both nonzero implies
that the associated coefficient families $(\langle f,x_{i}\rangle)_{i\in I}$
and $(\langle g,x_{i}\rangle)_{i\in I}$ are. By assumption, these coefficient
families are disjointly supported. Hence they are linearly independent, and
accordingly, $f$ and $g$ are linearly independent. In particular,
$f+g\not\in\mathbb{T}(f-g)$. On the other hand, the disjointness assumption
entails for all $i\in I$

$\left|\langle f+g,x_{i}\rangle\right|=\left|\langle
f,x_{i}\rangle\right|+|\left\langle g,x_{i}\rangle\right|=\left|\langle
f-g,x_{i}\rangle\right|~{},$

which contradicts the phase retrieval property.

For part (c) we note that the assumptions on the two families allow to recover
the coefficient family $(\langle f,x_{i}\rangle)_{i\in I}$ from the
coefficient family $(\langle f,y_{j}\rangle)_{j\in J}$, and the same holds for
the families of coefficient moduli. Also, these observations remain true if
the families $(x_{i})_{i\in I}$ and $(y_{j})_{j\in J}$ are interchanged. These
facts then imply the equivalence in (c). ∎

We next turn to Hilbert space tensor products and their realization via
Hilbert-Schmidt operators. First some notation. Let
$\mathcal{H}_{1},\mathcal{H}_{2}$ denote nontrivial Hilbert spaces, and
$HS(\mathcal{H}_{2},\mathcal{H}_{1})$ the space of Hilbert-Schmidt operators
$A:\mathcal{H}_{2}\to\mathcal{H}_{1}$, endowed with the usual scalar product

$\langle A,B\rangle={\rm trace}(AB^{*})~{}.$

Given $x\in\mathcal{H}_{1},y\in\mathcal{H}_{2}$, let

$x\otimes y\in HS(\mathcal{H}_{2},\mathcal{H}_{1})~{}:~{}\mathcal{H}_{2}\ni
z\mapsto\langle z,y\rangle x~{}.$

With this terminology in place, we can formulate the following basic
observation.

######  Lemma 5.

Let $(x_{i})_{i\in I}\subset\mathcal{H}_{1}$, $(y_{j})_{j\in
J}\subset\mathcal{H}_{2}$ denote two systems of vectors with the phase
retrieval property. Then $(x_{i}\otimes y_{j})_{(i,j)\in I\times J}$ has the
phase retrieval property for $HS(\mathcal{H}_{2},\mathcal{H}_{1})$.

###### Proof.

Let $A,B\in HS(\mathcal{H}_{2},\mathcal{H}_{1})$ with

$\forall(i,j)\in I\times j~{}:~{}|\langle A,x_{i}\otimes
y_{j}\rangle|=|\langle B,x_{i}\otimes y_{j}\rangle|$

Note that the operator scalar products are computed as

$\langle A,x_{i}\otimes y_{j}\rangle=\langle Ay_{j},x_{i}\rangle~{}.$

Fixing $j\in J$, this gives for all $i\in I$:

$|\langle Ay_{j},x_{i}\rangle|=|\langle By_{j},x_{i}\rangle|~{}.$

The phase retrieval property for $(x_{i})_{i\in I}$ then entails

$Ay_{j}=v(j)By_{j}~{},~{}\forall j\in J~{},$

with suitable $v(j)\in\mathbb{T}$. We next prove that $v$ is constant on the
set

$S=\\{j\in J~{}:~{}Ay_{j}\not=0\\}~{}.$

To see this, we fix $i\in I$, and get for all $j\in J$ that

$|\langle y_{j},A^{*}x_{i}\rangle|=|\langle y_{j},B^{*}x_{i}\rangle|~{}.$

Hence the phase retrieval property of $(y_{j})_{j\in J}$ furnishes a map
$u:I\to\mathbb{T}$ with

$A^{*}x_{i}=u(i)B^{*}x_{i}~{},~{}\forall i\in I~{}.$ (4)

Now assume that $j,j^{\prime}\in S$ are given. Then there exists $i\in I$ such
that

$\langle Ay_{j},x_{i}\rangle\langle Ay_{j^{\prime}},x_{i}\rangle\not=0~{},$
(5)

by Lemma 4 (b), and definition of $S$. This entails

$\langle Ay_{j},x_{i}\rangle=v(j)\langle By_{j},x_{i}\rangle=v(j)u(i)\langle
Ay_{j},x_{i}\rangle~{},$

and the choice of $i$ allows to cancel $\langle Ay_{j},x_{i}\rangle$ from this
equation to obtain $v(j)=\overline{u(i)}$. The same reasoning can be applied
to $v(j^{\prime})$, leading to $v(j^{\prime})=\overline{u(i)}=v(j)$. Plugging
this observation into equation (4) provides $v\in\mathbb{T}$ satisfying

$\forall j\in J~{}:~{}Ay_{j}=vBy_{j}$

and Lemma 4 (a) now applies to yield $A=vB$. ∎

We now turn to groups and representations. Before we state the main result of
this section, we highlight several representation-theoretic facts relevant to
the problem at hand (see Remark 6 and Lemma 8.) Given a locally compact group
$G$, we let $\widehat{G}$ denote its unitary dual, i.e. the set of equivalence
classes of irreducible unitary representations, and $\pi\in\widehat{G}$ will
be shorthand for “$\pi$ is irreducible”. Throughout the paper, we will
repeatedly factor out suitably chosen normal subgroups. One such subgroup of
particular interest is the projective kernel of a unitary representation
$\pi$, which is given by

$K_{\pi}=\\{x\in G:\pi(x)\in\mathbb{T}\cdot{\rm
id}_{\mathcal{H}_{\pi}}\\}~{}.$

This is a closed normal subgroup of $G$ containing the kernel of $\pi$.

######  Remark 6.

Let $G$ denote a locally compact group, and let $\pi$ be a unitary
representation of $G$. Assume that $K\lhd G$ is a normal subgroup contained in
the projective kernel of $\pi$. Then $\pi$ induces a projective representation
$\overline{\pi}$ of the quotient group $G/K$, by picking a cross-section
$\sigma:G/K\to G$ and letting $\overline{\pi}(xK)=\pi(\sigma(xK))$. Now Lemma
4(c) shows that the system $(\pi(x)\eta)_{x\in G}$ does phase retrieval iff
the system $(\pi(\sigma(x))_{xK\in G/K}$ does.

This observation applies to two cases of particular interest: Firstly, and
trivially, we may always factor out normal subgroups $H$ contained in the
kernel, and pass the action down to a representation of $G/H$. Moreover, if
$\pi$ is irreducible and $Z<G$ is a central subgroup, then
$\pi(Z)\subset\mathbb{T}\cdot{\rm id}_{\mathcal{H}_{\pi}}$ follows by Schur’s
lemma. Therefore $\pi$ does phase retrieval iff the associated projective
representation of $G/Z$ does.

######  Remark 7.

At this point, let us shortly explain why we focus on irreducible
representations. This focus has a long tradition in representation theory, due
to the fact that typically, an understanding of the irreducible cases is
fundamental to an understanding of more complex representations. The general
intuition is that a decomposition into irreducible subrepresentations allows
to reduce the general problem into a family of subproblems, which can then be
addressed independently.

In the context of the phase retrieval, this intuition works to a limited
degree. If $(\pi,\mathcal{H}_{\pi})$ is a representation, and
$\mathcal{H}_{\pi}$ decomposes into two closed invariant subspaces,
$\mathcal{H}_{\pi}=\mathcal{K}_{1}\oplus\mathcal{K}_{2}$, then any vector
$\eta\in\mathcal{H}_{\pi}$ doing phase retrieval for $\pi$ gives rise to two
vectors $\eta_{1}\in\mathcal{K}_{1},\eta_{2}\in\mathcal{K}_{2}$ such that the
restriction of $\pi$ to $\mathcal{K}_{i}$ does phase retrieval for
$\mathcal{K}_{i}$, by taking $\eta_{i}$ to be the orthogonal projection of
$\eta$ onto $\mathcal{K}_{i}$.

The standard representation-theoretic intuition would suggest that this
observation also works for the converse direction, at least under suitable
assumptions on the subrepresentations associated to the $\mathcal{K}_{i}$.
Specifically, assuming that the subrepresentations are irreducible and
inequivalent, one might be tempted to expect that a vector $\eta_{i}$ doing
phase retrieval for each $\mathcal{K}_{i}$ gives rise to a vector
$\eta=\eta_{1}+\eta_{2}$ that does phase retrieval for $\mathcal{H}_{\pi}$.
This is generally false, as the following example illustrates.

Pick an irreducible representation $\sigma$ such that the contragredient
representation $\overline{\sigma}$ is inequivalent to $\sigma$. Such
representations exist, e.g. for the Heisenberg groups from Examples 1 (when
$n>2$) and 2. Assume that $\eta_{1}\in\mathcal{H}_{\sigma}$ is a vector doing
phase retrieval for $\sigma$. If we then realize $\overline{\sigma}$ on
$\mathcal{H}_{\sigma}$ in the canonical manner, the associated matrix
coefficients are related by

$V_{\eta}^{\overline{\sigma}}f=\overline{V_{\eta}^{\sigma}f}~{};$

this can be understood as the defining relation of the contragredient
representation. On the one hand, this shows that $\eta$ also does phase
retrieval for $\overline{\sigma}$. However, if we now define
$\pi=\sigma\oplus\overline{\sigma}$, and use the vector
$\eta_{0}=\eta\oplus\eta$, we get

$\left|V_{\eta_{0}}^{\pi}(f\oplus
0)\right|=\left|V^{\sigma}_{\eta}f\right|=\left|V_{\eta_{0}}^{\pi}(0\oplus
f)\right|~{},$

hence $\eta_{0}$ does not do phase retrieval.

To summarize: Understanding the phase retrieval problem for irreducible
representations will not lead to automatic solutions for arbitrary
representations, but it remains a fundamental first step, that we address in
the following for nilpotent groups.

We next need a representation-theoretic lemma that is quite probably folklore.
It can be proved by direct calculations using the realization of induced
representations via cross-sections, see e.g. page 79 of [11].

######  Lemma 8.

Let $Z<N<G$ be locally compact groups, with $N$ normal and closed in $G$, and
$Z$ central and closed in $G$. Let $\tau$ be a representation of $N$ with the
property that the restriction $\tau|_{Z}$ is a multiple of a character
$\chi\in\widehat{Z}$. Then the restriction $\left({\rm
ind}_{N}^{G}\tau\right)|_{Z}$ is also a multiple of $\chi$.

We aim to prove the existence of vectors doing phase retrieval, for an
irreducible representation $\pi$ of a simply connected, connected nilpotent
Lie group $G$. The basic idea is to use induction over $\dim(G)$, and to
employ Mackey’s theory of induced representations in the process. For the
induction step to work, we need a somewhat stronger induction hypothesis on
the vectors doing phase retrieval. The relevant additional property is defined
next:

######  Definition 9.

Let $\pi$ denote a unitary representation of the locally compact group $G$.
$\eta\in\mathcal{H}_{\pi}$ has the full support property if for all
$0\not=f\in\mathcal{H}_{\pi}$ the set

$\\{x\in G:V_{\eta}f(x)=0\\}\subset G$

has Haar measure zero.

######  Remark 10.

Just as for the case of phase retrieval in Remark 6, the full support property
can equivalently be verified for the associated projective representation
$\overline{\pi}$ of $G/K$, where $K$ is any normal subgroup contained in the
projective kernel of $\pi$. This observation makes use of the fact that the
Haar measures of $G/K$ and $G$ are related by Weil’s integral formula.

######  Remark 11.

While the full support property has a strictly auxiliary role in our
considerations, it seems of independent representation-theoretic interest. We
are currently not aware of any sources investigating this property in any
systematic way.

As the prime example of a vector with the full support property, we mention
the Gaussian window

$\eta(x)=e^{-\pi x^{2}}~{},(x\in\mathbb{R})~{}.$

It is well-known that the image of the short-time Fourier transform
$STFT_{\eta}$ associated to $\eta$, defined by

$STFT_{\eta}f(x+iy)=\langle f,M_{y}T_{x}\eta\rangle~{},$

consists of functions whose product with a suitable, fixed choice of two-
dimensional Gaussian is entire (see e.g. Proposition 3.4.1 of [9]). But
clearly $STFT_{\eta}f(x+iy)=V_{\eta}f(x,y,0)$, where the right-hand side
denotes the matrix coefficient associated to the Schrödinger representation.
Hence the fact that nonzero entire functions are nonzero a.e. (together with
Remark 10) allows to conclude that $\eta$ has the full support property for
the Schrödinger representation.

Note also that any vector $\eta$ with the full support property for the
Schrödinger representation automatically does phase retrieval; recall from
Example 2 that a.e. nonvanishing of $V_{\eta}\eta$ was sufficient for the
latter. Hence the Gaussian hass both desired properties.

We are now in position to establish the first main result of this paper.

######  Theorem 12.

Let $G$ denote a simply connected, connected nilpotent Lie group, and $\pi$ an
irreducible representation of $G$. Then there exists a vector
$\eta\in\mathcal{H}_{\pi}$ that does phase retrieval, and has the full support
property. In particular, $\pi$ does phase retrieval.

###### Proof.

We proceed by induction over ${\rm dim}(G).$ Noting first that the abelian
case is trivial, and secondly, that the case of the Heisenberg group is
settled, we may assume that $G$ is a nonabelian group of dimension $n>3$.

Furthermore, we may assume that $K={\rm Ker}(\pi)$ is a zero-dimensional
subgroup of $G.$ Otherwise, the Lie algebra $\mathfrak{k}$ of $K$ has positive
dimension, and the connected component of the identity
$K_{0}=\exp(\mathfrak{k})$ in $K$ is a normal subgroup of positive dimension.
Now the inductive hypothesis yields a vector with phase retrieval and full
support property for the associated representation of the quotient group
$G/K$, and thus for $\pi$ (see Remarks 6, 10.)

Hence from now on the kernel of $\pi$ is assumed to be zero-dimensional. This
implies that the center $Z(G)$ of $G$ is necessarily one-dimensional. Pick
$Z\in\mathfrak{g}$ such that $\exp(\mathbb{R}Z)=Z(G)$. By Kirillov’s lemma [7,
Lemma 1.1.12], there exist $X,Y\in\mathfrak{g}$ with the following properties:
$\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathbb{R}X$, $[X,Y]=Z$,
$\mathfrak{g}_{0}$ is an ideal in $\mathfrak{g}$, and the centralizer of $Y$.
In particular, the associated subgroup $G_{0}=\exp(\mathfrak{g}_{0})$ is a
normal subgroup, with $G=G_{0}\exp(\mathbb{R}X)$. Furthermore,
$A=\exp(\mathbb{R}Y)\exp(\mathbb{R}Z)$ is central in $G_{0}$, and normal in
$G$.

For better readability, the remainder of the proof is structured into several
steps.

Step 1: There exists an irreducible representation $\tau$ of $G_{0}$ such that
$\pi\simeq{\rm ind}_{G_{0}}^{G}(\tau)$.

To see this, consider the dual action of $G$ on the dual group $\widehat{A}$,
defined for arbitrary $g\in G$ and $a\in A,$ by
$\left[g\star\chi\right]\left(a\right)=\chi\left(g^{-1}ag\right).$ Let
$G_{\chi}=\left\\{g\in G:g\star\chi=\chi\right\\}$ be the stabilizer of $\chi$
and let

$\widetilde{G}_{\chi}=\left\\{\tau\in\widehat{G_{\chi}}:\tau|_{A}\text{ is a
multiple of }\chi\right\\}.$

For $\tau\in\widetilde{G}_{\chi},$
$\pi_{\tau}=\mathrm{ind}_{G_{\chi}}^{G}\left(\tau\right)$ is an irreducible
representation of $G.$ In fact, according to the theory of Mackey [11, Theorem
4.22], every irreducible representation of $G$ is of this form. Hence, since
$\pi$ is assumed to be irreducible, there exists $\tau,$ such that
$\pi\simeq\mathrm{ind}_{G_{\chi}}^{G}\left(\tau\right)$ for suitable
$\chi\in\widehat{A}$ and $\tau\in\widetilde{G}_{\chi}$. $\tau$ is irreducible,
since $\pi$ is. To complete the proof of Step 1, it suffices to show that
$G_{\chi}=G_{0}$. Since $A$ is central in $G_{0}$, the conjugation action of
$G_{0}$ on $A$ is trivial. The same then holds for the dual action of $G_{0}$
on $\widehat{A}$, and this establishes $G_{0}\subset G_{\chi}.$ Moreover,
since $\pi$ is irreducible, $\pi|_{\exp(\mathbb{R}Z)}$ is the multiple of a
character, and by assumptions on the kernel of $\pi$, this character is
necessarily, nontrivial. Furthermore, Lemma 8 implies that this character
coincides with $\chi$ on $Z(G)$. Hence, by suitably normalizing $Z$ and $Y$,
when necessary, we may assume without loss of generality that
$\pi(\exp(zZ))=e^{2\pi iz}~{}.$ The relation $[X,Y]=Z$ then entails

$\exp(-xX)\exp(yY)\exp(xX)=\exp(yY)\exp(-xyZ)~{},$

and this leads to the following string of equalities:

$(\exp(xX)\star\chi)(\exp(yY))=\chi(\exp(yY)\exp(-xyZ))=e^{-2\pi
ixy}\chi(\exp(yY))~{}.$

These observations imply that $\exp(xX)\star\chi=\chi$ can only hold for
$x=0$. Since every $g\in G$ can be factored uniquely as $g=\exp(xX)g_{0}$,
with $g_{0}\in G_{0}\subset G_{\chi}$ and $x\in\mathbb{R}$, we derive finally
that $G_{0}=G_{\chi}$.

Step 2: Choosing an explicit realization of $\pi$. We will now introduce
coordinates on $G$, and express $\pi$ more explicitly in terms of these
coordinates, and a suitable choice of realization. Thus far, we have shown
that $\pi\simeq{\rm ind}_{G_{0}}^{G}(\tau)$, where $G_{0}\lhd G$, and $\tau$
is an irreducible representation, restricting to a multiple of a character
$\chi$ on $A=\exp(\mathbb{R}Y)\exp(\mathbb{R}Z)$. Furthermore,
$\chi(\exp(zZ))=e^{2\pi iz}$. Let $\mathfrak{g}_{0}\subset\mathfrak{g}$ denote
the Lie algebra of $G_{0}$, and let $W\subset\mathfrak{g}_{0}$ denote a vector
space complement to $\mathbb{R}X\oplus\mathbb{R}Y\subset\mathfrak{g}_{0}.$
Then every $g\in G$ has the unique factorization

$g=\exp(zZ)\exp(yY)\exp(w)\exp(xX)~{},$

with $x,y,z\in\mathbb{R}$ and $w\in W$. Consequently, every element in the
closed and normal subgroup $G_{0}$ also admits a unique factorization of the
type $g_{0}=\exp(zZ)\exp(yY)\exp(w).$

As a consequence of the factorizations, we obtain that $G$ is the inner
semidirect product $G=G_{0}\rtimes\exp(\mathbb{R}X)$, which allows to use a
specific realization of induced representations (see e.g. page 79 of [11],
Realization III for Semidirect Products). The representation space can be
taken as $\mathcal{H}_{\pi}=L^{2}(\mathbb{R},\mathcal{H}_{\tau})$, the space
of weakly Borel-measurable $\mathcal{H}_{\tau}$-valued functions on
$\mathbb{R}$ with square-integrable norms, and the cited source provides the
following explicit formulae

$\left[\pi\left(g\right)f\right]\left(t\right)=\begin{cases}f\left(t-x\right)\text{
if }g=\exp\left(xX\right)\\\ e^{-2\pi ity}f\left(t\right)\text{ if
}g=\exp\left(yY\right)\\\
\tau\left(\exp\left(e^{ad\left(-tX\right)}w\right)\right)f\left(t\right)\text{
if }g=\exp\left(w\right)~{},w\in W\end{cases}.$ (6)

It is worth noting that $\pi$ acts partially like a Schrödinger representation
via $\exp(xX)$ and $\exp(yY).$

Step 3: Fixing the vector $\eta$. By Remark 11, there exists $\eta_{1}\in
L^{2}(\mathbb{R})$ with full support and phase retrieval properties for the
Schrödinger representation. Furthermore, the inductive hypothesis provides
$\eta_{2}\in\mathcal{H}_{\tau}$ with full support and phase retrieval
properties for $\tau$. We then define $\eta\in
L^{2}(\mathbb{R},\mathcal{H}_{\tau})$ by $\eta(t)=\eta_{1}(t)\eta_{2}$, and
claim that it has the full support and phase retrieval properties for $\pi$.

Step 4: Explicit formulae for matrix coefficients. Fix any nonzero $f\in
L^{2}(\mathbb{R},\mathcal{H}_{\tau})$. By abuse of notation, we will
systematically use

$V_{\eta}f(y,w,x)~{},~{}(y,w,x)\in\mathbb{R}\times W\times\mathbb{R}$

instead of

$V_{\eta}(\exp(yY)\exp(w)\exp(xX))$

We will also omit the central variable $zZ$ from our discussion, in view of
Remarks 6 and 10.

We then get from (6) that

$\displaystyle V_{\eta}f(y,w,x)$ $\displaystyle=$ $\displaystyle\langle
f,\pi(\exp(yY)\exp(w)\exp(xX))\eta\rangle$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}}\langle
f\left(t\right),\tau\left(\exp\left(e^{-ad\left(tX\right)}w\right)\right)\eta\left(t-x\right)\rangle
e^{2\pi iyt}dt$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\langle
f\left(t\right),\tau\left(\exp\left(e^{-ad\left(tX\right)}w\right)\right)\eta_{2}\rangle\overline{e^{2\pi
iyt}\eta_{1}(t-x)}dt$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\langle
f\left(t\right),\tau\left(\exp(-tX)\exp(w)\exp(tX)\right)\eta_{2}\rangle\overline{e^{2\pi
iyt}\eta_{1}(t-x)}dt~{}.$

Now, using $\varrho$ to denote the Schrödinger representation, as well as

$F:W\times\mathbb{R}\to\mathbb{C}~{},~{}F(w,t)=\langle
f\left(t\right),\tau\left(\exp(-tX)\exp(w)\exp(tX)\right)\eta_{2}\rangle$ (7)

we may rewrite this formula as

$(V_{\eta}f)(y,w,x)=\left(V_{\eta_{1}}^{\varrho}F(w,\cdot)\right)(y,x)~{}.$
(8)

In addition, we make the observation that

$F(w,t)=\left(V_{\eta_{2}}^{\tau}f(t)\right)\left(\exp(-tX)\exp(w)\exp(tX)\right)~{}.$
(9)

Step 5: $\eta$ does phase retrieval. Now assume that $f,h\in
L^{2}(\mathbb{R},\mathcal{H}_{\tau})$ are such that $|V_{\eta}f|=|V_{\eta}h|$
holds. With $F$ from (7) and $H$ defined analogously as

$H:W\times\mathbb{R}\to\mathbb{C}~{},~{}H(w,t)=\langle
h\left(t\right),\tau\left(\exp(-tX)\exp(w)\exp(tX)\right)\eta_{2}\rangle$

equation (8) gives for all $w\in W$

$\left|V_{\eta_{1}}^{\varrho}(F(w,\cdot))\right|=\left|V_{\eta_{1}}^{\varrho}(H(w,\cdot))\right|~{}.$

Since $\eta_{1}$ does phase retrieval, this implies the existence of a map

$u:W\to\mathbb{T}~{},~{}H(w,\cdot)=u(w)F(w,\cdot)~{}~{}\forall w\in W.$ (10)

It follows for almost all $t\in\mathbb{R}$ that

$|H(\cdot,t)|=|F(\cdot,t)|~{}.$

By construction, $\exp(W)\subset G_{0}$ is a system of representatives modulo
the central subgroup $\exp(\mathbb{R}Z)\exp(\mathbb{R}Y)$ of $G_{0}$, which is
contained in the projective kernel of $\tau$. Hence equation (9) (and its
analog for $H$) allows to rewrite the previous equation as follows, for almost
all $t\in\mathbb{R}$:

$\forall g_{0}\in
G_{0}~{}:~{}\left|V_{\eta_{2}}^{\tau}(f(t))(\exp(-tX)g_{0}\exp(tX))\right|=\left|V_{\eta_{2}}^{\tau}(h(t))(\exp(-tX)g_{0}\exp(tX))\right|~{}.$

For fixed $t\in\mathbb{R}$, conjugation by $\exp(tX)$ is a bijection of
$G_{0}$ onto itself, hence we may simplify this equation to

$\left|V_{\eta_{2}}^{\tau}(f(t))\right|\equiv\left|V_{\eta_{2}}^{\tau}(h(t))\right|~{}.$

Now we can appeal to the phase retrieval property of $\eta_{2}$ to conclude
the existence of a map

$v:\mathbb{R}\to\mathbb{T}~{},~{}h(t)=v(t)f(t)~{}$ (11)

with the equality holding for almost every $t\in\mathbb{R}$. Note that this
also entails, for almost every $t$, that

$H(\cdot,t)=v(t)G(\cdot,t)~{}.$

Defining the Borel set

$C_{f}=\\{t\in\mathbb{R}~{}:~{}f(t)\not=0\\}\subset\mathbb{R}~{},$ (12)

(11) shows that the proof of phase retrieval boils down to showing that $v$ is
constant on $C_{f}$. Note that by removing a set of measure zero from $C_{f}$
we can assume that (11) holds for all $t\in C_{f}$.

At this point the full support property of $\eta_{2}$ enters the picture.
Picking any $t\in C_{f}$, equation (7) and the full support property of
$\eta_{2}$ implies that the set

$W_{f}(t)=\\{w\in W:F(w,t)\not=0\\}$

has full measure. Note that this step also used the fact that the automorphism
of $G_{0}$ induced by conjugation with $\exp(tX)$ preserves the Haar measure
of $G_{0}$, and Remark 10.

Hence, picking $t_{1},t_{2}\in C_{f}$ and any $w$ in $W_{f}(t_{1})\cap
W_{f}(t_{2})$ (which is of full measure, hence nonempty), we obtain from (10)

$\displaystyle v(t_{i})F(w,t_{i})=H(w,t_{i})=u(w)F(w,t_{i})~{}.$

By choice of $w$, we may cancel $F(w,t_{i})$ on both sides, to obtain
$v(t_{1})=u(w)=v(t_{2})$. This concludes the proof of phase retrieval.

Step 6: $\eta$ has the full support property. Let $0\not=f\in
L^{2}(\mathbb{R},\mathcal{H}_{\tau})$ be given, and let $C_{f}$ be as in (12).
We will prove that the set

$\left\\{(y,w,x)\in\mathbb{R}\times
W\times\mathbb{R}:V_{\eta}f(y,w,x)=0\right\\}$

has measure zero. Then an appeal to Remark 10 finishes the proof.

Given any $w\in W$, equation (8) together with the full support property of
$\eta_{1}$ shows that

$\left\\{(y,x)\in\mathbb{R}\times\mathbb{R}:V_{\eta}f(y,w,x)=0\right\\}$

has measure zero iff

$\|F(w,\cdot)\|_{L^{2}(\mathbb{R})}\not=0~{},$

with the function $F$ from (7).

Thus by Fubini’s theorem, the full support property for $\eta$ is shown once
we establish that

$\left\\{w\in W:\int_{\mathbb{R}}|F(w,t)|^{2}dt=0\right\\}$

has measure zero. For this purpose, introduce the set

$B_{f}=\\{(w,t)\in W\times C_{f}:F(w,t)=0\\}~{},$

with $C_{f}$ defined in Step 5. We also introduce the auxiliary function

$\tilde{F}(w,t)=\left(V_{\eta_{2}}^{\tau}f(t)\right)\left(\exp(w)\right)$

and associated set

$\tilde{B}_{f}=\\{(w,t)\in W\times C_{f}:\tilde{F}(w,t)=0\\}~{}.$

Given $t\in\mathbb{R}$, consider the map $\alpha(t):W\to W$ defined as

$\alpha(t)(w)=P_{W}\left(e^{-ad(tX)}w\right)$

where $P_{W}$ is the projection onto $W$ inside $\mathfrak{g}_{0}$, along the
complement $\mathbb{R}Z\oplus\mathbb{R}Y$. Then by definition of $\alpha(t)$,
we have

$\exp(\alpha(t)w)^{-1}\exp(-tX)\exp(w)\exp(tX)\in Z(G_{0})~{},$

and since $\tau$ restricted to $Z(G_{0})$ is a character, this gives rise to
the equation

$|F(w,t)|=|\tilde{F}(\alpha(t)w,t)|~{}.$

We also note that $\alpha(t)$ is a diffeomorphism, with inverse map given by
$\alpha(-t)$.

These observations allow to write

$B_{f}=\\{(w,t)\in W\times C_{f}:(\alpha(t)w,t)\in\tilde{B}_{f}\\}~{}.$ (13)

The whole point of introducing $\tilde{B}_{f}$ was that the assumption that
$\eta_{2}$ has the full support property, and the definition of $C_{f}$, imply
that $\tilde{B}_{f}$ is a set of measure zero, by Fubini’s theorem. It then
follows by (13), a change of variable involving the $\alpha(t)$, and yet
another application of Fubini’s theorem, that $B_{f}$ has measure zero. A
final application of Fubini’s theorem then yields a set $W^{\prime}\subset W$
of full measure such that $F(w,\cdot)$ vanishes almost nowhere on $C_{f}$, for
all $w\in W^{\prime}$. Since $C_{f}$ is of positive measure, this finally
implies

$\int_{\mathbb{R}}|F(w,t)|^{2}dt>0$

for all $w\in W^{\prime}$, and the proof is finished. ∎

##  3\. Finite nilpotent groups

This section is devoted to establishing analogs of Theorem 12 for finite
nilpotent groups. While we will not be able to cover all finite nilpotent
groups, we will exhibit large classes of finite nilpotent groups, for which an
analog can be formulated and proved, using an inductive method that is quite
similar to that employed for Lie groups.

The following theorem contains the largest previously known class of finite
groups for which phase retrieval is established:

######  Theorem 13.

Let $G$ denote a finite nilpotent group of nilpotency class 2, which means
that $G$ is nonabelian, and $G/Z(G)$ is abelian. Then every irreducible
representation of $G$ does phase retrieval.

This theorem is a consequence of Theorem 1.7 from [12], which states that
every projective irreducible representation of a finite abelian group does
phase retrieval, combined with the natural correspondence between projective
representations of finite abelian groups and unitary representations of class
2 nilpotent groups. Our results will significantly extend this class of
examples.

Recall that a $p$-group is a group $G$ with cardinality $p^{r}$, for some
positive exponent $r>0$ and prime number $p$. If $p^{k}$ is the highest power
of some prime number $p$ dividing the order of $G,$ then a subgroup of $G$ of
order $p^{k}$ is called a $p$-Sylow subgroup of $G.$ The order of a group $G$
is the smallest integer $n$ satisfying $x^{n}=e$ for all $x\in G$. Clearly,
the order of a $p$-group is again a power of $p$.

The treatment of general finite nilpotent groups is easily reduced to that of
$p$-groups, by the following well-known observation (see e.g. Theorems 5.2.4
and 8.4.2 of [16]).

######  Theorem 14.

Every finite nilpotent group $G$ is the direct product of its $p$-Sylow
subgroups. As a consequence, every irreducible representation of $G$ is the
outer tensor product of irreducible representations of its $p$-Sylow
subgroups.

The following Lemma is a direct consequence of Lemma 5, which immediately
gives a solution for the phase retrieval problem for tensor product
representations in terms of solutions for the factors. Since it immediately
generalizes to finitely many tensor factors, this lemma reduces the phase
retrieval problem for irreducible representation of general finite nilpotent
groups to the $p$-group case, on which we shall subsequently concentrate.

######  Lemma 15.

For $k=1,2,$ let $\pi_{k}$ be a unitary representation of some finite group
$G_{k}$ and suppose that each $\pi_{k}$ does phase retrieval and is realized
is some Hilbert space $\mathcal{H}_{\pi_{k}}.$ Next, let
$\pi=\pi_{1}\otimes\pi_{2}$ be the outer tensor product of $\pi_{1}$ and
$\pi_{2}$. If $\eta_{1}$ does phase retrieval for $\pi_{1}$ and if $\eta_{2}$
does phase retrieval for $\pi_{2}$ then
$\eta_{1}\otimes\eta_{2}\in\mathcal{H}$ does phase retrieval for $\pi.$

Before we turn to the study of phase retrieval properties for $p$-groups, we
will need to establish several auxiliary notions and results. To a large part,
these notions are motivated by the desire to adapt the inductive approach
developed for the Lie group case to the $p$-group setting. They mostly concern
the inductive construction of irreducible representations. But we will also
need a finite group analog to the full support property, which is necessarily
of a more quantitative nature. We start by formulating this analog:

######  Definition 16.

Let $(\pi,\mathcal{H}_{\pi})$ denote a unitary representation of a finite
group $G$. Given a vector $\eta\in\mathcal{H}_{\pi}$, we let

$p_{0}(\pi,\eta)=\max_{f\in\mathcal{H}_{\pi}\setminus\\{0\\}}\left(1-\frac{|{\rm
supp}(V_{\eta}f)|}{|G|}\right)$

as well as

$p_{0}(\pi)=\min\\{p_{0}(\pi,\eta):\eta\in\mathcal{H}_{\pi}\setminus\\{0\\}\\}~{}.$

The following remark formulates an analog to Remark 6 for the quantity
$p_{0}$:

######  Remark 17.

Let $\pi$ be a representation of a finite group $G$, and $K\lhd G$ is a normal
subgroup contained in the projective kernel of $\pi$. Picking any system
$W\subset G$ of representatives modulo $K$, it is easy to see that

$p_{0}(\pi,\eta)=\max_{f\in\mathcal{H}_{\pi}\setminus\\{0\\}}\left(1-\frac{\left|{\rm
supp}\left((V_{\eta}f)|_{W}\right)\right|}{|W|}\right)$

######  Remark 18.

$p_{0}(\pi,\eta)$ can be interpreted as the maximal proportion of zeros
occurring in the matrix coefficients $V_{\eta}f$ for an arbitrary nonzero
vector $f$. This quantity is therefore related to sampling sets: For any set
$A\subset G$ satisfying $|A|>p_{0}(\pi,\eta)|G|$, the sampled coefficient map

$V_{\eta}|_{A}:f\mapsto(V_{\eta}f)|_{A}$

has trivial kernel by definition of $p_{0}(\pi,\eta)$, hence it is injective.

This property is also related to the full spark property, studied in the
representation theoretic context in [14, 13]. In fact, for any representation
$\pi$ having the full spark property, one can derive

$p_{0}(\pi)=\frac{d_{\pi}-1}{|G|}~{}.$

A related result was derived by Malikiosis in [13], who showed that, for
suitable choices of $\eta\in\mathbb{C}^{n}$ the system
$(\pi(k,l,0)\eta)_{k,l\in\mathbb{Z}_{n}}$ has the full spark property, where
$\pi$ is the Schrödinger representation from Example 1, associated to the
finite cyclic group $\mathbb{Z}_{n}$ of order $n$. Since
$W=\\{(k,l,0):k,l=0,\ldots,n-1\\}\subset\mathbb{H}_{n}$ is a system of
representatives modulo the center of $\mathbb{H}$, which is contained in the
projective kernel of $\pi$, Remark 17 allows to determine

$p_{0}(\pi)=\frac{n-1}{n^{2}}~{}.$ (14)

######  Remark 19.

In our subsequent arguments (see Lemma 29 below), an upper estimate for the
quantity $p_{0}(\tau)$ for a certain representation of a suitably chosen
subgroup $G_{0}$ of $G$ will play an important role, as a sufficient condition
for phase retrieval. It is noteworthy that the phase retrieval property of
$\pi$ also entails a necessary lower estimate for $p_{0}(\pi)$, as follows:
Assuming that the vector $\eta\in\mathcal{H}_{\pi}$ does phase retrieval for
$\pi$, Lemma 4(b) implies that for any nonzero $f,g\in\mathcal{H}_{\pi}$, the
pointwise product $(V_{\eta}f)\cdot(V_{\eta}g)$ is not identically zero.
Letting $A={\rm supp}(V_{\eta}f)\subset G$ and $g=\pi(y)f$ for $y\in G$ yields
via

$V_{\eta}g(x)=\langle\pi(y)f,\pi(x)\eta\rangle=V_{\eta}f(y^{-1}x)$

that $yA\cap A\not=\emptyset$, or equivalently: $y\in A\cdot A^{-1}$. Hence we
have shown

$A\cdot A^{-1}=G~{},$

for the support $A$ of any nonzero matrix coefficient $V_{\eta}f$. This
observation, together with Remark 18, allows to derive the estimate

$p_{0}(\pi,\eta)\leq 1-|G/K|^{-1/2}~{},$

where $K\lhd G$ denotes the projective kernel of $\pi$.

For the formulation of the next lemma, we introduce a new piece of
terminology: Given a finite dimensional Hilbert space $\mathcal{H}$, we call a
set $M\subset\mathcal{H}$ real Zariski closed if there exist mappings

$f_{1},\ldots,f_{k}:\mathcal{H}\to\mathbb{R}$

that depend polynomially on the real and imaginary parts of arbitrarily chosen
complex linear coordinates on $\mathcal{H}$, and such that

$M=\\{x\in\mathcal{H}~{}:~{}f_{1}(x)=\ldots=f_{k}(x)=0\\}~{}.$

We call $M$ real Zariski open if its complement is real Zariski closed.

######  Lemma 20.

Let $(\pi,\mathcal{H}_{\pi})$ denote a finite-dimensional representation of
the finite group $G$. Let

$M_{1}=\\{\eta\in\mathcal{H}_{\pi}:\eta\mbox{ does phase retrieval }\\}$

and

$M_{2}=\\{\eta\in\mathcal{H}_{\pi}:p_{0}(\pi,\eta)=p_{0}(\pi)\\}~{}.$

Then $M_{1}$ and $M_{2}$ are real Zariski open. If $\pi$ does phase retrieval,
then $M_{1}\cap M_{2}$ is nonempty and real Zariski open. In particular
$M_{1}\cap M_{2}\subset\mathcal{H}_{\pi}$ is of full Lebesgue measure.

###### Proof.

For the statement concerning $M_{1}$ we refer to Lemma 2.5 of [4]. For
$M_{2}$, we identify $\mathcal{H}_{\pi}$ with $\mathbb{C}^{d_{\pi}}$ (using
any choice of basis in $\mathcal{H}_{\pi}$), identify $\mathbb{C}^{G}$ with
$\mathbb{C}^{|G|}$ by a similar procedure, and use these identifications to
associate to the linear operator
$V_{\eta}:\mathbb{C}^{d_{\pi}}\to\mathbb{C}^{G}$ the describing matrix
$\mathcal{A}_{\eta}\in\mathbb{C}^{|G|\times d_{\pi}}$. Given any matrix
$\mathcal{A}\in\mathbb{C}^{|G|\times d_{\pi}}$, and an arbitrary set
$B\subset\\{1,\ldots,|G|\\}$, we let $\mathcal{A}^{B}$ denote the submatrix
obtained by picking the lines of $\mathcal{A}$ corresponding to indices
contained in $B$. Then

$p_{0}(\pi,\eta)\leq\frac{k}{|G|}$

is equivalent to the statement that, for arbitrary sets
$B\subset\\{1,\ldots,|G|\\}$ of cardinality $k+1$, the submatrix
$\mathcal{A}_{\eta}^{B}$ has rank $d_{\pi}$. But the set

$\mathcal{Z}_{k}=\\{\mathcal{A}\in\mathbb{C}^{|G|\times d_{\pi}}:\forall
B\subset\\{1,\ldots,|G|\\}\mbox{ with }|B|=k+1~{},~{}{\rm
rank}(\mathcal{A}^{B})=d_{\pi}\\}$

can be described in terms of $d_{\pi}\times d_{\pi}$-subdeterminants, hence is
real Zariski-open. Now we have that

$\displaystyle p_{0}(\pi,\eta)=p_{0}(\pi)$ $\displaystyle\Leftrightarrow$
$\displaystyle p_{0}(\pi,\eta)\leq p_{0}(\pi)$ $\displaystyle\Leftrightarrow$
$\displaystyle\mathcal{A}_{\eta}\in\mathcal{Z}_{\pi_{0}(\pi)|G|}~{}.$

Since the map $\eta\mapsto A_{\eta}$ is conjugate linear, we see that $M_{2}$
is conjugate linear preimage of a real Zariski open set, hence real Zariski
open itself. Now if $\pi$ does phase retrieval, then $M_{1}$ is nonempty,
whereas $M_{2}$ is nonempty by definition. Nonempty real Zariski open sets are
open and dense in the standard topology; in fact they are of full Lebesgue
measure. Then $M_{1}\cap M_{2}$ is nonempty, and real Zariski-open, hence it
inherits the remaining properties as well. ∎

We will now start to introduce the central device of our treatment of
$p$-groups, namely a version of Kirillov’s lemma for this class of groups. For
this purpose, we introduce further notation. Let $G$ denote a nonabelian
$p$-group. Let $[w,y]=w^{-1}y^{-1}wy$ be the commutator of $w,y\in G$.
Furthermore, write $w^{y}=y^{-1}wy$ for the conjugation product of $w$ and
$y$. We then have the well-known relations

$[xz,y]=[x,y]^{z}[z,y]~{},~{}[x,zy]=[x,y][x,z]^{y}~{},$ (15)

which can be verified by direct calculation.

We introduce

$Z_{1}(G)=\\{y\in G:yZ(G)\in Z(G/Z(G))\\}~{}.$

Since $G$ is nilpotent, $G/Z(G)$ is as well. In particular, $G/Z(G)$ has
nontrivial center, which implies that $Z(G)\subsetneq Z_{1}(G)$, unless $G$ is
abelian. The following elementary lemma will be used repeatedly:

######  Lemma 21.

The mapping

$[\cdot,\cdot]:G\times Z_{1}(G)\to Z(G)~{},~{}(x,y)\mapsto[x,y]$

is a well-defined bihomomorphism, i.e., for all $x\in G,y\in Z_{1}(G)$, the
maps $[x,\cdot]$ and $[\cdot,y]$ are homomorphisms into $Z(G)$.

###### Proof.

Let $x\in G,y\in Z_{1}(G)$. The fact that $yZ(G)$ is central in $G/Z(G)$
implies

$[x,y]Z(G)=[xZ(G),yZ(G)]=e_{G}Z(G)~{},$

hence $[x,y]\in Z(G)$, and $[\cdot,\cdot]:G\times Z_{1}(G)\to Z(G)$ is well-
defined. Furthermore, (15) yields for $x,z\in G$

$[xz,y]=[x,y]^{z}[z,y]=[x,y][z,y]~{},$

since the conjugation action is trivial on $[x,y]\in Z(G)$. If $z\in
Z_{1}(G)$, we similarly get

$[x,zy]=[x,y][x,z]^{y}=[x,y][x,z]=[x,z][x,y]~{},$

since $Z(G)$ is abelian. ∎

The following is the announced $p$-group analog of Kirillov’s lemma for
nilpotent Lie groups [7, Lemma 1.1.12].

######  Lemma 22.

Let $G$ be a non-abelian $p$-group with cyclic center $Z\left(G\right)$. Then
there exist $z\in Z(G),y\in Z_{1}(G)$ and $x\in G$ as well as an integer
$r\geq 1$ such that the following hold:

  1. (a)

$z=[x,y]$, $\langle z\rangle$ is a subgroup of $Z(G$) with $\langle
z\rangle=[G,y]\cong\mathbb{Z}_{p^{r}}$.

  2. (b)

$A=\langle y,Z(G)\rangle$ is an abelian normal subgroup of $G$ with
$A/Z(G)=\langle yZ(G)\rangle\cong\mathbb{Z}_{p^{r}}$.

  3. (c)

Define $G_{0}=\\{w:[w,y]=e_{G}\\}$. Then $G_{0}$ is a normal subgroup of $G$
with $G/G_{0}=\langle xG_{0}\rangle\cong\mathbb{Z}_{p^{r}}$. $A<G_{0}$ is
central.

###### Proof.

Proof of Part (a): Pick any $y\in Z_{1}(G)\setminus Z(G)$. By Lemma 21,
$[G,y]$ is a subgroup contained in the center $Z(G).$ Since the center of $G$
is assumed to be cyclic, $[G,y]$ is cyclic as well, hence
$[G,y]\cong\mathbb{Z}_{p^{r}}$ for some natural number $r$, since $G$ is a
$p$-group. The case $r=0$ is equivalent to $[G,y]=\\{e\\}$, or $y\in Z(G)$,
contrary to the choice of $y$. The existence of $z$ and $x$ is then obvious.

Proof of Part (b): It is clear that $A$ is an abelian subgroup of $G$. Fixing
$w\in G$, and appealing to Lemma 21, we obtain

$w^{-1}yw=[w,y^{-1}]y=\underbrace{[w,y]^{-1}}_{\in Z(G)}y\in A~{}.$

This implies that $\langle y,Z(G)\rangle$ is normal. To complete the proof of
Part (b), we show that $yZ(G)$ has order $p^{r}$ in $G/Z(G)$. In order to see
this, we use Lemma 21 to compute $[x,y^{l}]=[x,y]^{l}=z^{l}\not=e_{G}$ for
$1\leq l<p^{r}$, by choice of $z$. This shows in particular $y^{l}\not\in
Z(G)$.

On the other hand, assuming $y^{p^{r}}\not\in Z(G)$ would entail the existence
of $w\in G$ with $e_{G}\not=[w,y^{p^{r}}]=[w,y]^{p^{r}}~{}.$ This would imply
that $[w,y]\in[G,y]$ has order $>p^{r}$, which contradicts (a).

Proof of Part (c): Note that by definition, $G_{0}$ is the kernel of the
homomorphism $[\cdot,y]:G\to Z(G)$, which has image $[G,y]$, generated by
$z=[x,y]$. Hence $G_{0}$ is a normal subgroup, and $\langle
xG_{0}\rangle=G/G_{0}\cong\langle z\rangle\cong\mathbb{Z}_{p^{r}}$ follows by
the first homomorphism theorem. $y$ is central in $G_{0}$ by definition of the
latter, and $Z(G)\subset G_{0}$ is central even in $G$, hence $A=\langle
y,Z(G)\rangle$ is central in $G_{0}$. ∎

######  Remark 23.

We note that the case $[G,y]\subsetneq Z(G)$ may well occur. For a concrete
example, fix a prime number $p$ and consider
$G=\mathbb{Z}_{p}\times\mathbb{Z}_{p}\times\mathbb{Z}_{p^{2}}$, with group law

$(k,l,m)\cdot(k^{\prime},l^{\prime},m^{\prime})=(k+k^{\prime},l+l^{\prime},m+m^{\prime}-pl^{\prime}k)~{}.$

The center of this group is given by
$\\{0\\}\times\\{0\\}\times\mathbb{Z}_{p^{2}}$, whereas
$[G,y]=\langle(0,0,p)\rangle$.

The notation of the lemma suggests that the group $H=\langle x,y,z\rangle$,
for $x,y,z$ from Lemma 22 is isomorphic to a Heisenberg group. Note that the
analogous statement in the Lie group case, i.e. that the closed subgroup
associated to the Lie subalgebra generated by $X,Y,Z$ is isomorphic to the
three-dimensional Heisenberg group, holds true. The next example shows however
that the $p$-group case is more diverse. Nevertheless the Schrödinger
representations from Example 1 constitute important basic building blocks in
our inductive strategy to establish phase retrieval for general $p$-groups,
and the following example is intended to shed some light on this phenomenon.

######  Example 24.

Let $H=\mathbb{H}_{p^{r}}$ denote a finite Heisenberg group associated to the
cyclic group $\mathbb{Z}_{p^{r}}$, as given by (1) and (2). Define
$x=(1,0,0),y=(0,1,0)$ and $z=(0,0,-1)$. Then one has $Z(H)=\langle
z\rangle=Z_{0}$,

$Z_{1}(H)=H=\langle x,y,z\rangle=\langle x,y\rangle~{},$

and $[x,y]=z$, as well as all the other properties from Lemma 22. It is
tempting to assume that the relations in this lemma already characterize $H$
up to isomorphism.

But this is not the case. For the construction of a counterexample, fix a
prime number $p$ and an integer $r\geq 1$. Consider the group
$G=\mathbb{Z}_{p^{2r}}\times\mathbb{Z}_{p^{r}}$, with group law

$(k,l)(k^{\prime},l^{\prime})=(k+(1-lp^{r})k^{\prime},l+l^{\prime})~{},0\leq
k,k^{\prime}<p^{2r}~{},0\leq l,l^{\prime}<p^{r}~{}.,$ (16)

where the operations on the right-hand side are taken modulo $p^{2r},p^{r}$,
respectively. Using

$(1-lp^{r})(1-l^{\prime}p^{r})\equiv 1-(l+l^{\prime})p^{r}~{}~{}\mbox{ mod
}p^{2r}~{},$

one easily verifies that (16) is a well-defined group law. We let
$x=(0,1),y=(1,0)$ and $z=(p^{r},0)$, and obtain that $Z(G)=\langle
z\rangle=Z_{0}$,

$Z_{1}(G)=G=\langle x,y,z\rangle=\langle x,y\rangle=\langle
y\rangle\rtimes\langle x\rangle~{},$

as well as $[x,y]=z$.

Again, all remaining properties noted in Lemma 22 hold for $x,y,z$ also.
Nonetheless, $G$ is clearly not isomorphic to the Heisenberg group $H$, since
it contains elements of order $p^{2r}$.

In order to exhibit similarities of the newly constructed group to the
Heisenberg group case, we now determine certain irreducible unitary
representations of $G$. Letting $A=\langle
y,z\rangle=\mathbb{Z}_{p^{2r}}\times\\{0\\}\cong\mathbb{Z}_{p^{2r}}$, we let
$\xi\in\mathbb{T}$ be a root of unity of order $p^{2r}$, and define the
associated character $\chi_{\xi}\in\widehat{A}$ as

$\chi_{\xi}(k,0)=\xi^{k}~{},~{}k=0,\ldots,p^{2r}-1~{}.$

The dual action of $x$ on the characters is given by

$x^{l}\ast\chi_{\xi}(k,0)=\chi_{\xi}((0,-l)(k,0)(0,l))=\xi^{k(1+lp^{r})}=\chi_{\xi^{1+lp^{r}}}(k,0)~{}.$

Now let $\xi_{0}$ denote an arbitrary primitive root of order $p^{2r}$. Then
we have for $0\leq l<p^{r}$ that

$\xi_{0}^{(1+lp^{r})}=\xi_{0}\Longleftrightarrow\xi_{0}^{lp^{r}}=1\Longleftrightarrow
l=0~{},$

i.e., the stabilizer of $\xi_{0}$ under the dual action is trivial.

We now consider $\pi={\rm ind}_{A}^{G}\chi_{\xi_{0}}$. By Mackey’s theorem,
$\pi$ is an irreducible representation, and since $\langle y,z\rangle$ is a
normal subgroup, we can realize $\pi$ on $l^{2}(G/A)=l^{2}(\langle x\rangle)$
by the formula from page 79 of [11], which yields

$\left[\pi(y^{k}x^{l})\varphi\right](x^{j})=\chi_{\xi_{0}}(x^{-j}y^{k}x^{j})\varphi(x^{j-l})~{}.$

Since $\left[x,y\right]=z,$ $x^{-j}\left(y^{k}\right)x^{j}=y^{k}z^{-kj}$, and
we obtain

$\displaystyle\left[\pi(y^{k}x^{l})\varphi\right](x^{j})$ $\displaystyle=$
$\displaystyle\chi_{\xi_{0}}(y^{k}z^{-kj})\varphi(x^{j-l})$ $\displaystyle=$
$\displaystyle\xi_{0}^{k}\xi_{1}^{-kj}\varphi(x^{j-l})$

where $\xi_{1}=\xi_{0}^{p^{r}}$ is a primitive root of order $p^{r}$. This
observation may be rephrased as

$\pi(y^{k}x^{l})=\alpha(k,l)M_{k}T_{l}~{},$

with $\alpha(k,l)\in\mathbb{T}$. This shows that, even though $G$ and $H$ are
clearly non-isomorphic, the operators in the realization of $\pi$ differ from
a Schrödinger representation only by scalar factors, and Lemma 4 (c) entails
that the vectors doing phase retrieval for $\pi$ are precisely the ones that
do phase retrieval for the Schrödinger representation.

The next definition introduces a bookkeeping device, intended to describe the
construction of an arbitrary irreducible representations by repeated induction
steps.

######  Definition 25.

Let $G$ denote a $p$-group, and $\pi$ an irreducible, faithful representation
of $G$. $(G_{0},r,\tau)$ is called a K-triple for $\pi$ if $r\in\mathbb{N}$,
$G_{0}\lhd G$, $\tau$ is a representation of $G_{0}$ such that $\pi\simeq{\rm
ind}_{G_{0}}^{G}\tau$, and there exist $x\in G$, $y\in Z_{1}(G)\setminus Z(G)$
and $z\in Z(G)$ with the following properties:

  1. (i)

$[x,y]=z$, and $\langle z\rangle=[G,y]\cong\mathbb{Z}_{p^{r}}$.

  2. (ii)

$G_{0}$ is the centralizer of $y$.

  3. (iii)

$G/G_{0}=\langle xG_{0}\rangle\cong\mathbb{Z}_{p^{r}}$.

The K-triple is called split, if $x$ can be chosen such that $x^{p^{r}}=e$.

It is easy to see that a K-triple $(G_{0},r,\tau)$ is split if and only if the
short exact sequence

$1\to G_{0}\to G\to G/G_{0}\to 1$

splits, i.e. when $G$ is the inner semidirect product $G_{0}\rtimes\langle
x^{\prime}\rangle$ for a suitable choice of $x^{\prime}$. Here the “only
if”-part is clear. For the converse, assume that a K-triple $(G_{0},r,\tau)$
is given, and that in addition $G=G_{0}\rtimes\langle x^{\prime}\rangle$ for a
suitable choice of $x^{\prime}$. This entails in particular that $x^{\prime}$
has order $p^{r}$. Since $G_{0}$ is by definition the kernel of the
homomorphism $[\cdot,y]:G\to\langle z\rangle$, we get that
$s\mapsto[(x^{\prime})^{s},y]\in\langle z\rangle$ is onto, hence
$z^{\prime}=[x^{\prime},y]$ fulfills $\langle z^{\prime}\rangle=\langle
z\rangle$. This shows that $x^{\prime},y,z^{\prime}\in G$ have the properties
required of $x,y,z$ in Definition 25 to show that $(G_{0},r,\tau)$ is a split
K-triple.

The following lemma provides a central result for our inductive approach. It
is the $p$-group analog to Step 1 in the proof of Theorem 12, and its proof is
remarkably similar to the proof for the Lie group setting.

######  Lemma 26.

Let $G$ denote a nonabelian $p$-group, and $\pi$ a faithful irreducible
representation of $G$. Then there exists a K-triple for $\pi$.

###### Proof.

Since $\pi$ is irreducible, $\pi(Z(G))$ is a (finite) subgroup of the torus.
It follows that $\pi(Z(G))$ is cyclic, and by faithfulness of $\pi$, this
entails that $Z(G)$ is cyclic. Hence Lemma 22 is applicable, and we obtain the
group elements $x,y,z$, as well as the normal subgroup $G_{0}$ and
$r\in\mathbb{N}$ such that the properties (i)-(iii) from Definition 25 are
fulfilled. It therefore remains to prove the existence of $\tau$. Note that
$G_{0}$ is the centralizer of $y$ by its definition in part (c) of the lemma.

Let $A=\langle y,Z(G)\rangle$. Then $A$ is an abelian normal subgroup of $G$.
Note that this fact can be concluded from the choice according to Lemma 22,
but it also follows from $z=[x,y]\in Z(G)$ and the choice of $G_{0}$ as
centralizer of $y$. In a similar way, $A$ is seen to be central in $G_{0}$.

For any given unitary character $\chi$ of $A,$ $G$ acts on $\chi$ via the dual
action which is defined as follows. For arbitrary $g\in G$ and $a\in A,$
$\left[g\star\chi\right]\left(a\right)=\chi\left(g^{-1}ag\right).$ Let
$G_{\chi}=\left\\{g\in G:g\star\chi=\chi\right\\}$ be the stabilizer of $\chi$
and let

$\widetilde{G}_{\chi}=\left\\{\tau\in\widehat{G_{\chi}}:\tau|_{A}\text{ is a
multiple of }\chi\right\\},$

where $\widehat{G}_{\chi}$ denotes the set of (equivalence classes of)
irreducible representations of $G_{\chi}$. For $\tau\in\widetilde{G}_{\chi},$
$\pi_{\tau}=\mathrm{ind}_{G_{\chi}}^{G}\left(\tau\right)$ is an irreducible
representation of $G.$ In fact, according to the theory of Mackey [11, Theorem
4.22], every irreducible representation of $G$ is of this form.

Hence $\pi\simeq\mathrm{ind}_{G_{\chi}}^{G}\left(\tau\right)$ for suitable
$\chi\in\widehat{A}$ and $\tau\in\widetilde{G}_{\chi}$. The proof is finished
when we have shown that $G_{\chi}=G_{0}$. By condition (ii), $A$ is central in
$G_{0}$, which means that the conjugation action of $G_{0}$ on $A$ is trivial.
The same then follows for the dual action on $\widehat{A}$, which establishes
$G_{0}\subset G_{\chi}$.

In order to show the reverse inclusion, we take a closer look at the dual
action of $x$ and its powers. By Lemma 8 applied with $Z=Z(G)$ and $N=G_{0}$,
the characters obtained by restricting either $\pi$ or $\tau$ to $Z(G)$ must
coincide. By assumption, $\pi$ is faithful, hence $\tau|_{Z_{0}}$ is faithful.
Since $Z_{0}=\langle z\rangle$, we get that $\tau(z^{k})=\xi^{k}$, for a
suitable $p^{r}$th root of unity $\xi\in\mathbb{T}$. In fact, faithfulness
requires that $\xi$ is a primitive root of unity of order $p^{r}$, i.e.
$\xi^{k}=1$ holds iff $p^{r}$ divides $k$, for all $k\in\mathbb{Z}$.

Since $y\in Z_{1}(G)$, we can appeal to Lemma 21 to compute
$x^{-m}yx^{m}=[x^{m},y^{-1}]y=z^{-m}y$ which leads to

$(x^{m}\star\chi)(y)=\chi(z^{-m})\chi(y)=\xi^{-m}\chi(y)~{}.$

Hence $x^{m}\star\chi=\chi$ entails $\xi^{-m}=1$, and since $\xi$ is a
primitive root of order $p^{r}$, this only happens for $m\in p^{r}\mathbb{Z}$.

By (iii), every $g\in G$ can be factored uniquely as $g=x^{m}g_{0}$, with
$g_{0}\in G_{0}\subset G_{\chi}$ and $0\leq m<p^{r}$. This implies that

$\chi=g\star\chi=x^{m}\star\chi$

holds iff $m=0$, or equivalently, iff $g\in G_{0}$. This proves
$G_{0}=G_{\chi}$ and thus $\pi\simeq{\rm ind}_{G_{0}}^{G}\tau$. ∎

We next provide an explicit description of the representation $\pi$ in terms
of the data provided by a K-triple. It is again quite analogous to the Lie
group case, but this time the analogy requires an additional assumption (which
is automatic in the Lie group setting), namely that the group extension $1\to
G_{0}\to G\to G/G_{0}\to 1$ splits.

######  Lemma 27.

Let $G$ denote a $p$-group, and $\pi$ an irreducible, faithful representation
of $G$, with a split K-triple $(G_{0},r,\tau)$. Let $x,y,z\in G$ denote the
group elements associated to the triple according to Definition 25, with the
property $x^{p^{r}}=e$.

  1. (a)

Let $W\subset G_{0}$ denote a system of representatives modulo
$\langle\\{y\\}\cup Z(G)\rangle$. Then every element $g\in G$ has a unique
factorization

$g=z^{\prime}y^{k}wx^{l}~{},~{}z^{\prime}\in Z(G),~{}k,l\in\\{0,\ldots
p^{r}-1\\}~{},w\in W~{}.$

  2. (b)

The representation ${\rm ind}_{G_{0}}^{G}\tau$ acts in the Hilbert space
$l^{2}\left(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau}\right)$ by the formula

$\left[\pi(y^{k}wx^{l})\varphi\right](j)=\xi_{1}^{k}\xi_{2}^{-kj}\tau(x^{-j}wx^{j})\varphi(j-l)~{},$

where $\xi_{1}\in\mathbb{T}$ is a suitable root of unity, and $\xi_{2}$ is a
primitive root of unity of order $p^{r}$.

###### Proof.

Part (a) follows from properties (i)-(iii), and the choice of $W$.

For part (b), we use the semidirect product structure $G=G_{0}\rtimes\langle
x\rangle$, and again appeal to the realization of induced representations for
semidirect products, elaborated on [11, Page 79]. We identify the quotient
$G/G_{0}$ with the cyclic group $\mathbb{Z}_{p^{r}}$, using the identification
$x^{j}G_{0}\mapsto j$. Letting $w\in W$ and $\varphi\in
l^{2}\left(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau}\right)$, we get by the cited
result that

$\left[\pi\left(y^{k}wx^{l}\right)\varphi\right]\left(j\right)=\tau\left(x^{-j}\left(y^{k}w\right)x^{j}\right)\varphi\left({j-l}\right)$

for $\left(k,w,l\right)\in\mathbb{Z}_{p^{r}}\times W\times\mathbb{Z}_{p^{r}}$,
$j\in\mathbb{Z}_{p^{r}}$. Since $\left[x,y\right]=z,$ we get
$x^{-j}y^{k}x^{j}=y^{k}z^{-kj}$ and

$\displaystyle\pi\left(y^{k}wx^{l}\right)\varphi\left(x^{j}\right)$
$\displaystyle=$
$\displaystyle\tau\left(x^{-j}y^{k}x^{j}\right)\circ\tau\left(x^{-j}wx^{j}\right)\varphi\left(j-l\right)$
(17) $\displaystyle=$
$\displaystyle\chi\left(y^{k}z^{-kj}\right)\cdot\tau\left(x^{-j}\left(w\right)x^{j}\right)\varphi\left(j-l\right)$
$\displaystyle=$
$\displaystyle\chi\left(y^{k}\right)\cdot\chi\left(z^{-kj}\right)\cdot\tau\left(x^{-j}wx^{j}\right)\varphi\left(j-l\right).$

Here $\chi:A\to\mathbb{T}$ denotes the character obtained by restricting the
irreducible representation $\tau$ to the central subgroup $A$. Since $\pi$ is
faithful, Lemma 8 implies that $\tau$ is faithful on $Z(G)$. Hence, as $z$ has
order $p^{r}$, there exists a primitive root of unity $\xi_{2}$ of order
$p^{r}$ such that

$\tau(z^{l})=\xi_{2}^{l}~{}.$

Furthermore $\chi(y^{k})=\xi_{1}^{k}$ for a suitable root of unity, whose
order is at most the order of $y$. ∎

######  Lemma 28.

Let $G$ denote a $p$-group, and $\pi$ an irreducible, faithful representation
of $G$, with K-triple $(G_{0},r,\tau)$. Let $W\subset G_{0}$ denote a system
of representatives modulo $\langle\left\\{y\right\\}\cup Z(G)\rangle$, where
$y\in Z_{1}(G)$ denotes the group element associated to the triple according
to Definition 25. Then there exist mappings $\alpha_{j}:W\to W$ and
$\beta_{j}:W\to\langle\left\\{y\right\\}\cup Z(G)\rangle$,
$j=0,\ldots,p^{r}-1$ such that

$\forall j=0,\ldots,p^{r}-1,~{}\forall w\in
W~{}:~{}x^{-j}wx^{j}=\beta_{j}(w)\alpha_{j}(w)~{}.$

###### Proof.

Since $\langle y,z\rangle$ is normal in $G$, it is invariant under conjugation
with $x^{j}$. Hence this conjugation induces an automorphism of the quotient
group $G_{0}/\langle\left\\{y\right\\}\cup Z(G)\rangle$, which in turn yields
the factorization stated in the lemma. ∎

We can now formulate a sufficient criterion for $\pi$ to do phase retrieval,
based on suitable assumptions on the associated K-triple. The reasoning is
rather similar to Step 5 in the proof of Theorem 12.

######  Lemma 29.

Let $G$ denote a $p$-group, and $\pi$ an irreducible, faithful representation
of $G$, with a split K-triple $(G_{0},r,\tau)$. Assume that $\tau$ does phase
retrieval, and that $\pi$ is given by the realization from Lemma 27.

  1. (a)

Assume that $p_{0}(\tau)<1/2$. If $\eta_{1}\in l^{2}(\mathbb{Z}_{p^{r}})$ has
the phase retrieval property for the Schrödinger representation, and
$\eta_{2}\in\mathcal{H}_{\tau}$ has the phase retrieval property for $\tau$,
and additionally $p_{0}(\tau,\eta_{2})<1/2$, the vector

$\eta\in
l^{2}\left(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau}\right)~{},~{}\eta(j)=\eta_{1}(j)\cdot\eta_{2}$

does phase retrieval property for $\pi$. Such vectors exist.

  2. (b)

Pick any system $W\subset G_{0}$ of representatives modulo $\langle\\{y\\}\cup
Z(G)\rangle$, where $x,y\in G$ are associated to the K-triple according to
Definition 25. Assume that

$\forall w\in W~{}:~{}\alpha_{1}(w)=w$

holds, where $\alpha_{1}:W\to W$ is the bijection from Lemma 28. If
$\eta_{1}\in l^{2}(\mathbb{Z}_{p^{r}})$ has the phase retrieval property for
the Schrödinger representation, and $\eta_{2}\in\mathcal{H}_{\tau}$ has the
phase retrieval property for $\tau$, the vector

$\eta\in
l^{2}\left(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau}\right)~{},~{}\eta(j)=\eta_{1}(j)\cdot\eta_{2}$

has the phase retrieval property for $\pi$.

###### Proof.

We first derive formulae for matrix coefficients, which are valid for both
cases under consideration. Given $\eta=\eta_{1}\cdot\eta_{2}$ with
$\eta_{1}\in l^{2}(\mathbb{Z}_{p^{r}})$ and $\eta_{2}\in\mathcal{H}_{\tau}$,
the matrix coefficient of $f\in l^{2}(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau})$
is computed in the following two-step procedure: Using Lemma 27 for all
$k,l=0,\ldots,p^{r}-1$, with suitable choices of $x,y,z\in G$ and $W\subset
G_{0}$ as in the cited lemma, we have for all $w\in W$ and all
$k,l=0,\ldots,p^{r}-1,$

$V_{\eta}f(y^{k}wx^{l})=\sum_{j=0}^{p^{r}-1}\langle
f(j),\tau(x^{-j}wx^{j})\eta_{2}\rangle\xi_{1}^{-k}\xi_{2}^{kj}\overline{\eta_{1}(j-l)}~{}.$
(18)

This can be rephrased as the result of consecutive computations of matrix
coefficients with respect to $\tau$ and a (slightly modified version of) the
Schrödinger representation. For this purpose, we let

$F:W\times\mathbb{Z}_{p^{r}}\to\mathbb{C}~{},F(w,j)=\langle
f(j),\tau(x^{-j}wx^{j})\eta_{2}\rangle$ (19)

and note that

$F(w,j)=V_{\eta_{2}}^{\tau}(f(j))(x^{-j}wx^{j})~{}.$ (20)

Then equation (18) can be rewritten as

$V_{\eta}f(y^{k}wx^{l})=V_{\eta_{1}}^{\varrho}(F(w,\cdot))(k,l)$ (21)

where

$V_{\eta_{1}}^{\varrho}g(k,l)=\sum_{j=0}^{p^{r}-1}g(j)\xi_{1}^{-k}\xi_{2}^{kj}\overline{\eta_{1}(j-l)}~{}.$

This is just a matrix coefficient with respect to the Schrödinger
representation from Example 1, modified by multiplication with $\xi_{1}^{-k}$,
and by replacing the powers of the standard primitive $p^{r}$th root of unity,
given by $e^{2\pi i/p^{r}}$, by powers of $\xi_{2}$, which is a primitive root
of order $p^{r}$ as well, by Lemma 27(b). Hence, $V_{\eta_{1}}^{\varrho}g$
coincides with the windowed Fourier transform of $g$ with respect to
$\eta_{1}$ up to phase factors (independent of $g$) and a permutation of
coefficients. By Lemma 4 (c), these operations do not affect the phase
retrieval property, hence our choice of $\eta_{1}$ guarantees the phase
retrieval property for $V_{\eta_{1}}^{\varrho}$ as well.

Now assume that $f,h\in l^{2}(\mathbb{Z}_{p}^{r},\mathcal{H}_{\tau})$ are such
that $|V_{\eta}f|\equiv|V_{\eta}g|$ holds. We use $F$ from (19) and define $H$
analogously as

$H:W\times\mathbb{Z}_{p^{r}}\to\mathbb{C}~{},H(w,j)=\langle
h(j),\tau(x^{-j}wx^{j})\eta_{2}\rangle$ (22)

equation (21) gives for all $w\in W$

$\left|V_{\eta_{1}}^{\varrho}(F(w,\cdot))\right|=\left|V_{\eta_{1}}^{\varrho}(H(w,\cdot))\right|~{}.$

Since $\eta_{1}$ does phase retrieval, we obtain the existence of a map

$u:W\to\mathbb{T}~{},~{}H(w,\cdot)=u(w)F(w,\cdot)~{},~{}\forall w\in W~{}.$

This implies for all $j\in\mathbb{Z}_{p^{r}}$ that

$|H(\cdot,j)|\equiv|F(\cdot,j)|~{}.$

$W\subset G_{0}$ is a system of representatives modulo the central subgroup
$\langle\\{y\\}\cup Z(G)\rangle$ of $G_{0}$, and the latter is contained in
the projective kernel of $\tau$. Hence equation (20) (and its analog for $H$)
allows to rewrite the previous equation as follows, for all
$j\in\mathbb{Z}_{p^{r}}$:

$\forall g_{0}\in
G_{0}~{}:~{}\left|V_{\eta_{2}}^{\tau}(f(j))(x^{-j}g_{0}x^{j})\right|=\left|V_{\eta_{2}}^{\tau}(h(j))(x^{-j}g_{0}x^{j})\right|~{}.$

For fixed $j\in\mathbb{Z}_{p^{r}}$, conjugation by $x^{j}$ is an automorphism
of $G_{0}$, hence we may simplify this equation to

$\left|V_{\eta_{2}}^{\tau}(f(j))\right|=\left|V_{\eta_{2}}^{\tau}(h(j))\right|~{}.$

Now the phase retrieval property of $\eta_{2}$ implies the existence of a map

$v:\mathbb{Z}_{p^{r}}\to\mathbb{T}~{},\forall
j\in\mathbb{Z}_{p^{r}}:~{}h(j)=v(j)f(j)~{}.$ (23)

Note that this also entails, for all $j$, that

$H(\cdot,j)\equiv v(j)F(\cdot,j)~{}.$

Defining the set

$C_{f}=\\{j\in\mathbb{Z}_{p^{r}}~{}:~{}f(j)\not=0\\}~{},$ (24)

(23) shows that the proof of phase retrieval boils down to showing that $v$ is
constant on $C_{f}$. This is achieved if we can show that for all
$j,j^{\prime}\in C_{f}$ there exists $w\in W$ such that

$F(w,j)\cdot F(w,j^{\prime})\not=0~{},$

since then the reasoning finishing step 5 in the proof of Theorem 12 can be
easily adapted to the finite setting: We obtain that

$v(j)F(w,j)=H(w,j)=u(w)F(w,j)~{},~{}v(j^{\prime})F(w,j^{\prime})=u(w)F(w,j^{\prime})$

and $F(w,j),F(w,j^{\prime})$ can be cancelled to give
$v(j)=u(w)=v(j^{\prime})$.

It is only at this stage that we need to distinguish the parts (a) and (b) of
the lemma. In the case of part (a), we recall that $F(\cdot,j)$ is the
restriction to $W$ of a matrix coefficient with respect to the action of the
representation $\tau$ on $\eta_{1}$. By choice of $W$ and $\eta_{1}$, and in
view of Remark 17, the proportions of zero entries in $F(\cdot,j)$ and
$F(\cdot,j^{\prime})$ is therefore $<1/2$, for $j,j^{\prime}\in C_{f}$. Hence
the pidgeonhole principle yields the desired $w\in W$ with
$F(w,j)F(w,j^{\prime})\not=0$.

In case (b) we have $\alpha_{j}(w)=w$ for all $j\in\mathbb{Z}_{p^{r}}$. Here
all $F(\cdot,j)$ are restrictions of matrix coefficients to $W$, with respect
to the action of $\tau$ on $\eta_{2}$. Now the phase retrieval property of
$V_{\eta_{2}}^{\tau}$ entails via Lemma 4 (b) that $F(\cdot,j)\cdot
F(\cdot,j^{\prime})\not\equiv 0$ holds for $j,j^{\prime}\in C_{f}$.

We close the proof by observing that in case (a), the existence of a vector
$\eta_{2}$ fulfilling the assumptions of (a) is guaranteed by Lemma 20. ∎

The next definition extends the bookkeeping device provided by K-triples, in
order to account for repeated applications of Lemma 29.

######  Definition 30.

Let $G$ be a $p$-group, and $\pi$ an irreducible representation. A full
sequence of K-triples associated to $\mathbf{\pi}$ is a sequence of triples
$(H_{i},r_{i},\tau_{i})$ with the following properties: Let $\tau_{0}=\pi$,
$K_{0}={\rm ker}(\pi_{0}),H_{0}=G/K_{0}$, as well as $K_{i}={\rm
ker}(\tau_{i})$ for $i=1,\ldots,k$. Then, for all $0\leq i<k$,
$(H_{i+1},r_{i+1},\tau_{i+1})$ is a K-triple for the representation
$\overline{\tau}_{i}$ of $H_{i}/K_{i}$ induced by $\tau_{i}$, and
$H_{k}/K_{k}$ is abelian.

######  Remark 31.

Repeated application of Lemma 26 shows that every irreducible representation
$\pi$ has an associated full sequence of K-triples.

The sizes of the quotient groups are related by the estimate

$|H_{i+1}/K_{i+1}|\leq|H_{i+1}|=p^{-r_{i+1}}|H_{i}/K_{i}|~{},$

which results from the fact that $H_{i+1}<H_{i}/K_{i}$ has index
$p^{r_{i+1}}$. We further note that $H_{k}$ must be nontrivial: Recall that
$\overline{\tau}_{k-1}\simeq{\rm ind}_{H_{k}}^{H_{k-1}/K_{k-1}}\tau_{k}$ is
required to be irreducible. In the case of a trivial subgroup $H_{k}$, this
representation is just the left regular representation of the (nontrivial)
group $H_{k-1}/K_{k-1}$, which is never irreducible.

Hence $H_{k}$ has cardinality at least $p$. As a consequence we get that the
length $k$ of a full sequence of K-triples fulfills

$1+k\leq\log_{p}|G|~{}.$ (25)

Repeated applications of Lemma 29 requires control over properties of the
associated matrix coefficients, as quantified in $p_{0}(\pi)$. Since the
quantity $p_{0}(\pi)$ is of independent interest (see Remark 18), the
following lemma is as well.

######  Lemma 32.

Let $G$ denote a nonabelian $p$-group, and $\pi$ a faithful, irreducible
representation of $G$. Let $(G_{0},r,\tau)$ denote a split K-triple associated
to $\pi$. Then we have the estimate

$p_{0}(\pi)\leq p_{0}(\tau)+\frac{p^{r}-1}{p^{2r}}~{}.$

###### Proof.

We use the realization of $\pi$ from Lemma 27. Fix a system $W\subset G_{0}$
of representatives modulo $\langle\\{y\\}\cup Z(G)\rangle$. Given a nonzero
$f\in l^{2}(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau})$, we intend to estimate the
proportions of zeros in the family
$(V_{\eta}f(y^{k}wx^{l}))_{k,l=0,\ldots,p^{r}-1,w\in W}$. Since this is the
restriction of $V_{\eta}f$ to a system of representatives modulo the center of
$G$, the proportion of zeros of $V_{\eta}f$ coincides with this quantity.

We pick $\eta_{1}\in l^{2}(\mathbb{Z}_{p^{r}})$ and
$\eta_{2}\in\mathcal{H}_{\tau}$, and let $\eta=\eta_{1}\cdot\eta_{2}\in
l^{2}(\mathbb{Z}_{p^{r}},\mathcal{H}_{\tau})$. We recall some notations and
observations from the proof of Lemma 29, specifically

$F:W\times\mathbb{Z}_{p^{r}}\to\mathbb{C}~{},F(w,j)=\langle
f(j),\tau(x^{-j}wx^{j})\eta_{2}\rangle$ (26)

and the related formulas

$F(w,j)=V_{\eta_{2}}^{\tau}(f(j))(x^{-j}wx^{j})~{},$ (27)

and

$V_{\eta}f(y^{k}wx^{l})=V_{\eta_{1}}^{\varrho}(F(w,\cdot))(k,l)~{}.$ (28)

We assume that $\eta_{1},\eta_{2}$ are chosen such that

$p_{0}(\sigma,\eta_{1})=p_{0}(\sigma)=\frac{p^{r}-1}{p^{2r}}~{},~{}p_{0}(\tau,\eta_{2})=p_{0}(\tau)~{},~{}$

where $\sigma$ denotes the Schrödinger representation from Example 1, and the
second equality for $p_{0}(\sigma)$ follows by equation (14). Note that this
quantity is also an upper bound for the proportion of zeros in
$V_{\eta_{1}}^{\varrho}(g)$ for nonzero vector $g\in
l^{2}(\mathbb{Z}_{p^{r}})$, by the same reasoning as in the proof of Lemma 29:
The matrix coefficients $V_{\eta_{1}}^{\varrho}g$ and $V_{\eta_{1}}^{\sigma}g$
differ up to a multiplication with phase factors and a permutation, none of
which affects the proportion of zeros.

Let

$c_{0}=\frac{|\\{w\in W:F(w,\cdot)\equiv 0\\}|}{|W|}~{}.$

In view of (27) and (28) we have that

$V_{\eta}^{\pi}f(y^{k}wx^{l})=0$

may occur for fixed $w\in W$ and suitable $k,l$ either if $F(w,\cdot)\equiv
0$, in which case it holds for all $k,l\in\mathbb{Z}_{p^{r}}$, or if
$F(w,\cdot)\not\equiv 0$ holds. In the latter case, the proportion of zeros in
$V_{\eta}^{\pi}f(y^{k}wx^{l})$ for fixed $w$ can be bounded using
$p_{0}(\eta_{2},\sigma)$. These observations lead to the overall estimate

$\displaystyle|\\{(k,w,l):V_{\eta}^{\pi}f(y^{k}wx^{l})=0\\}|$
$\displaystyle\leq$ $\displaystyle|\\{w\in W:F(w,\cdot)=0\\}|\cdot
p^{2r}+|\\{w\in W:F(w,\cdot)\not=0\\}|\cdot p^{2r}p_{0}(\eta_{2},\sigma)~{}.$

Dividing both sides of this inequality by $|G/Z(G)|=|W|p^{2r}$ yields

$\frac{|\\{(k,w,l):V_{\eta}^{\pi}f(y^{k}wx^{l})=0\\}|}{|W|p^{2r}}\leq
c_{0}+(1-c_{0})\frac{p^{r}-1}{p^{2r}}\leq c_{0}+\frac{p^{r}-1}{p^{2r}}~{}.$

It therefore remains to prove

$c_{0}\leq p_{0}(\eta_{2},\tau)=p_{0}(\eta_{2})~{}.$ (29)

To see this, pick any $j_{0}\in\mathbb{Z}_{p^{r}}$ such that $f(j_{0})\not=0$;
$j_{0}$ exists since we assume $f$ to be nonzero. We can then estimate

$\displaystyle|\\{w\in W:F(w,\cdot)\equiv 0\\}|$ $\displaystyle\leq$
$\displaystyle|\\{w\in W:F(w,j_{0})=0\\}|$ $\displaystyle=$
$\displaystyle|\\{w\in
W:V_{\eta_{2}}^{\tau}(f(j_{0}))(x^{-j_{0}}wx^{j_{0}})=0\\}|$
$\displaystyle\leq$ $\displaystyle|W|\cdot p_{0}(\tau,\eta_{2})~{},$

which proves (29), and finishes the proof of the lemma. ∎

A straightforward inductive application of Lemma 29 now gives the following
result, which formulates a sufficient criterion for phase retrieval that can
be checked using a full sequence of K-triples.

######  Corollary 33.

Let $G$ denote a $p$-group. Assume that $\pi$ is an irreducible representation
of $G$ possessing a full sequence of K-triples $(H_{i},r_{i},\tau_{i})$, for
$i=1,\ldots,k$, that are all split. If

$\sum_{i=2}^{k}\frac{p^{r_{i}}-1}{p^{2r_{i}}}<\frac{1}{2}~{},$

then $\pi$ does phase retrieval.

###### Proof.

A downwards induction establishes

$p_{0}(\tau_{j})\leq\sum_{i=j+1}^{k}\frac{p^{r_{i}}-1}{p^{2r_{i}}}<\frac{1}{2}~{},$

for all $j=1,\ldots,k$. Hence Lemma 29 (a) can be applied $k$ times, and
yields the desired statement. ∎

As an application of the corollary, we show that a simple size estimate on
$|G|$ suffices to guarantee phase retrieval for all irreducible
representations of $G$, if $G$ has exponent $p$.

######  Corollary 34.

Assume that the $p$-group $G$ has exponent $p$ and fulfills $|G|\leq
p^{2+p/2}$. Then every irreducible representation of $G$ does phase retrieval.

###### Proof.

First note that, $p$ being a prime, the exponent $p$ property is equivalent to
stating that every nontrivial group element has order $p$. Clearly this
property is inherited by quotients and subgroups. As a consequence, every
K-triple is split and of type $(G_{0},1,\tau)$, and this observation extends
to all K-triples occurring in a full sequence associated to an irreducible
representation $\pi$.

Let $k$ denote the length of such a full sequence of K-triples. By equation
(25) and the assumption on $|G|$ we have

$k\leq\log_{p}|G|-1\leq 1+p/2~{}.$

Using that all exponents in the sequence of $K$-triples equal one, the
computation

$\displaystyle\sum_{i=2}^{k}\frac{p-1}{p^{2}}$ $\displaystyle=$
$\displaystyle(k-1)\frac{p-1}{p^{2}}$ $\displaystyle\leq$
$\displaystyle(\log_{p}|G|-2)\cdot\frac{p-1}{p^{2}}\leq\frac{p}{2}\cdot\frac{p-1}{p^{2}}<\frac{1}{2}~{}.$

shows that Corollary 33 applies, and yields the result. ∎

######  Remark 35.

It is quite possible that the requirement concerning split K-triples reflects
a shortcoming of our proof method, rather than being an objective obstacle for
the phase retrieval property. Understanding the precise relationship between
these properties remains to be investigated in future work.

Given that it is currently unclear which irreducible representations $\pi$ of
a general $p$-group have an associated sequences of K-triples that are also
split, the true scope of Corollary 33 is not yet fully understood. However,
Corollary 34 exhibits one general class to which Corollary 33 applies, and it
is already quite sizeable: According to [17], the number of $p$-groups of
order $p^{8}$ and exponent $p$, for primes $p>7$, is given by the formula

$p^{4}+2p^{3}+20p^{2}+147p+(3p+29)\gcd(p-1,3)+5\gcd(p-1,4)+1246~{},$

and Corollary 34 guarantees the phase retrieval properties for arbitrary
irreducible representations of these groups, as soon as $p\geq 13$. The
enumeration and systematic construction of $p$-groups with specific properties
is still an ongoing endeavour, and therefore the emergence of further general
classes which can be treated with Corollary 33 seems quite plausible.

In any case, we expect that Corollary 33 can be employed for the systematic
construction of $p$-groups and associated representations with particular
interesting properties. One such property is the nilpotency class of $G$:
Observe that the size restriction $|G|\leq p^{2+p/2}$ entails a restriction on
the nilpotency class. However, we conjecture that for any prime number $p$
there exist $p$-groups $G$ of arbitrarily large nilpotency class and/or size,
and irreducible representations of $G$ doing phase retrieval. A proof of this
conjecture via Corollary 33 would be achieved by constructing sequences of
split K-triples of arbitrary length $k$, with sufficiently large exponents
$r_{1},\ldots,r_{k}$.

## Concluding remarks

While the statements of our main results were existence results for vectors
doing phase retrieval, the inductive methods employed to prove these results
provide access to the explicit construction of such vectors.

In the Lie group case, this can be formulated in a rather satisfactory way. If
one follows the inductive construction of an arbitrary irreducible
representation by repeated applications of Kirillov’s lemma and induction of
representations, the canonical identification of Hilbert spaces
$L^{2}(\mathbb{R},L^{2}(\mathbb{R}^{k}))\cong L^{2}(\mathbb{R}^{k+1})$
ultimately provides a realization of $\pi$ on $\mathbb{R}^{m}$ for a suitable
$m\geq 1$. If one traces the inductive construction of the vector $\eta$ doing
phase retrieval in Step 3, one can employ a Gaussian for $\eta_{1}$ at each
step. In the identification $L^{2}(\mathbb{R},L^{2}(\mathbb{R}^{k}))\cong
L^{2}(\mathbb{R}^{k+1})$ one obtains $\eta(t,u)=\eta_{1}(t)\eta_{2}(u)$, and a
simple induction yields that $\eta$ is a Gaussian, as well.

This leads to the fundamental observation that every irreducible
representation of an arbitrary simply connected, connected nilpotent Lie group
has an explicitly computable realization on $L^{2}(\mathbb{R}^{m})$, for
suitable $m$, in which Gaussian vectors do phase retrieval.

In the finite group case, the setting is somewhat more involved, even assuming
that all elements in a full sequences of K-triples are already known to be
split: Note that the corresponding induction step in Lemma 29 requires
choosing a vector $\eta_{2}$ that guarantees phase retrieval and
$p_{0}(\eta_{2},\tau)<1/2$, and combining it with a vector $\eta_{1}$ doing
phase retrieval for the Schrödinger representation. Here the proof of Lemma 32
estimating $p_{0}(\eta_{2},\tau)$ relies on the same procedure to construct
$\eta$ from $\eta_{1}$ and $\eta_{2}$. That means that the vectors $\eta$
simultaneously guaranteeing the estimates in Lemma 32 and phase retrieval can
be constructed inductively as well, as soon as one can solve the following,
currently open problem: Are there explicit constructions of vectors $\eta$ for
the Schrödinger representation which simultaneously have the phase retrieval
and full spark property?

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