# iPCA and stability of star quivers

Cole Franks and Visu Makam

###### Abstract.

Integrated principal components analysis, or iPCA, is an unsupervised learning
technique for grouped vector data recently defined by Tang and Allen [38].
Like PCA, iPCA computes new axes that best explain the variance of the data,
but iPCA is designed to handle corrupting influences by the elements within
each group on one another - e.g. data about students at a school grouped into
classrooms. Tang and Allen showed empirically that regularized iPCA finds
useful features for such grouped data in practice. However, it is not yet
known when unregularized iPCA generically exists. For contrast, PCA (which is
a special case of iPCA) typically exists whenever the number of data points
exceeds the dimension. We study this question and find that the answer is
significantly more complicated than it is for PCA. Despite this complexity, we
find simple sufficient conditions for a very useful case - when the groups are
no more than half as large as the dimension and the total number of data
points exceeds the dimension, iPCA generically exists. We also fully
characterize the existence of iPCA in case all the groups are the same size.
When all groups are not the same size, however, we find that the group sizes
for which iPCA generically exists are the integral points in a non-convex
union of polyhedral cones. Nonetheless, we exhibit an algorithm to decide
whether iPCA generically exists that is polynomial in the node dimensions
(based on the affirmative answer for the saturation conjecture by [31]) as
well as a very simple randomized algorithm.

At its core, our approach identifies connections between iPCA and stability
notions for star quivers, thus bringing tools from invariant theory and quiver
representations to the table. Approaching invariant theory and quiver
representations from a computational perspective is a very rich endeavor and
has found many important applications in complexity theory and algebraic
statistics [26, 21, 7, 18, 1, 10] in the last decade. Our work identifies
another interesting and important algorithmic problem in the invariant theory
of quivers, i.e., given a dimension vector $\alpha$ and a weight $\sigma$ for
a quiver $Q$, decide if $\alpha$ is $\sigma$-semistable/polystable/stable.
While current techniques are insufficient to give a polynomial time algorithm
in any reasonable generality, we are able to leverage several interesting
features of the representations of star quivers (the setting relevant to iPCA)
to obtain algorithms that are polynomial time in the node dimensions by
appealing to powerful results in optimization. In the future, we hope to build
on these techniques to give polynomial time algorithms for deciding $\sigma$
stabilities of a dimension vector in greater generality.

###### Contents

  1. 1 Introduction
  2. 2 Invariant theory and MLE for Gaussian group models
  3. 3 Quiver representations and stability
  4. 4 Relating iPCA to stability of star quivers
  5. 5 Schofield stability for star quivers
  6. 6 Polytopal characterization of Schofield stability
  7. A From complex iPCA to real iPCA

##  1\. Introduction

Connections between stability notions in invariant theory and maximum
likelihood estimation for certain Gaussian models was discovered recently in
[1]. As one often finds with such an unexpected and deep connection, this
allows for the infusion of techniques from invariant theory in statistics.
This has already led to new and exciting results in statistics in a short
period of time, see [10, 11, 19, 20], with more results sure to come in the
future as it remains an active area of research.

Following in this spirit, we exploit the connection between invariant theory
and maximum likelihood estimation to characterize and efficiently decide the
existence of iPCA, a recently published unsupervised learning method.
Interestingly, the nature of the problems we pursue with respect to iPCA
identifies a new algorithmic direction in the invariant theory of quivers
giving evidence that such a connection can be beneficial in both directions.
In particular, we obtain new results in quiver representations. Thus, our
results are of interest to both statisticians and quiver theorists alike. We
present our main results in a fashion that can be appreciated by a wide
audience, yet we do not shy away from rigorously presenting the techniques we
develop in the language of quiver representations - we would like this paper
to serve as a motivation for quiver theorists to pursue the algorithmic
problems we identify and build on our results. We include precise references
to the literature for quiver theory that enables any reader to understand the
relevant results from quivers should they choose to do so. To this end, our
introduction begins with a very accessible discussion of iPCA and our results,
followed by a short indication of our approach via quivers, the details of
which will form the core of the rest of the paper.

###  1.1. Integrated Principal Components Analysis

Integrated principal components analysis, or iPCA, is an unsupervised learning
technique for grouped vector data recently defined by Tang and Allen [38].
Like PCA, iPCA computes new axes that best explain the variance of the data,
but iPCA is designed to handle corrupting influences by the elements within
each group on one another. In particular, the axes chosen by iPCA remain the
same even when each data point is replaced by an unknown linear combination of
the data points within its group. Suppose the $n$ data points in
$x_{1},\dots,x_{n}\in\mathbb{R}^{p}$ are split into $k$ groups, with the
$i^{th}$ group of size $q_{i}$. Let $B_{i}$ be the $p\times q_{i}$ matrix
whose columns are the data points in the $i^{th}$ group. See Fig. 1

The iPC scores $u_{i}\in\mathbb{R}^{d}$ are defined as the eigenvectors (in
increasing order of eigenvalue) of the positive-definite matrix
$\widehat{\Theta}$ defined as follows. For reasons to be explained shortly, we
call this the _ipca likelihood_.

(1)
$\displaystyle\widehat{\Theta},\widehat{\Theta}_{1},\dots,\widehat{\Theta}_{k}$
$\displaystyle=\arg\max f(\Theta,\Theta_{1},\dots,\Theta_{m})$ (2)
$\displaystyle\text{ where }f(\Theta,\Theta_{1},\dots,\Theta_{k})$
$\displaystyle=n\log|\Theta|+p\sum_{i=1}^{m}\log|\Theta_{i}|-\sum_{i=1}^{k}\operatorname{Tr}(\Theta
B_{i}\Theta_{i}B_{i}^{\dagger}).$

Here $|X|$ denotes the determinant of $X$. The minimization ranges over
$p\times p$ positive definite matrices $\Theta$ and $q_{i}\times q_{i}$
positive definite matrices $\Theta_{i}$. For comparison, the PCA axes are the
eigenvectors of the matrix $\widehat{\Theta}$ obtained by restricting each
$\Theta_{i}$ to be the $q_{i}\times q_{i}$ identity matrix in the above
optimization problem. We call
$\widehat{\Theta},\widehat{\Theta}_{1},\dots,\widehat{\Theta}_{k}$ the _iPCA
precision matrices_. Usually the main focus is in estimating $\Theta$; the
remaining parameters are thought of as nuisance parameters.

$\begin{array}[]{ccccc}\underset{B_{1}}{\left[\begin{array}[]{ccc}|&\dots&|\\\
x_{1}&\dots&x_{q_{1}}\\\
|&\dots&|\end{array}\right]},&\underset{B_{2}}{\left[\begin{array}[]{ccc}|&\dots&|\\\
x_{q_{1}+1}&\dots&x_{q_{1}+q_{2}}\\\
|&\dots&|\end{array}\right]},&\dots&,\underset{B_{k}}{\left[\begin{array}[]{ccc}|&\dots&|\\\
x_{n-q_{k}+1}&\dots&x_{n}\\\
|&\dots&|\end{array}\right]}&\left.\vphantom{\left[\begin{array}[]{ccc}|&\dots&|\\\
x_{1}&\dots&x_{q_{1}}\\\ |&\dots&|\end{array}\right]}\right\\}p\end{array}$
Figure 1. Data points grouped as matrices.

Like PCA, the iPCA precision matrices can be interpreted as a maximum
likelihood estimator (MLE) for the precision matrix of a centered Gaussian
random variable. Whereas the PCA axes are eigenvectors of the MLE
$\widehat{\Theta}$ for the precision matrix assuming the data points are drawn
independently from a centered Gaussian, the iPCA precisions
$\widehat{\Theta},\widehat{\Theta}_{1},\dots,\widehat{\Theta}_{k}$ are the MLE
for the precision matrix of a Gaussian with certain dependencies within each
group. More precisely, we assume that each matrix $B_{i}$ is drawn
independently from a centered Gaussian with precision matrix given by the
Kronecker product $\Theta\otimes\Theta_{i}$ for a $p\times p$ positive
definite matrix $\Theta$ and a $q_{i}\times q_{i}$ positive definite matrix
$\Theta_{i}$ (viewing $B_{i}$ as a vector in $\mathbb{R}^{p\cdot q_{i}}$).
Equivalently, $B_{i}$ is distributed as the $p\times q_{i}$ matrix
$\Theta^{-1/2}Y\Theta_{i}^{-1/2}$ where $Y$ is a $p\times q_{i}$ matrix with
independent, standard normal entries. Under this assumption,
$\widehat{\Theta}\otimes\widehat{\Theta}_{i}$ is the maximum likelihood
estimator for $\Theta\otimes\Theta_{i}$.

The maximum likelihood interpretation of iPCA captures several estimation
problems of interest. The case $m=1$ is the _matrix normal model_ , or the
estimation of a covariance/precision matrix that can be written as a Kronecker
product. The matrix normal model is used to model matrix-variate data that
arises naturally in numerous applications like gene microarrays, spatio-
temporal data, and brain imaging [42]. On the other hand, when $q_{i}=1$ for
all $1\leq i\leq m$, one finds that $\widehat{\Theta}$ is proportional to
_Tyler’s M estimator_ for the shape of an elliptical distribution [40]. iPCA
interpolates between these two situations. We can interpret iPCA as an
estimator for the precision matrix $\Theta$ that is robust to each data point
in the group being replaced by a linear combination of the other data points
in its group (i.e. replacing $B_{i}$ by $B_{i}\Theta_{i}^{-1/2}$) which can be
chosen adversarially after seeing the data. As illustrated by Allen and Tang
[38], such corruptions can obscure the true precision matrix from PCA and
other methods for integrating groups of data. For a detailed discussion on the
advantages and usefulness of iPCA and its strong performance in case studies,
we refer the reader to [38] where iPCA was used as a preprocessing method in a
machine learning pipeline for predicting patients’ cognition and Alzheimer’s
diagnosis.

###  1.2. Existence and uniqueness of iPCA

Despite the practical utility of iPCA, the existence and uniqueness of iPCA is
poorly understood. Like PCA, iPCA need not exist if the data is too
pathological. However, it is known that PCA exists for _generic_ data whenever
there are as many samples as there are features. In contrast, it was not known
in which dimensions and for which sizes of groups iPCA exists for generic
data. In the extreme case when all the groups are of size one (Tyler’s M
estimator), it is known that iPCA generically exists whenever the number $n$
of data points is strictly larger than the dimension $d$ [40]. On the other
extreme, the number of samples required for the MLE to generically exist
uniquely for the matrix normal model was found exactly in [10], and is on the
order of $p/q+q/p$. In [38], Tang and Allen largely bypass the question of
existence and uniqueness of iPCA by considering instead a regularized version
of Eq. 2.

In this paper we characterize the existence and uniqueness of iPCA. We say
iPCA generically exists for dimension $p$ and group sizes $q_{1},\dots,q_{k}$
if the iPCA precisions defined in Eq. 2 exist and are unique for generic
$B_{1},\dots,B_{k}$. In particular, if iPCA generically exists uniquely then
the iPCA precisions exist with probability $1$ if $B_{1},\dots,B_{k}$ is
chosen according to any distribution on
$\operatorname{Mat}(p,q_{1})\times\dots\times\operatorname{Mat}(p,q_{k})$ that
has a density with respect to the Lebesgue measure. The phrases ‘iPCA
generically exists uniquely’ and ‘Eq. 2 is generically bounded/unbounded
above’ are defined similarly. Our first result is a simple sufficient
condition for the generic existence and uniqueness of iPCA.

######  Theorem 1.1.

iPCA generically exists uniquely for $p,q_{1},\dots,q_{k}$ provided

$q_{i}\leq\frac{p}{2}\text{ and }2p\leq n.$

As iPCA does not exist whenever $p>n$ or whenever $q_{i}>p$ for some group
$i$, the theorem is tight up to constant factors. When all the groups sizes
are the same, we obtain the following necessary and sufficient conditions for
the generic unique existence of iPCA.

######  Theorem 1.2.

Fix $k$. Let $\Gamma(p,q)=kq^{2}+p^{2}-kpq$. Let $r={\rm gcd}(p,q)$. Then

  * •

$\Gamma(p,q)\leq r^{2}$ implies iPCA generically exists for $(p,q,\dots,q)$.

  * •

$\Gamma(p,q)<0$ or $\Gamma(p,q)\in\\{0,r^{2}\\}$ with $r=1$ implies iPCA
generically exists _uniquely_ for $(p,q,\dots,q)$.

  * •

In all other cases, i.e., $\Gamma(p,q)>r^{2}$, Eq. 2 is generically unbounded
above (iPCA does not exist).

When the group have differing sizes, the characterization is more complex.
Nonetheless, well-known polyhedral characterizations of quiver stability imply
a polyhedral characterization for the existence and uniqueness of iPCA. The
conditions for existence and uniqueness will require checking firstly that the
dimension vector resides in a certain cone, which will be a union of
polyhedral cones, and secondly that certain divisibility conditions are
satisfied. To this end, we define the _region of generic unique existence_ as
the set

$S=\\{p,q_{1},\dots,q_{k}\in\mathbb{R}^{k+1}:\text{iPCA generically exists
uniquely for }p,q_{1},\dots,q_{k}\\}\subset\mathbb{R}^{k+1}$

and the _cone of generic unique existence_ as $\mathbb{R}_{\geq
0}S=\\{av:a\geq 0,v\in S\\}$.

######  Theorem 1.3.

Suppose $q_{1}+\dots+q_{k}=n$. Let $d=n/p$.

  1. (1)

iPCA generically exists for $p,q_{1},\dots,q_{k}$ if and only if Eq. 2
generically bounded above.

  2. (2)

There is a convex polyhedral cone $\Sigma_{d,k}\subset\mathbb{R}^{k+1}$ such
that iPCA exists for $p,q_{1},\dots,q_{k}$ if and only if
$(p,q_{1},\dots,q_{k})\in\Sigma_{d,k}.$

  3. (3)

The cone of generic unique existence for iPCA is a union of finitely many
convex polyhedral cones. Moreover, generic existence and uniqueness of iPCA
for $p,q_{1},\dots,q_{k}$ can be decided in
$\operatorname{poly}(p,n,q_{1},\dots,q_{k})$.

######  Remark 1.4 (Time complexity).

As the algorithm in Theorem 1.3 depends polynomially on $q_{1},\dots,q_{k}$
rather than the total length of their binary encodings, it is not technically
a polynomial time algorithm. However, it _is_ polynomial time in the input
size to the iPCA estimation problem which is a vector of dimension
$n\cdot(\sum_{i}q_{i})$. For this reason we abuse notation and refer to this
algorithm as polynomial time in the remainder of this work.

The polyhedral characterization given by the theorem suggests that
$p,q_{1},\dots,q_{k}$ need not span a convex cone, but rather a union of
convex cones which may or may not be convex. Fig. 2 shows a non-convex
example.

Though the above theorem gives an efficient deterministic algorithm to decide
generic existence and uniqueness of iPCA, this algorithm is rather
complicated. We also provide a simple randomized algorithm to decide generic
existence and uniqueness of iPCA. Computing the iPCA precisions is an instance
of an algorithmic problem called _operator scaling_ , which is known to be
solvable to an arbitrary degree of precision in polynomial time [21].

Figure 2. The intersection of the cone of generic unique existence for $k=5$
in $\mathbb{R}^{6}$ with a random three-dimensional subspace through the
origin.

######  Remark 1.5 (Consistency of iPCA).

Interpreting iPCA as a maximum likelihood estimator, it is natural to ask
under what circumstances the maximum likelihood estimator $\hat{\Theta}$ is
likely to be close to the true $\Theta$. This question was studied in [38] for
regularized versions of iPCA. While this is not the focus of the present
paper, the works [19, 20] provided bounds for error rates for the matrix
normal model and Tyler’s M estimator, respectively, using tools from
geodesically convex optimization. We conjecture that their techniques applied
in this setting will yield error bounds comparable to those of PCA under
slightly stronger hypotheses than Theorem 1.1. Namely, we conjecture that for
$p=\tilde{\Omega}(\max\\{q_{1},\dots,q_{n}\\})$ and $n=\tilde{\Omega}(p)$ we
have $\|I-\Theta^{-1/2}\hat{\Theta}\Theta^{-1/2}\|_{op}=\tilde{O}(\sqrt{p/n})$
with high probability in $n$. The $\tilde{\Omega},\tilde{\Omega}$ hide
polylogarithmic factors in $q_{i}$ and $p$, respectively.

###  1.3. iPCA and stability for star quivers

Our central observation is that the existence and uniqueness of iPCA can be
interpreted in terms of _quiver representations_. A quiver representation is a
collection of vector spaces together with a collection of linear maps between
the vector spaces. The spaces and directions of linear maps are represented by
a directed multigraph. The iPCA problem arises from the so-called _star
quiver_ (see the picture below). Given data
$B_{1}\in\operatorname{Mat}_{p,q_{1}},\dots,B_{m}\in\operatorname{Mat}_{p,q_{m}}$,
we form a quiver representation in a natural way. The vertices are the spaces
$\mathbb{R}^{p},\mathbb{R}^{q_{1}},\dots,\mathbb{R}^{q_{m}}$, and for
$i\in[m]$ there is a directed edge from $\mathbb{R}^{q_{i}}$ to
$\mathbb{R}^{p}$ corresponding to the linear map $B_{i}$:

(3)
${\mathbb{R}^{p}}$${\mathbb{R}^{q_{1}}}$${\mathbb{R}^{q_{2}}}$${\dots}$${\mathbb{R}^{q_{k}}}$$\scriptstyle{B_{1}}$$\scriptstyle{B_{2}}$$\scriptstyle{B_{k}}$

The existence and uniqueness of iPCA is captured by stability notions on the
associated quiver representations. Stability is a technical notion from
geometric invariant theory, whose definition we postpone to Section 3. A
surprising and extremely useful link between maximum likelihood estimation and
stability notions in invariant theory was discovered recently [1], which we
will recall in Section 2.

Postponing the exact definitions, we indicate briefly a few things to give the
reader an idea of the broad strokes of our approach and the techniques and
ideas we draw from. For a quiver $Q$, a dimension vector $\alpha$ and a weight
$\sigma$, a natural algorithmic question to ask is whether a generic
$\alpha$-dimensional representation of $Q$ is
$\sigma$-semistable/polystable/stable. In the case of iPCA, we consider the
star quiver as above and take the dimension vector
$\alpha=(p,q_{1},\dots,q_{k})$ as indicated above with the weight
$\sigma=(-n,p,\dots,p)$. We find that that
$\sigma$-semistability/polystability/stability in this setting is equivalent
to having Eq 2 generically bounded above/iPCA generically existing/iPCA
generically existing uniquely.

In addition to all the general results from quiver representations that we
use, we wish to point out a few important features in this particular setting
that are useful to us. With completely unrelated motivations, Schofield had
identified a special weight (which we call the Schofield weight) for any
dimension vector. The first crucial observation is that for the dimension
vector $(p,q_{1},\dots,q_{k})$ for the star quiver, the Schofield weight is
$\sigma=(-n,p,\dots,p)$, the precise weight relevant to iPCA. The second
important idea is an overutilization of symmetry. While we naturally view the
iPCA data as a representation of the star quiver, it can just as naturally be
viewed as a representation of its opposite quiver (i.e., the star quiver with
all arrows reversed). Well known to quiver theorists is a non-trivial way to
relate the star quiver and its opposite through the use of reflection
functors. Combining reflection functors with other obvious symmetries allows
us to reduce the number of cases to consider significantly, in particular
allows us to prove Theorem 1.1. The reader familiar with castling transforms
and prehomogeneous spaces will appreciate the similarity to the classication
of prehomogenous spaces of tensors, see e.g., [41]. Finally, we note that
generic semistability can be studied via semi-invariants. In the case of
interest for iPCA, one can relate the semi-invariants to extremely well
studied numbers in algebraic combinatorics called Littlewood-Richardson
coefficients. In particular, Knutson and Tao’s seminal work on the saturation
conjecture and other results that followed provides us with the tools needed
for us to exhibit algorithms that run in polynomial time in the node
dimensions.

We also feel it is worth mentioning that while iPCA is naturally a problem
over the field of real numbers, most known results on quiver representations
and stability hold for algebraically closed fields such as the field of
complex numbers. This creates a rather subtle difficulty in applying the
results on quiver representations directly to iPCA. In fact, this is a problem
that one is very likely to encounter while trying to relate stability or
invariant theory to statistics in any setting in the manner discovered in [1].
To overcome this problem, one must appeal to some deep and technical results
on algebraic groups. Hence, that discussion is kept entirely in the appendix
for the sake of brevity and clarity.

###  1.4. Related work

The independent work [4], which was released during the preparation of this
draft, also provides an algorithm for the generic existence and uniqueness of
iPCA.

###  1.5. Organization and notation

In Section 2 we discuss the relationship between Gaussian group models and
stability in invariant theory, and in particular how iPCA can expressed in
this language. In Section 3 we review the notions we need about the stability
of quivers. In Section 4 we relate the generic existence of iPCA to the
Scofield stability of star quivers, and then in Section 5 we prove results
about Schofield stability and use them to finally prove Theorem 1.1 and
Theorem 1.2. In Section 6 we describe the polytopal characterization of the
existence of iPCA and our algorithmic results, proving Theorem 1.3.

We use the following symbols. The tuple $(p,q_{1},\dots,q_{k})$ will denote
the dimensions in iPCA, and $n:=\sum_{i}q_{i}$. We use
$B_{i}:\mathbb{R}^{q_{i}}\to\mathbb{R}^{p}$ for the linear maps in iPCA, and
$\Theta$, $\Theta_{i}$ for the solution to iPCA (inverses of what appear in
Allen-Tang). Finally we use $\Sigma_{d,n}$ for the polyhedron describing
semistability under the weight $(-n,p,\dots,p)$, or equivalently the weight
$(-d,1,\dots,1)$.

##  2\. Invariant theory and MLE for Gaussian group models

Invariant theory is the study of symmetries captured by group actions. From
its very beginnings, there has been significant focus on computation in the
subject of invariant theory and as a consequence, invariant theory finds many
applications. We refer to [5] and references therein for a comprehensive
introduction to the subject and its applications. In this section, we will
recall some basic notions in invariant theory and draw the connection to
maximum likelihood estimation that was discovered recently in [1].

For this section, let $K=\mathbb{R}$ or $\mathbb{C}$ be the ground field. The
basic setting is as follows. A representation of a group $G$ is an action of
$G$ on a (finite-dimensional) vector space $V$ (over the field $K$) by linear
transformations. Equivalently, a representation can be thought of as a group
homomorphism $\rho:G\rightarrow\operatorname{GL}(V)$. In particular, an
element $g\in G$ acts on $V$ by the linear transformation $\rho(g)$. We write
$g\cdot v$ or $gv$ to mean $\rho(g)v$. Throughout this paper, we will only
consider the setting where $G$ is a linear algebraic group (over the ground
field $K$), i.e., $G$ is an (affine) variety, the multiplication and inverse
maps are morphism of varieties, and the action is a rational action (or
rational representation), i.e., $\rho:G\rightarrow\operatorname{GL}(V)$ is a
morphism of algebraic groups.

The $G$-orbit of $v\in V$ is

$O_{v}:=\\{gv\ |\ g\in G\\}\subseteq V,$

and we denote by $\overline{O_{v}}$ the closure of the orbit $O_{v}$.

######  Remark 2.1.

To define the closure, we need to define a topology on $V$. In this paper, we
will only use the fields $K=\mathbb{R}$ or $\mathbb{C}$ and so we use the
standard Euclidean topology on $V$ for orbit closures, unless otherwise
specified. This is not standard. In literature, the topology is usually taken
as the Zariski topology. Thankfully, in the setting of rational actions of
reductive groups, for $K=\mathbb{C}$, the orbit closure with respect to the
Euclidean topology agrees with the orbit closure with respect to the Zariski
topology.

For a point $v\in V$, the subgroup $G_{v}:=\\{g\in G\ |\ gv=v\\}$ is called
the stabilizer. We now define stability notions in invariant theory that play
a central role in this paper.

######  Definition 2.2.

Let $K=\mathbb{R}$ or $\mathbb{C}$, and let $G$ be an algebraic group (over
$K$) with a rational action on a vector space $V$ (over $K$), i.e.,
$\rho:G\rightarrow\operatorname{GL}(V)$. Let $\Delta$ denote the kernel of the
homomorphism $\rho$. Give $V$ the standard Euclidean topology. Then, for $v\in
V$, we say $v$ is

  * •

unstable if $0\in\overline{O_{v}}$;

  * •

semistable if $0\notin\overline{O_{v}}$;

  * •

polystable if $v\neq 0$ and $O_{v}$ is closed;

  * •

stable if $v$ is polystable and the quotient $G_{v}/\Delta$ is finite.

Observe that a point is unstable if and only if it is not semistable and also
note that stable $\implies$ polystable $\implies$ semistable.

For any action of $G$ on $V$, there is a natural diagonal action on the direct
sum $V^{m}$ defined by $g\cdot(v_{1},\dots,v_{m})=(gv_{1},\dots,gv_{m})$ for
all $g\in G$ and $v_{i}\in V$. Moreover, note that for any group action
$\rho:G\rightarrow\operatorname{GL}(V)$, the notions of semistable, polystable
and stable are the same whether we consider the action of $G$ or the action of
the group $\rho(G)$ on $V$ (by matrix-vector multiplication).

The set of all unstable points is called the null cone, a central object in
the study of computational invariant theory. When $K=\mathbb{C}$, the
invariant ring is non-trivial precisely when a generic point is semistable,
which is equivalent to the existence of a semistable point, i.e., the null
cone is not all of $V$. The null cone membership is the problem of deciding
whether a vector is in the null cone, i.e., whether it is unstable. This
problem has received enormous amount of attention in recent years, see [2]
(and references therein) for a comprehensive overiew. Similarly, the problem
of deciding whether a given vector is polystable has also been studied due to
its applications to degree lower bounds for invariant rings, see [9, 7].

###  2.1. MLE for Gaussian group models and invariant theory

A curious and unexpected connection between stability notions defined above
and maximum likelihood estimation was discovered in [1]. We now explain this
connection in more detail and in the subsequent subsection, we describe
explicitly the relation to iPCA.

Suppose $K=\mathbb{R}$ or $\mathbb{C}$. We consider multivariate Gaussian
models on $K^{n}$. In the case of $K=\mathbb{R}$, we take our Gaussian
distributions to be centered (i.e. mean zero), so that the distribution is
defined by simply giving its precision matrix which is a positive definite
matrix. In the case of $K=\mathbb{C}$, we take our Gaussian distributions to
be circularly symmetric, so that the distribution is again defined by simply
giving its precision matrix which is a positive definite Hermitian matrix. Let
${\rm PD}_{n}$ denote the space of positive definite (Hermitian when
$K=\mathbb{C}$) matrices with entries in $K$. By the above discussion, a
subset of ${\rm PD}_{n}$ defines a Gaussian model.

Let $\rho:G\rightarrow\operatorname{GL}_{n}$ be a representation of a group
$G$. Then, we define a corresponding Gaussian group model
$\mathcal{M}_{G}:=\\{\rho(g)^{\dagger}\rho(g)\ |\ g\in G\\}\subseteq{\rm
PD}_{n}$.111Note that adjoint is the same as transpose for a matrix with real
entries. Informally speaking, the maximum likelihood estimation problem
decomposes into two parts. Firstly, estimate the maximum likelihood up to a
scalar factor by computing the maximum likelihood estimator in a subgroup of
matrices of determinant $1$, denoted $G_{SL}$. After this is done, finding the
correct scalar factor becomes a trivial one-dimensional optimization problem
with a closed form solution. This approach leads to the following result for
Gaussian group models proved in [1] (we state a more general, but equivalent
form of their result; see also [10]).

######  Theorem 2.3 ([1]).

Let $K=\mathbb{R}$ or $\mathbb{C}$, and let $V$ be a finite dimensional
Hilbert space, i.e., a vector space with a positive definite inner product
(Hermitian when $K=\mathbb{C}$). Let $\rho:G\rightarrow\operatorname{GL}(V)$
be a rational action of $G$ on $V$. Suppose
$\rho(G)\subseteq\operatorname{GL}(V)$ is a Zariski closed subgroup, closed
under adjoints and non-zero scalar multiples. Let
$G_{\operatorname{SL}}\subseteq G$ be a subgroup such that
$\rho(G_{\operatorname{SL}})=\rho(G)\cap(\operatorname{SL}(V))$ and let $Y\in
V^{m}$ be an $m$-tuple of samples. Then, for the (diagonal) action of
$G_{\operatorname{SL}}$ and the model $\mathcal{M}_{G}$, we have

  * •

$Y$ is semistable $\Longleftrightarrow$ The likelihood $l_{Y}$ is bounded from
above;

  * •

$Y$ is polystable $\Longleftrightarrow$ an MLE exists;

  * •

$Y$ is stable $\implies$ there is a unique MLE. If $K=\mathbb{C}$, the
converse also holds, i.e., there is a unique MLE $\implies Y$ is stable.

The fact that the converse of the last part does not hold for $K=\mathbb{R}$
is a bit disappointing. However, a partial result in that regard which is very
useful is the following:

######  Lemma 2.4.

Keep the notation of Theorem 2.3. Let $K=\mathbb{R}$. If there is a unique MLE
given $Y$, then the image of the stabilizer
$(\rho(G_{\operatorname{SL}}))_{Y}=\rho((G_{\operatorname{SL}})_{Y})$ is
compact.

###### Proof.

Suppose there is a unique MLE given $Y$. Suppose $g\cdot Y$ is a point of
minimal norm in the $G_{\operatorname{SL}}$-orbit of $Y$. Then, the MLEs are
in bijection with $\rho((G_{\operatorname{SL}})_{g\cdot
Y})/\rho((G_{\operatorname{SL}})_{g\cdot Y})\cap O(V)$, where $O(V)$ is the
subgroup of $V$ that preserves the inner product, see [1]. In particular, we
have a unique MLE if and only if $\rho((G_{\operatorname{SL}})_{g\cdot
Y})\subseteq O(V)$. Since $\rho((G_{\operatorname{SL}})_{g\cdot
Y})=(\rho(G_{\operatorname{SL}}))_{g\cdot Y}$ is a closed subgroup of $O(V)$
(indeed stabilizers are closed subgroups), we see that it is compact because
$O(V)$ is compact. As $\rho(G_{\operatorname{SL}})_{g\cdot Y}$ is conjugate to
$\rho(G_{\operatorname{SL}})_{Y}$, we get that the latter is compact as well.
∎

###  2.2. iPCA as a Gaussian group model

In this subsection, we place iPCA in the framework of Gaussian group models so
that we can study existence and uniqueness of iPCA by studying the
corresponding invariant theory. Later on, we will see that the invariant
theory that is related to iPCA is the invariant theory of quivers, in
particular the star quiver, see Section 4.

Let $K=\mathbb{R}$ or $\mathbb{C}$. Let $\alpha=(p,q_{1},\dots,q_{k})$ and
recall that $n=\sum_{i}q_{i}$. Consider the group
$G=\operatorname{GL}(\alpha)=\operatorname{GL}_{p}\times\prod_{i=1}^{k}\operatorname{GL}_{q_{i}}$.
Consider the action $\rho=\rho_{\alpha}$ of $G$ on
$\oplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$ given by

(4)
$(g,h_{1},\dots,h_{k})\cdot(B_{1},\dots,B_{k})=(gB_{1}h_{1}^{-1},gB_{2}h_{2}^{-1},\dots,gB_{k}h_{k}^{-1}).$

Identify $\oplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$ with $K^{pn}$ as
follows. For matrices
$(B_{1},\dots,B_{k})\in\oplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$, place
them (horizontally) next to each other to get a matrix of size $p\times n$.
Then, vectorize this matrix, i.e., stack its columns. With this
identification, the representation $\rho:G\rightarrow\operatorname{GL}_{pn}$
given by

$\rho(g,h_{1},\dots,h_{k})=g\otimes{\rm
diag}((h_{1}^{-1})^{\top},\dots,(h_{k}^{-1})^{\top}),$

where $\otimes$ denotes the Kronecker (or tensor) product of matrices and
${\rm diag}((h_{1}^{-1})^{\top},\dots,(h_{k}^{-1})^{\top})$ denotes a block
diagonal matrix whose diagonal blocks are
$(h_{1}^{-1})^{\top},\dots,(h_{k}^{-1})^{\top}$.

For a choice of
$\sigma=(\sigma_{0},\sigma_{1},\dots,\sigma_{k})\in\mathbb{Z}^{k+1}$, we
define the subgroup

$\operatorname{GL}(\alpha)_{\sigma}:=\\{(g,h_{1},\dots,h_{k})\in\operatorname{GL}(\alpha)\
|\ \det(g)^{\sigma_{0}}\cdot\prod_{i=1}^{k}\det(h_{i})^{\sigma_{i}}=1\\}.$

######  Lemma 2.5.

Let $\sigma=(-n,p,p,\dots,p)$. Then
$\rho_{\alpha}(G)\cap\operatorname{SL}_{pq}=\rho_{\alpha}(\operatorname{GL}(\alpha)_{\sigma})$.

###### Proof.

This is clear since
$\det(\rho_{\alpha}(g))=\det(g)^{n}\cdot\prod_{i=1}^{k}\det(h_{i})^{-p}$. ∎

######  Proposition 2.6.

Let $K=\mathbb{R}$ or $\mathbb{C}$. Let $\alpha=(p,q_{1},\dots,q_{k})$,
$G=\operatorname{GL}(\alpha)$ and consider the action $\rho_{\alpha}$ of $G$
on ${\rm
Rep}(\Delta[k],\alpha))=\oplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$. Let
$\mathcal{M}_{\alpha}$ denote the corresponding Gaussian group model. Let
$\sigma=(-n,p,p,\dots,p)$. Then, for $B\in{\rm Rep}(\Delta[k],\alpha))$, we
have

  * •

$B$ is $\operatorname{GL}(\alpha)_{\sigma}$-semistable $\Longleftrightarrow
l_{B}$ is bounded from above;

  * •

$B$ is $\operatorname{GL}(\alpha)_{\sigma}$-polystable $\Longleftrightarrow$
an MLE exists;

  * •

$B$ is $\operatorname{GL}(\alpha)_{\sigma}$-stable $\implies$ there is a
unique MLE. Further, if $K=\mathbb{C}$, the converse also holds, i.e., there
is a unique MLE $\implies B$ is $\operatorname{GL}(\alpha)_{\sigma}$-stable.
Moreover, when $K=\mathbb{R}$, there is a unique MLE $\implies$ the stabilizer
$(\rho_{\alpha}(\operatorname{GL}(\alpha)_{\sigma}))_{B}$ is compact.

The notation ${\rm Rep}(\Delta[k],\alpha))$ is certainly uninitiated but will
become clear when we discuss quivers. But we use this notation to maintain
consistency in the paper.

###### Proof.

This follows from Theorem 2.3 and Lemma 2.4 since we can take
$\operatorname{GL}(\alpha)_{\sigma}$ as $G_{\operatorname{SL}}$ by Lemma 2.5.
∎

Now, a simple computation shows that the model $\mathcal{M}_{\alpha}$ is given
by

(5) $\mathcal{M}_{\alpha}=\\{\Theta\otimes{\rm
diag}(\Theta_{1},\dots,\Theta_{k})\ |\ \Theta\in{\rm PD}_{p},\Theta_{i}\in{\rm
PD}_{q_{i}}\\}.$

Allen and Tang [38] describe iPCA as the maximum likelihood estimation for the
Gaussian model $\mathcal{M}_{\alpha}$. Moreover, in [38], the log-likelihood
function of $\mathcal{M}_{\alpha}$ is computed.

Let $K=\mathbb{R}$. Let $\alpha=(p,q_{1},\dots,q_{k})$ and let
$n=\sum_{i}q_{i}$. Let $\operatorname{GL}(\alpha),\rho_{\alpha},{\rm
Rep}(\Delta[k],\alpha))$ and $\mathcal{M}_{\alpha}$ be as above. Then, for
$B=(B_{1},\dots,B_{k})\in{\rm Rep}(\Delta[k],\alpha))$, the log-likelihood
function (see [Eq. 12, [38]] is

$l_{B}(\Theta\otimes{\rm
diag}(\Theta_{1},\dots,\Theta_{k}))=n\log|\Theta|+p\sum_{i=1}^{k}\log|\Theta_{i}|-\sum_{i=1}^{k}\operatorname{Tr}(\Theta
B_{i}\Theta_{i}B_{i}^{\dagger})$

There is a formula for the log-likelihood function of a Gaussian group model
is given in [1]. The interested reader can verify that specializing that to
the Gaussian group model $\mathcal{M}_{\alpha}$ you recover the same formula.

For sample data $B=(B_{1},\dots,B_{k})\in{\rm Rep}(\Delta[k],\alpha))$, the
iPCA precisions are equal to maximum likelihood estimators for $B$ with
respect to the model $\mathcal{M}_{\alpha}$. More precisely, the function Eq 2
of which the iPCA precisions are the optima is the log-likelihood function of
$\mathcal{M}_{\alpha}$ and hence the iPCA precision matrices correspond to
MLEs. Hence, existence and uniqueness of iPCA is the same as existence and
uniqueness of MLE for the model $\mathcal{M}_{\alpha}$.

Summarizing the above discussion, we get:

######  Corollary 2.7.

Let $K=\mathbb{R}$. Let $\alpha=(p,q_{1},\dots,q_{k})$. Let
$\operatorname{GL}(\alpha),\rho_{\alpha},{\rm Rep}(\Delta[k],\alpha))$ and
$\mathcal{M}_{\alpha}$ be as above. Then, for sample data in ${\rm
Rep}(\Delta[k],\alpha))$:

  * •

Eq 2 is the log-likelihood function for $\mathcal{M}_{\alpha}$ is generically
bounded above.

  * •

iPCA precision matrices are precisely the MLE w.r.t to $\mathcal{M}_{\alpha}$.

######  Remark 2.8.

If we take $K=\mathbb{C}$, then maximum likelihood estimation for the complex
model $\mathcal{M}_{\alpha}$ can be seen as a complex version of iPCA. Our
results on the real model (on existence and uniqueness of MLE) also hold for
this complex model. Indeed, our proof strategy is to find the answers for the
complex model and then transfer the results to the real model.

##  3\. Quiver representations and stability

From a purely algebraic standpoint, the theory of quiver representations was
built to be a natural and rich generalization of linear algebra. Quiver
representations have numerous applications to various areas in algebra and
geometry, e.g., cluster algebras [16, 29], Schubert calculus [12, 36, 13],
non-commutative algebraic geometry [35] and symplectic resolutions [23] to
name a few. In the last decade, connections between quivers and computational
complexity were discovered [25] and this led to progress on non-commutative
rational identity testing and more, see [21, 7, 26, 6, 8, 32, 22]. With regard
to statistics, exact sample size thresholds for matrix normal models were
obtained in [10] by utilizing a connection to the invariant theory of
Kronecker quivers (via Theorem 2.3) that was first explained in [1].

In this section, we will discuss the basic notions of quiver representations
and stability of quiver representations. Then, we will explain in detail the
connection between iPCA and star quivers, which will be used in the subsequent
sections to prove our main results on existence and uniqueness of iPCA.

Let $K$ denote the ground field, we will only consider $K=\mathbb{R}$ or
$\mathbb{C}$ in this paper. A quiver $Q$ is a directed acyclic graph222In
literature, quivers are not always taken to be directed acyclic. But for our
purposes, we will make this a standing assumption throughout not only because
this is the case that is relevant to us, but also because the theory is better
understood in this case., i.e. a set of vertices denoted $Q_{0}$ and a set of
arrows $Q_{1}$. For each arrow $a\in Q_{1}$, we denote by $ta$ and $ha$, the
tail vertex and head vertex of the arrow. We will demonstrate all the basic
notions and definitions in the crucial example (below) of the star quiver
$\Delta[k]$ with vertices $x$ and $y_{1},\dots,y_{k}$ with $k$ arrows
$a_{1},\dots,a_{k}$ such that the arrow $a_{i}$ goes from $y_{i}$ to $x$ .

(6)
${x}$${y_{1}}$${y_{2}}$${\dots}$${y_{n}}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$$\scriptstyle{a_{k}}$

A representation $V$ of $Q$ is simply an assignment of a finite-dimensional
vector space $V(x)$ for each $x\in Q_{0}$ and a linear transformation
$V(a):V(ta)\rightarrow V(ha)$ for each arrow $a\in Q_{1}$. A morphism of
quiver representations $\phi:V\rightarrow W$ is a collection of linear maps
$\phi(x):V(x)\rightarrow W(x)$ for each $x\in Q_{0}$ subject to the condition
that for every $a\in Q_{1}$, the diagram below commutes.

${V(ta)}$${V(ha)}$${W(ta)}$${W(ha)}$$\scriptstyle{V(a)}$$\scriptstyle{\phi(ta)}$$\scriptstyle{\phi_{ha}}$$\scriptstyle{W(a)}$

A representation $V$ of $\Delta[k]$ is given by assigning vector spaces $V(x)$
to $x$ and and $V(y_{i})$ to $y_{i}$ for each $i$, and $n$ linear maps
$V(a_{1}),\dots,V(a_{k})$ where $V(a_{i}):V(y_{i})\rightarrow V(x)$. A
morphism between two representations $V$ and $W$ of $\Delta[k]$ is given by
linear maps $\phi(x):V(x)\rightarrow W(x)$ and
$\phi(y_{i}):V(y_{i})\rightarrow W(y_{i})$ for all $i$ such that $\phi(x)\circ
V(a_{i})=W(a_{i})\circ\phi(y_{i})$ for all $i$.

A subrepresentation $U$ of $V$ is a collection of subspaces $U(x)\subseteq
V(x)$ such that for every arrow $a\in Q_{1}$ the linear map $U(a)$ is simply a
restriction of $V(a)$. In particular, this means that the image of $U(ta)$
under $V(a)$ will need to be contained in $U(ha)$. For two representations $V$
and $W$, we define their direct sum $V\oplus W$ to be the representation that
assigns $V(x)\oplus W(x)$ to each vertex $x$ and the linear map
$\begin{pmatrix}V(a)&0\\\ 0&W(a)\end{pmatrix}$ for each arrow $a\in Q_{1}$.
Similarly, the notion of direct summand, image, kernel, co-image, etc. are all
defined in a straightforward way. We say $V$ is indecomposable if it cannot be
written as a direct sum of two (proper) subrepresentations. Otherwise, $V$ is
called decomposable. In summary, the category of quiver representations forms
an Abelian category. We refer to [15] for complete details.

The dimension vector of a representation $V$ is $\underline{\dim}(V)=(\dim
V(x))_{x\in Q_{0}}$. So, for a representation $V$ of $\Delta[k]$, its
dimension vector is

$\underline{\dim}(V)=(\dim(V(x)),\dim(V(y_{1})),\dim(V(y_{2})),\dots,\dim(V(y_{k}))).$

We will keep the convention throughout this paper that when we write a
dimension vector of $\Delta[k]$, we will first specify the dimension at $x$
and then the dimensions at $y_{1},y_{2},\dots,y_{k}$ in order.

For any representation $V$ of a quiver $Q$, for each $x\in Q_{0}$, picking a
basis for $V(x)$ identifies $V(x)$ with $K^{\dim(V(x))}$. Further, with this
identification, every linear map $V(a)$ is just a matrix of size
$\dim(V(ha))\times\dim(V(ta))$. Thus, we come to the following definition. For
any dimension vector $\alpha=(\alpha(x))_{x\in Q_{0}}\in\mathbb{N}^{Q_{0}}$
(where $\mathbb{N}=\\{0,1,2,\dots,\\}$), we define the representation space

${\rm Rep}(Q,\alpha)=\bigoplus_{a\in
Q_{1}}\operatorname{Mat}_{\alpha(ha),\alpha(ta)}.$

Any point $V=(V(a))_{a\in Q_{1}}\in{\rm Rep}(Q,\alpha)$ can be interpreted as
a representation of $Q$ with dimension vector $\alpha$ as follows: for each
$x\in Q_{0}$, assign the vector space $K^{\alpha(x)}$, and for each arrow
$a\in Q_{1}$, the matrix $V(a)$ describes a linear transformation from
$K^{\alpha(ta)}$ to $K^{\alpha(ha)}$. The base change group
$\operatorname{GL}(\alpha)=\prod_{x\in Q_{0}}\operatorname{GL}_{\alpha(x)}$
acts on ${\rm Rep}(Q,\alpha)$ in a natural fashion where
$\operatorname{GL}_{\alpha(x)}$ acts on the vector space $K^{\alpha(x)}$
assigned to vertex $x$ by changing basis. More concretely, for
$g=(g_{x})_{x\in Q_{0}}\in\operatorname{GL}(\alpha)$ and $V=(V(a))_{a\in
Q_{1}}\in{\rm Rep}(Q,\alpha)$, the point $g\cdot V\in{\rm Rep}(Q,\alpha)$ is
defined by the formula

$(g\cdot V)(a)=g_{ha}V(a)g_{ta}^{-1}.$

The $\operatorname{GL}(\alpha)$ orbits in ${\rm Rep}(Q,\alpha)$ are in $1-1$
correspondence with isomorphism classes of $\alpha$-dimensional
representations.

For the star quiver $\Delta[k]$, suppose we pick a dimension vector
$\alpha=(p,q_{1},\dots,q_{k})$, then the representation space

${\rm
Rep}(\Delta[k],\alpha))=\bigoplus_{i=1}^{n}\operatorname{Mat}_{p,q_{i}}.$

Now,
$\operatorname{GL}(\alpha)=\operatorname{GL}_{p}\times\operatorname{GL}_{q_{1}}\times\operatorname{GL}_{q_{2}}\times\dots\times\operatorname{GL}(q_{n})$,
and the action is given by the formula

$(g_{x},g_{y_{1}},\dots,g_{y_{n}})\cdot(B_{1},\dots,B_{n})=(g_{x}B_{1}g_{y_{1}}^{-1},\dots,g_{x}B_{n}g_{y_{n}}^{-1}),$

which when compared with Eq 4 suggests the connection between iPCA and the
star quiver.

######  Remark 3.1.

The space ${\rm Rep}(Q,\alpha)$ is a representation of
$\operatorname{GL}(\alpha)$ and its various subgroups. At the same time, we
refer to a point $V\in{\rm Rep}(Q,\alpha)$ also as a representation. We advise
the reader to keep in mind that we think of $V$ as a representation of the
quiver $Q$ and not of any group to avoid confusion. Moreover, if $V\in{\rm
Rep}(Q,\alpha)_{\mathbb{R}}=\bigoplus_{a\in
Q_{1}}\operatorname{Mat}_{\alpha(ha),\alpha(ta)}(\mathbb{R})$, then it can be
thought of as both a real and complex representation of $Q$.

###  3.1. Stability notions

We follow the conventions from [15] for consistency. For this section, we fix
$K=\mathbb{C}$. We intentionally refrain from defining stability notions in
the case of $K=\mathbb{R}$ to avoid confusion.

Let $Q$ be a quiver with no oriented cycles (self loops are counted as
oriented cycles). Let $\alpha$ be a dimension vector. Consider the subgroup
$\operatorname{SL}(\alpha)=\prod_{x\in
Q_{0}}\operatorname{SL}(\alpha(x))\subseteq\operatorname{GL}(\alpha)$. Then,
the invariant ring for the action of $\operatorname{SL}(\alpha)$ on ${\rm
Rep}(Q,\alpha)$ is called the ring of semi-invariants

$\operatorname{SI}(Q,\alpha)=K[{\rm
Rep}(Q,\alpha)]^{\operatorname{SL}(\alpha)}.$

For any $\sigma=(\sigma(x))_{x\in Q_{0}}\in\mathbb{Z}^{Q_{0}}$ (which we call
a weight), we have a character of $\operatorname{GL}(\alpha)$ which we also
denote $\sigma$ by abuse of notation. The character
$\sigma:\operatorname{GL}(\alpha)\rightarrow K^{\times}$ is given by
$\sigma((g_{x})_{x\in Q_{0}})=\prod_{x\in Q_{0}}\det(g_{x})^{\sigma(x)}$. The
ring of semi-invariants has a decomposition

${\rm SI}(Q,\alpha)=\bigoplus_{\sigma\in\mathbb{Z}^{Q_{0}}}{\rm
SI}(Q,\alpha)_{\sigma},$

where $\operatorname{SI}(Q,\alpha)_{\sigma}=\\{f\in{\rm SI}(Q,\alpha)\ |\
f(g\cdot x)=\sigma(g^{-1})f(x)\ \forall g\in\operatorname{GL}(\alpha)\\}$.

We define the _effective cone of weights_

$\Sigma(Q,\alpha):=\\{\sigma\in\mathbb{Z}^{Q_{0}}\ |\ {\rm
SI}(Q,\alpha)_{m\sigma}\neq 0\ \text{for some }m\in\mathbb{Z}_{>0}\\}.$

For a weight $\sigma$ and a dimension vector $\beta$, we define
$\sigma(\beta):=\sum_{x\in Q_{0}}\sigma(x)\beta(x)$. We point out here that
every $\sigma\in\Sigma(Q,\alpha)$ must satisfy $\sigma(\alpha)=0$. For each
$0\neq\sigma\in\Sigma(Q,\alpha)$, we consider the subring

$\operatorname{SI}(Q,\alpha,\sigma):=\oplus_{m=0}^{\infty}\operatorname{SI}(Q,\alpha)_{m\sigma}.$

For a sincere dimension vector $\alpha$ (i.e., $\alpha(x)\neq 0\ \forall x\in
Q_{0}$), it turns out that this subring can also be seen as an invariant ring,
i.e., $\operatorname{SI}(Q,\alpha,\sigma)=K[{\rm
Rep}(Q,\alpha)]^{\operatorname{GL}(\alpha)_{\sigma}}$ where
$\operatorname{GL}(\alpha)_{\sigma}=\\{g\in\operatorname{GL}(\alpha)\ |\
\sigma(g)=1\\}$. Note that $\operatorname{GL}(\alpha)_{\sigma}$ is a reductive
group. It is well-known that the associated projective variety ${\rm
Proj}(\operatorname{SI}(Q,\alpha,\sigma))$ defines a moduli space for the
$\alpha$-dimensional representations of $Q$, see [30].

We make a definition following King [30]. We follow the convention from [15]
which is consistent with our notational choices so far, but differs from
King’s original convention by a sign.

######  Definition 3.2 (King [30]).

Let $Q$ be a quiver with no oriented cycles, $V$ be a representation of $Q$
and $\sigma\in\mathbb{Z}^{Q_{0}}$ a weight such that
$\sigma(\underline{\dim}V)=0$.

  * •

$V$ is $\sigma$-semistable if $\sigma(\beta)\leq 0$ for all
$\beta\in\mathbb{Z}_{\geq 0}^{Q_{0}}$ such that $V$ contains a
subrepresentation of dimension $\beta$.

  * •

$V$ is $\sigma$-stable if $\sigma(\beta)<0$ for all $\beta\in\mathbb{Z}_{\geq
0}^{Q_{0}}$ (other than $0$ and $\underline{\dim}(V)$) such that $V$ contains
a subrepresentation of dimension $\beta$.

  * •

$V$ is $\sigma$-polystable if $V=V_{1}\oplus V_{2}\oplus\dots\oplus V_{k}$
such that $V_{i}$ are all $\sigma$-stable representations.

We now relate $\sigma$-stability notions to
$\operatorname{GL}(\alpha)_{\sigma}$-stability notions. The following result
is due to King. For a complete and self-contained proof, see the appendix in
[10]. Note however that in [10], the following result was stated for $\sigma$
indivisible, but it is clear from the proof that this is not necessary.

######  Theorem 3.3 (King [30]).

Let $Q$ be a quiver with no oriented cycles,
$\alpha\in\mathbb{Z}_{>0}^{Q_{0}}$ a sincere dimension vector and
$0\neq\sigma\in\mathbb{Z}^{Q_{0}}$ such that $-\sigma\notin\Sigma(Q,\alpha)$.
A representation $V\in{\rm Rep}(Q,\alpha)$ is $\sigma$-semistable (resp.
$\sigma$-polystable, $\sigma$-stable) if and only if $V$ is
$\operatorname{GL}(\alpha)_{\sigma}$-semistable (resp.
$\operatorname{GL}(\alpha)_{\sigma}$-polystable,
$\operatorname{GL}(\alpha)_{\sigma}$-stable).

######  Definition 3.4.

We say a dimension vector $\alpha$ is $\sigma$-semistable/polystable/stable if
a generic representation $V\in{\rm Rep}(Q,\alpha)$ is
$\sigma$-semistable/polystable/stable.

###  3.2. Canonical decomposition, Schur roots and Schofield’s result

In this subsection, we will recall some established results in the theory of
quiver representations that we will need. First a series of definitions. For a
representation $V$ of $Q$, we denote by ${\rm End}_{Q}(V)$ the endomorphism
ring of $V$ (in the category of quiver representations).

######  Definition 3.5 (Schur root).

Let $Q$ be a quiver and let $\alpha$ be a dimension vector. We say $\alpha$ is
a Schur root if it satisfies any of the following equivalent conditions:

  * •

${\rm End}_{Q}(V)=\mathbb{C}$ for some $V\in{\rm Rep}(Q,\alpha)$.

  * •

${\rm End}_{Q}(V)=\mathbb{C}$ for generic $V\in{\rm Rep}(Q,\alpha)$.

  * •

A generic $V\in{\rm Rep}(Q,\alpha)$ is indecomposable.

We indicate briefly why the three conditions in the definition above are
equivalent. The equivalence of the first two conditions follows from a
standard argument in algebraic geometry using the fact that fiber dimension is
upper semi-continuous. If ${\rm End}(V)=\mathbb{C}$, then $V$ must be
indecomposable because the endomorphism ring is local, so this establishes
that the second condition implies the third. The converse is non-trivial
because the endomorphism rings of indecomposable representations are local and
there are many other possiblities for local rings other than $\mathbb{C}$. We
refer to [28, Proposition 1] for a proof.

The concept of canonical decomposition is due to Kac [27, 28].

######  Definition 3.6 (Canonical decomposition).

Let $Q$ be a quiver and $\alpha$ a dimension vector. Then we write

$\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$

and call it the canonical decomposition if a generic $V\in{\rm Rep}(Q,\alpha)$
splits as $V=V_{1}\oplus V_{2}\oplus\dots\oplus V_{l}$ with $V_{i}$
indecomposable of dimension $\beta_{i}$.

Canonical decomposition will play an important role in our arguments in later
sections. The following result is also due to Kac, but we first need a
definition. For representations $V$ and $W$ of $Q$, we denote by ${\rm
Ext}(V,W)$ the first extension group. We do not want to digress too much here
about extension groups, so we refer the interested reader to [15]. For
dimension vectors $\gamma$ and $\delta$ of $Q$, we denote by ${\rm
ext}(\gamma,\delta)$ the generic value of $\dim({\rm Ext}(V,W))$ for
$(V,W)\in{\rm Rep}(Q,\gamma)\times{\rm Rep}(Q,\delta)$.

######  Theorem 3.7 (Kac [28]).

Let $Q$ be a quiver and $\alpha$ a dimension vector. Then
$\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ is the canonical
decomposition if and only if both the following conditions hold:

  * •

$\beta_{i}$ is a Schur root for all $i$;

  * •

${\rm ext}(\beta_{i},\beta_{j})=0$ for all $i\neq j$.

######  Remark 3.8.

It is again a consequence of the upper semi-continuity of fiber dimensions
that ${\rm ext}(\gamma,\delta)$ is the minimum value of $\dim({\rm Ext}(V,W))$
for $(V,W)\in{\rm Rep}(Q,\gamma)\times{\rm Rep}(Q,\delta)$.

We now turn to recalling a result of Schofield. First, some definitions. For a
quiver $Q$, we define the _Euler form_ , denoted $\left<-,-\right>$, on
$\mathbb{Z}^{Q_{0}}$ as $\left<\alpha,\beta\right>=\sum_{x\in
Q_{0}}\alpha(x)\beta(x)-\sum_{a\in Q_{1}}\alpha(ta)\beta(ha)$. The symmetrized
version, often called Cartan form is defined as
$(\alpha,\beta)=\left<\alpha,\beta\right>+\left<\beta,\alpha\right>$

######  Definition 3.9 (Schofield weight and Schofield stability).

Let $Q$ be a quiver and $\alpha$ a dimension vector. The Schofield weight
$\sigma$ is the weight defined by
$\sigma(\beta)=\left<\alpha,\beta\right>-\left<\beta,\alpha\right>$. Further,
we say that $\alpha$ is Schofield semistable/polystable/stable if $\alpha$ is
$\sigma$-semistable/polystable/stable for the Schofield weight $\sigma$.

Then reason for the above definition is clarified by the following theorem of
Schofield which will be crucial for us in many ways.

######  Theorem 3.10 (Schofield).

Let $Q$ be a quiver and $\alpha$ a dimension vector. Then $\alpha$ is a Schur
root if and only if $\alpha$ is Schofield stable.

##  4\. Relating iPCA to stability of star quivers

In this section, we will see that the interpretation of iPCA as a Gaussian
group model in Section 2.2 is quiver theoretic in nature. The goal of this
subsection is to relate (generic) existence and uniqueness of iPCA in a
precise manner to Schofield stability for star quivers. This is quite
straightforward for the complex version of iPCA, but we need some additional
results to make the transfer to real iPCA. We stress that we relate maximum
likelihood estimation for real iPCA directly to $\sigma$-stability (see
Theorem 4.3), which is a property of complex representations of quivers rather
than real representations of quivers! In the theory of quiver representations,
complex representations are better understood than real representations, the
underlying reason is an expected one – the field of complex numbers is
algebraically closed whereas the field of real numbers is not.

######  Remark 4.1.

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector for $\Delta[k]$, and
let $n=\sum_{i}q_{i}$. Then, the Schofield weight of $\alpha$ is the weight
$\sigma=(-n,p,p,\dots,p)$.

We observe the following:

######  Proposition 4.2.

Let $K=\mathbb{C}$. Let $p,q_{1},\dots,q_{k}\in\mathbb{Z}_{\geq 1}$ and let
$n=\sum_{i}q_{i}$, let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector of
$\Delta[k]$. Consider the complex model $\mathcal{M}_{\alpha}$. Then, for
generic data
$(B_{1},\dots,B_{k})\in\oplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}(\mathbb{C})$:

  * •

The likelihood function is bounded above if and only if $\alpha$ is Schofield
semistable;

  * •

MLE exists if and only if $\alpha$ is Schofield polystable;

  * •

MLE exists uniquely if and only if $\alpha$ is Schofield stable.

###### Proof.

The Schofield weight $\sigma$ is $(-n,p,p,\dots,p)$, the same weight from
Proposition 2.6. To show that $-\sigma\notin\Sigma(Q,\alpha)$, so that the
hypotheses of Theorem 3.3 are satisfied, observe that since there are no paths
from $x$ to $y_{i}$, we deduce that
$\operatorname{SI}(\Delta[k],\alpha)_{-d\sigma}=0$ for all
$d\in\mathbb{Z}_{\geq 1}$ from the determinantal description of semi-
invariants of quivers (see [17, 37, 13]). Now, the proposition follows from
Proposition 2.6 and Theorem 3.3. ∎

In Proposition 4.2 above, we related the complex model
$\mathcal{M}_{\alpha,\mathbb{C}}$ to $\sigma$-stability for the Schofield
weight $\sigma$. But we are more interested in the real model
$\mathcal{M}_{\alpha,\mathbb{R}}$ because the maximum likelihood estimation
for this real model is precisely (real) iPCA, the main object of study in this
paper. Hence, we prove the following result:

######  Theorem 4.3.

Let $p,q_{1},\dots,q_{k}\in\mathbb{Z}_{\geq 1}$ and let $n=\sum_{i}q_{i}$ and
let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector of $\Delta[k]$. Then,
for generic data
$B=(B_{1},\dots,B_{k})\in\oplus_{i=1}^{m}\operatorname{Mat}_{p,q_{i}}(\mathbb{R})$:

  * •

Eq 2 is bounded above if and only if $\alpha$ is Schofield semistable;

  * •

iPCA exists if and only if $\alpha$ is Schofield polystable;

  * •

iPCA exists uniquely if and only if $\alpha$ is Schofield stable.

Theorem 4.3 is not as straightforward as Proposition 4.2 and we need
additional insights to make the ‘transfer’ from $K=\mathbb{C}$ to
$K=\mathbb{R}$. This requires some technical results about varieties and
algebraic groups defined over $\mathbb{R}$. Since, we do not need these
technical details in later sections, we move them and the proof of Theorem 4.3
to Appendix A for the sake of readability.

##  5\. Schofield stability for star quivers

In this section, we will prove some of our main results on existence and
uniqueness of iPCA by studying Schofield stability. To do so, we will need to
introduce additional techniques from quiver representations, which we will
discuss when needed. We set $K=\mathbb{C}$ as the ground field for this entire
section.

###  5.1. Equivalence of Schofield semistability and Schofield polystability
for star quivers

First, we discuss a result that is specific to the quiver $\Delta[k]$, which
may not be true for more general quivers.

######  Proposition 5.1.

Let $p,q_{1},\dots,q_{k}\in\mathbb{Z}_{>0}$, and let
$\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector of the star quiver
$\Delta[k]$. Then

$\alpha\text{ is Schofield semistable}\Longleftrightarrow\alpha\text{ is
Schofield polystable}.$

To prove the proposition above, we need the following lemma.

######  Lemma 5.2.

Let $p,q_{1},\dots,q_{k}\in\mathbb{Z}_{>0}$, and let
$\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector of the star quiver
$\Delta[k]$. Let $\sigma$ be the Schofield weight of $\alpha$. Let
$\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ be the canonical
decomposition of $\alpha$. Then

$\alpha\text{ is
$\sigma$-polystable}\Longleftrightarrow\sigma\cdot\beta_{i}=0\ \forall i.$

###### Proof.

Suppose $\alpha$ is $\sigma$-polystable. Then, clearly, we must have
$\sigma\cdot\beta_{i}\leq 0$ for all $i$ since $\alpha$ is
$\sigma$-semistable. But then, we also have
$\sigma\cdot(\sum_{i}\beta_{i})=\sigma\cdot\alpha=0$. Thus, we must have
$\sigma\cdot\beta_{i}=0$ for all $i$.

Conversely, suppose $\sigma\cdot\beta_{i}=0$ for all $i$. Fix $j$. Since
$\sigma(x)<0$ and $\sigma(y_{i})>0$ for all $i$ and $\beta_{j}\neq 0$, the
support of $\beta_{j}$ must be of the form
$S=\\{x,y_{j_{1}},\dots,y_{ij_{k}}\\}$ for some $\emptyset\neq
J=\\{j_{1},\dots,j_{k}\\}\subseteq\\{1,2,\dots,k\\}$. Consider $\Delta[J]$ the
(full) subquiver of $\Delta[k]$ whose vertices are $\\{x\\}\cup\\{y_{j}\ |\
j\in J\\}$. Then, we can interpret $\beta_{j}$ as a dimension vector of
$\Delta[J]$. Since $\beta_{j}$ is a Schur root (of $\Delta[k]$ and hence of
$\Delta[J]$), it is $\mu$-stable for its Schofield weight $\mu$ by Theorem
3.10. But one easily checks that $\sigma\cdot\beta_{j}=0$ means that $\mu$ is
(up to a positive multiple) equal to $\sigma$ restricted to $\Delta[J]$, so
$\beta_{j}$ is $\sigma$-stable. Since $\beta_{j}$ is $\sigma$-stable for all
$j$, we conclude that $\alpha$ is $\sigma$-polystable. ∎

Now, we can prove Proposition 5.1.

###### Proof of Proposition 5.1.

Clearly if $\alpha$ is $\sigma$-polystable, then $\alpha$ is
$\sigma$-semistable. Conversely, suppose $\alpha$ is not $\sigma$-polystable.
Let $\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ be the
canonical decomposition. By Lemma 5.2, we see that $\sigma\cdot\beta_{i}\neq
0$ for some $i$. Now either $\sigma\cdot\beta_{i}>0$ or
$\sigma\cdot(\sum_{j\neq i}\beta_{j})>0$. In either case, this means that a
generic $V\in{\rm Rep}(Q,\alpha)$ has a direct summand $W$(and hence
subrepresentation) such that $\sigma\cdot\dim(W)>0$, so $V$ is not
$\sigma$-semistable. ∎

The above results raise an interesting question in quiver representations:

######  Problem 5.3.

Does there exist a quiver $Q$ and a dimension vector $\alpha$ such that
$\alpha$ is Schofield semistable, but not Schofield polystable? If so,
construct an example.

###  5.2. Opposite quivers

There are some operations we can perform on quivers while preserving
information about their generic stability/semistability properties in order to
extend the number of cases we can handle. One such operation is as follows.
Let $Q=(Q_{0},Q_{1})$ be a quiver, and let $Q^{{\rm opp}}$ denote the
_opposite quiver_ , where $Q^{{\rm opp}}_{0}=Q_{0}$ and $Q^{{\rm
opp}}_{1}=\\{a^{*}\ |\ a\in Q_{1}\\}$ where $ta^{*}=ha$ and $ha^{*}=ta$. In
other words, $Q^{{\rm opp}}$ is obtained from $Q$ by simply reversing the
orientations of all arrows. Given a representation $V$ of $Q$, we define a
representation $V^{{\rm opp}}$ of $Q^{{\rm opp}}$ as follows. For each $x\in
Q_{0}=Q^{{\rm opp}}_{0}$, set $V^{{\rm opp}}_{x}=V_{x}^{*}$ (the dual of
$V_{x}$) and for each $a^{*}\in Q^{{\rm opp}}_{1}$, let
$V(a^{*}):V(ta^{*})=V(ha)^{*}\rightarrow V(ta)^{*}=V(ha^{*})$ be the dual of
the linear transformation $V(a):V(ta)\rightarrow V(ha)$. It is easy to check
that the category of representations of $Q^{{\rm opp}}$ is the equivalent to
the dual (or opposite) category to the category of representations of $Q$ and
this anti-equivalence is given by the contravariant functor $V\mapsto V^{{\rm
opp}}$.

######  Lemma 5.4.

Let $V$ be a representation of a quiver $Q$. Let $\sigma\in\mathbb{Z}^{Q_{0}}$
be a weight. Then $V$ is $\sigma$-semistable/polystable/stable if and only if
$V^{{\rm opp}}$ is $(-\sigma)$-semistable/polystable/stable.

###### Proof.

The functor $V\mapsto V^{{\rm opp}}$ is contravariant and an anti-equivalence,
so it turns subrepresentations into quotients and vice versa (while preserving
the dimension vectors). Using this and Definition 3.2, the required result
follows easily. ∎

For a representation $V$ of $Q$, identify $V_{x}$ with
$\mathbb{C}^{\dim(V_{x})}$ for all $x\in Q_{0}$ by choosing bases. By giving
the standard inner product to each $\mathbb{C}^{\dim(V_{x})}$, we can identify
its dual again with $\mathbb{C}^{\dim(V_{x})}$. With these identifications,
the matrix representing $V(a^{*})$ is the conjugate transpose of the matrix
representing $V(a)$. To summarize, we have a bijective map

$\displaystyle\eta:{\rm Rep}(Q,\alpha)$ $\displaystyle\rightarrow{\rm
Rep}(Q^{{\rm opp}},\alpha)$ $\displaystyle(V(a))_{a\in Q_{1}}$
$\displaystyle\mapsto(V(a)^{{\dagger}})_{a^{*}\in Q^{{\rm opp}}_{1}}$

######  Lemma 5.5.

Let $Q$ be a quiver and $\alpha\in\mathbb{Z}^{Q_{0}}$ a dimension vector and
$\sigma\in\mathbb{Z}^{Q_{0}}$ a weight. Then $\alpha$ is
$\sigma$-semistable/polystable/stable for $Q$ if and only if $\alpha$ is
$(-\sigma)$-semistable/polystable/stable for the quiver $Q^{{\rm opp}}$.

###### Proof.

This follows from the fact that the map $\eta$ defined above is bijective and
converts $\sigma$-semistability/polystability/stability to
$(-\sigma)$-semistability/polystability/stability by Lemma 5.4. ∎

######  Remark 5.6.

Note that given a dimension vector $\alpha$ of a quiver $Q$, the Schofield
weight of $\alpha$ for $Q^{{\rm opp}}$ is the negative of the Schofield weight
of $\alpha$ for $Q$.

Recall the star quiver $\Delta[k]$. Its opposite $\Delta[k]^{{\rm opp}}$ is in
the figure below:

(7)
${x}$${y_{1}}$${y_{2}}$${\dots}$${y_{n}}$$\scriptstyle{a_{1}^{*}}$$\scriptstyle{a_{2}^{*}}$$\scriptstyle{a_{k}^{*}}$

The transition from a quiver to its opposite is not very powerful in itself
and seems like an exercise in linear algebra about dualizing vector spaces and
linear maps. However, there are other more interesting ways to go from
$\Delta[k]$ to $\Delta[k]^{{\rm opp}}$ called reflection functors. The
interplay of the above duality with reflection functors helps us completely
resolve generic existence and uniqueness of iPCA in the case where the legs
are equal, i.e., $q_{1}=q_{2}=\dots=q_{k}$. Now, we turn to discussing
reflection functors.

###  5.3. Reflection functors

Reflection functors are widely used in the theory of quiver representations
and were discovered by Bernstein, Gelfand and Ponomarev. Despite their
simplicity, reflection functors seem to be surprisingly powerful in numerous
contexts. We recall only what we need from reflection functors and refer the
interested reader to [15] for a more comprehensive treatment.

Given a quiver $Q=(Q_{0},Q_{1})$ and a vertex $x\in Q_{0}$, we define _the
quiver reflected at $x$_, denoted $r_{x}(Q)$, whose vertex set is $Q_{0}$ and
set of arrows is

$(r_{x}(Q))_{1}=\\{a\in Q_{1}\ |\ ha\neq x,ta\neq x\\}\cup\\{a^{*}\ |\ a\in
Q_{1},ha=x\text{ or }ta=x\\},$

and $ha^{*}=ta$ and $ta^{*}=ha$. In other words, one simply reverses all
arrows which “touch” the vertex $x$.

####  5.3.1. Reflection at a sink

Now, suppose $x$ is a sink of the quiver $Q$ (i.e., there are no arrows such
that $ta=x$, such as the vertex we have labeled $x$ in the star quiver
$\Delta[k]$) and let $a_{1},\dots,a_{l}$ be the arrows in $Q$ such that
$ha_{i}=x$, and let $y_{i}=ta_{i}$. Suppose $V$ is a representation of $Q$.
Then we have a map

$\phi(x)=(V(a_{1}),V(a_{2}),\dots,V(a_{l}))^{\top}:\bigoplus_{i=1}^{l}V(y_{i})\rightarrow
V(x).$

Thus we have the inclusion $\iota_{x}:{\rm
ker}(\phi_{x})\hookrightarrow\bigoplus_{i=1}^{l}V(y_{i})$. For
$j\in\\{1,2,\dots,l\\}$, let $\pi_{j}:\bigoplus_{i=1}^{l}V(y_{i})\rightarrow
V_{y_{j}}$ denote the projection to the $j^{th}$ summand. Then
$\pi_{j}\circ\iota_{x}:{\rm ker}(\phi_{x})\rightarrow V(y_{j})$.

We now define a representation $W=R_{x}^{+}(V)$ of $r_{x}(Q)$ as follows: For
$y\neq x$, set $W(y)=V(y)$ and set $W(x)={\rm ker}(\phi_{x})$. For $b\in
Q_{1}\setminus\\{a_{1},\dots,a_{k}\\}$, set $W(b)=V(b)$ and for
$j\in\\{1,2,\dots,l\\}$, we define $W(a_{j}^{*})=\pi_{j}\circ\iota_{x}$.

####  5.3.2. Reflection at a source

If $x$ is a source of the quiver $Q$ (i.e., there are no arrows such that
$ha=x$) and let $a_{1},\dots,a_{l}$ be the arrows in $Q$ such that $ta_{i}=x$,
and let $y_{i}=ha_{i}$. Suppose $V$ is a representation of $Q$. Then we have a
map

$\psi(x)=(V(a_{1}),V(a_{2}),\dots,V(a_{l})):V(x)\rightarrow\bigoplus_{i=1}^{l}V(y_{i})$

Thus, we have a surjection
$\eta_{x}:\bigoplus_{i=1}^{l}V(y_{i})\rightarrow{\rm coker}(\psi_{x})$. For
$j\in\\{1,2,\dots,l\\}$, let
$\zeta_{j}:V(y_{j})\hookrightarrow\bigoplus_{i=1}^{l}V(y_{i})$ denote the
inclusion of the $j^{th}$ summand. Then
$\eta_{x}\circ\zeta_{j}:V(y_{j})\rightarrow{\rm coker}(\psi_{x})$.

We now define a representation $W=R_{x}^{-}(V)$ of $r_{x}(Q)$ as follows: For
$y\neq x$, set $W(y)=V(y)$ and set $W(x)={\rm coker}(\psi_{x})$. For $b\in
Q_{1}\setminus\\{a_{1},\dots,a_{k}\\}$, set $W(b)=V(b)$ and for
$j\in\\{1,2,\dots,l\\}$, we define $W(a_{j}^{*})=\eta_{x}\circ\zeta_{j}$.

####  5.3.3. Reflection functors and canonical decomposition

It can be checked that $R_{x}^{+}$ and $R_{x}^{-}$ (when defined) are functors
from the category of representations of $Q$ to the category of representations
of $r_{x}(Q)$, the quiver $Q$ reflected at $x$. We will recall the important
results.

For $x\in Q_{0}$, let $S_{x}$ denote the representation that is defined as
$(S_{x})_{x}=K$ and $(S_{x})_{y}=0$ for all $x\neq y\in Q_{0}$. We need not
describe the maps between vertices, because all of them must be $0$.

######  Lemma 5.7.

Let $Q$ be a quiver and let $x\in Q_{0}$ be a sink. Let $V$ be an
indecomposable representation of $Q$ which is not isomorphic to $S_{x}$. Then
$R_{x}^{+}(Q)$ is an indecomposable representation of $r_{x}(Q)$ and moreover
$R_{x}^{-}(R_{x}^{+}(V))=V$.

Let $Q$ be a quiver and $x\in Q_{0}$ be a sink. Let $\mathcal{C}_{Q,x}$ denote
the subcategory of representation of the quiver $Q$ which do not have a direct
summand isomorphic to $S_{x}$. Similarly, let $\mathcal{C}_{r_{x}(Q),x}$
denote the subcategory of representations of $r_{x}(Q)$ which do not have a
direct summand isomorphic to $S_{x}$. Then, we have an equivalence of
categories

${\mathcal{C}_{Q,x}}$${\mathcal{C}_{r_{x}(Q),x}}$$\scriptstyle{R_{x}^{+}}$$\scriptstyle{R_{x}^{-}}$

given by $R_{x}^{+}$ and $R_{x}^{-}$. Moreover, for $V\in\mathcal{C}_{Q,x}$ of
dimension $\alpha$, we observe that the dimension of $R_{x}^{+}(V)$ is
$r_{x}(\alpha)$. For a vertex $x$ in a quiver $Q$, let $\epsilon_{x}$ denote
the dimension vector which is $1$ at $x$ and $0$ elsewhere.

######  Lemma 5.8.

Let $Q$ be a quiver and let $x\in Q_{0}$ be a sink. Let
$\alpha\neq\epsilon_{x}$ be a Schur root for $Q$. Then $r_{x}(\alpha)$ is a
Schur root for $r_{x}(Q)$.

###### Proof.

Let $V$ be an indecomposable representation of dimension $\alpha$ such that
${\rm End}_{Q}(V)=\mathbb{C}$. Such a $V$ exists because $\alpha$ is a Schur
root for $Q$, see Definition 3.5. Since $V$ is indecomposable, it does not
have a direct summand isomorphic to $S_{x}$, so $V\in\mathcal{C}_{Q,x}$. Hence
$R_{x}^{+}(V)$ is a representation of $r_{x}(Q)$ of dimension $r_{x}(\alpha)$
with a whose endomorphism ring is $\mathbb{C}$, so we conclude that
$r_{x}(\alpha)$ is a Schur root for $r_{x}(Q)$. ∎

######  Proposition 5.9.

Let $Q$ be a quiver and let $x\in Q_{0}$ be a sink. Let $\alpha$ be a
dimension vector of $Q$. Suppose its canonical decomposition is
$\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ (as a dimension
vector of $Q$) such that $\beta_{i}\neq\epsilon_{x}$ for all $i$. Then

$r_{x}(\alpha)=r_{x}(\beta_{1})\oplus r_{x}(\beta_{2})\oplus\dots\oplus
r_{x}(\beta_{l})$

is the canonical decomposition of $r_{x}(\alpha)$ as a dimension vector of
$r_{x}(Q)$.

###### Proof.

First, we observe that $r_{x}(\beta_{i})$ are all Schur roots for $r_{x}(Q)$
by Lemma 5.8. It suffices to show that ${\rm
ext}_{r_{x}(Q)}(r_{x}(\beta_{i}),r_{x}(\beta_{j}))=0$ for all $i\neq j$
because then we get the required conclusion by Theorem 3.7. Fix $i$ and $j$.
Since $\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ is the
canonical decomposition for $Q$, we get that ${\rm
ext}_{Q}(\beta_{i},\beta_{j})=0$. In particular, this means that for generic
representations $V_{i}$ and $V_{j}$ of dimension $\beta_{i}$ and $\beta_{j}$
such that ${\rm Ext}_{Q}(V_{i},V_{j})=0$. Since $\beta_{i}$ and $\beta_{j}$
are Schur roots, we can assume $V_{i}$ and $V_{j}$ are indecomposable and in
particular this implies that $V_{i},V_{j}\in\mathcal{C}_{Q,x}$. Thus, since
$R_{x}^{+}$ gives an equivalence of categories from $C_{Q,x}$ to
$C_{r_{x}(Q),x}$, we get that ${\rm
Ext}_{r_{x}(Q)}(r_{x}(V_{i}),r_{x}(V_{j}))=0$, so we conclude that ${\rm
ext}_{r_{x}(Q)}(r_{x}(\beta_{i}),r_{x}(\beta_{j}))=0$ (see Remark 3.8). ∎

######  Remark 5.10.

Analogous statements can be made for $R_{x}^{-}$ when $x$ is a source of the
quiver $Q$. We will not write them out explicitly, but we will make use of
them whenever needed.

####  5.3.4. Reflection functors for star quivers

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector for $\Delta[k]$.
Observe that $x$ is a sink. Now, observe that
$r_{x}(\alpha)=(n-p,q_{1},q_{2},\dots,q_{k})$.

######  Lemma 5.11.

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector for $\Delta[k]$ and
suppose $p<n$. Then, a generic representation $V\in{\rm
Rep}(\Delta[k],\alpha)$ does not contain $S_{x}$ as a direct summand.

###### Proof.

This is straightforward and left to the reader. ∎

######  Lemma 5.12.

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector of $\Delta[k]$.
Suppose $\sigma\in\mathbb{Z}^{Q_{0}}$ is such that

  * •

$\sigma\cdot\alpha=0$;

  * •

$\sigma(x)<0$;

  * •

$\sigma(y_{1})=\sigma(y_{2})=\dots=\sigma(y_{k})>0$.

Then up to a positive multiple $\sigma$ is the Schofield weight of $\alpha$.

###### Proof.

This is straightforward and left to the reader. ∎

######  Corollary 5.13.

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector for $\Delta[k]$ such
that $p<n$. Then $\alpha$ is Schofield semistable/polystable/stable for
$\Delta[k]$ if and only if $r_{x}(\alpha)$ is Schofield
semistable/polystable/stable for $\Delta[k]^{{\rm opp}}$.

###### Proof.

We make a few observations before we get into the proof. First, the Schofield
weight for $\alpha$ as a representation of $\Delta[k]$ is
$\sigma=(-n,p,p,\dots,p)$ and the Schofield weight for
$r_{x}(\alpha)=(n-p,q_{1},\dots,q_{k})$ as a representation of
$\Delta[k]^{{\rm opp}}$ is $\widehat{\sigma}=(n,p-n,p-n,\dots,p-n)$. Second,
if $\alpha=\beta_{1}\oplus\dots\oplus\beta_{l}$ is the canonical decomposition
of $\alpha$ for $\Delta[k]$, then
$r_{x}(\alpha)=r_{x}(\beta_{1})\oplus\dots\oplus r_{x}(\beta_{l})$ is the
canonical decomposition of $r_{x}(\alpha)$ for $\Delta[k]^{\rm opp}$, which
follows from Lemma 5.11 and Proposition 5.9. Third, Schofield semistability is
the same as Schofield polystability for $\Delta[k]$ by Proposition 5.1.
Analogously, one can establish that Schofield semistability is the same as
Schofield polystability for $\Delta[k]^{{\rm opp}}$; we leave the details to
the reader. So, we only need to argue the cases of polystability and
stability.

First, we tackle the case of stability. Suppose $\alpha$ is Schofield stable,
i.e., $\sigma$-stable for $\Delta[k]$. Then since $\alpha$ is a Schur root by
Theorem 3.10, and $\alpha\neq\epsilon_{x}$, we deduce that $r_{x}(\alpha)$ is
a Schur root for $\Delta[k]^{{\rm opp}}$ by Lemma 5.8. Thus $r_{x}(\alpha)$ is
Schofield stable for $\Delta[k]^{{\rm opp}}$ by Theorem 3.10. Conversely,
suppose $\alpha$ is not $\sigma$-stable for $\Delta[k]$, then it is not Schur,
so its canonical decomposition contains at least two factors, none of which
are $\epsilon_{x}$ by Lemma 5.11. But then by the argument above, the
canonical decomposition for $r_{x}(\alpha)$ for $\Delta[k]^{{\rm opp}}$ also
contains at least two factors, so $r_{x}(\alpha)$ is not a Schur root and
hence not Schofield stable.

Next, we tackle the case of polystability. Suppose $\alpha$ is Schofield
polystable, i.e., $\sigma$-polystable for $\Delta[k]$. Then the canonical
decomposition $\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ is
such that each $\beta_{i}$ is $\sigma$-stable and $\beta_{i}\neq\epsilon_{x}$.
In particular, observe that this means $\sigma\cdot\beta_{i}=0$ and this
implies that the Schofield weight of $\beta_{i}$ for $Q$ is $\sigma$ up to a
positive multiple by Lemma 5.12. One also verifies easily that
$\sigma\cdot\beta_{i}=0\implies\widehat{\sigma}\cdot r_{x}(\beta_{i})=0$,
which means that the Schofield weight of $r_{x}(\beta_{i})$ for $r_{x}(Q)$ is
$\widehat{\sigma}$ up to a positive multiple by the analog of Lemma 5.12 for
$\Delta[k]^{{\rm opp}}$. Since $r_{x}(\beta_{i})$ is a Schur root, it is
Schofield stable, i.e., $\widehat{\sigma}$-stable. Hence $r_{x}(\alpha)$ is
$\widehat{\sigma}$-polystable, i.e., Schofield polystable. ∎

######  Corollary 5.14.

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector for $\Delta[k]$ such
that $p<n$. Then $\alpha$ is Schofield semistable/polystable/stable for
$\Delta[k]$ if and only if $r_{x}(\alpha)=(n-p,q_{1},\dots,q_{k})$ is
Schofield semistable/polystable/stable for $\Delta[k]$.

###### Proof.

This follows from Corollary 5.13, Lemma 5.5 and Remark 5.6. ∎

In the above, we focused on reflection at the sink $x$ in the quiver
$\Delta[k]$. We can also look at reflections at the sources $y_{i}$, i.e.,
$R_{y_{i}}^{-}$. Applying $R_{y_{i}}^{-}$ reverses only the orientation of one
arrow. But then we observe that $R_{y_{i}}^{-}$ and $R_{y_{j}}^{-}$ commute
and we define $R_{y}^{-}=R_{y_{1}}^{-}\circ R_{y_{2}}^{-}\circ\dots\circ
R_{y_{k}}^{-}$. Now, $R_{y}^{-}$ is also a functor that takes representations
of $\Delta[k]$ to representations of $\Delta[k]^{{\rm opp}}$. Using similar
arguments as above, one can prove the following (whose proof we leave to the
reader):

######  Proposition 5.15.

Let $\alpha=(p,q_{1},\dots,q_{k})$ be a dimension vector for $\Delta[k]$ such
that $q_{i}<p$ for all $i$. Then $\alpha$ is Schofield
semistable/polystable/stable for $\Delta[k]$ if and only if
$(p,p-q_{1},\dots,p-q_{k})$ is Schofield semistable/polystable/stable for
$\Delta[k]$.

###  5.4. Proofs of Theorem 1.1 and Theorem 1.2

In order to prove Theorem 1.1, we need to recall a result of Kac. We say a
dimension vector $\alpha$ of a quiver $Q$ is connected if the directed graph
induced by its support is connected.

######  Theorem 5.16 ([27]).

Let $Q=(Q_{0},Q_{1})$ be a quiver. Let $\alpha$ be a connected dimension
vector such that $(\alpha,\epsilon_{x})\leq 0$ for all $x\in Q_{0}$.

  * •

If $(\alpha,\epsilon_{x})<0$ for some $x\in Q_{0}$, then $\alpha$ is a Schur
root.

  * •

If $(\alpha,\epsilon_{x})=0$ for all $x\in Q_{0}$, then we can write
$\alpha=c\beta$ uniquely for $c\in\mathbb{Z}_{\geq 1}$ and $\beta$
indivisible. Further;

    * –

If $c=1$, then $\alpha$ is a Schur root

    * –

If $c>1$, then $\alpha=\beta^{\oplus c}$ is the canonical decomposition of
$\alpha$.

Now we can prove Theorem 1.1.

###### Proof of Theorem 1.1.

Consider the dimension vector $\alpha=(p,q,q,\dots,q)$ of $\Delta[k]$. Observe
that $(\alpha,\epsilon_{x})=2p-\sum_{i}q_{i}\leq 0$ and
$(\alpha,\epsilon_{y_{i}})=2q_{i}-p\leq 0$. It is easy to see that the only
time we have $(\alpha,\epsilon_{x})$ = $(\alpha,\epsilon_{y_{i}})=0$ for all
$i$ is if $(p,q_{1},\dots,q_{k})=c(2,1,1,1,1)$. Unless $\alpha=(2c,c,c,c,c)$,
we have that $\alpha$ is a Schur root by part ($1$) of Theorem 5.16, hence
Schofield stable by Theorem 3.10, and so iPCA generically exists uniquely by
Theorem 4.3.

In the case $\alpha=(2c,c,c,c,c)$, by part ($2$) of Theorem 5.16, we deduce
that if $c=1$, then $\alpha$ is a Schur root and hence iPCA generically exists
uniquely by Theorem 3.10 and Theorem 4.3 (as argued above). If $c>1$, then
since the canonical decomposition is $(2c,c,c,c,c)=(2,1,1,1,1)^{\oplus c}$ by
part ($2$) of Theorem 5.16 and $(2,1,1,1,1)$ is Schur, so Schofield stable.
This means that $\alpha=(2c,c,c,c,c)$ is Schofield polystable (but not
Schofield stable), so we conclude by Theorem 4.3 that iPCA generically exists,
but we do not have generic uniqueness. ∎

Now, we turn to proving Theorem 1.2. For $p,q\in\mathbb{Z}_{>1}$, define
$\alpha(p,q)=(p,q,\dots,q)\in\mathbb{Z}^{k+1}$ to be interpreted as a
dimension vector of $\Delta[k]$. Then, by Corollary 5.14, if $p<kq$, then
$\alpha(p,q)$ is Schofield semistable/polystable/stable for $\Delta[k]$ if and
only if $\alpha(kq-p,q)$ is Schofield semistable/polystable/stable for
$\Delta[k]$. Similarly, by Proposition 5.15, if $q<p$, then $\alpha(p,q)$ is
Schofield semistable/polystable/stable for $\Delta[k]$ if and only if
$\alpha(p,p-q)$ is Schofield semistable/polystable/stable for $\Delta[k]$.

In view of the above paragraph we want to define an equivalence relation
$\sim_{k}$ on $\mathbb{Z}_{>1}^{2}$. If $p<kq$, then $(p,q)\sim_{k}(kq-p,q)$
and if $q<p$, then $(p,q)\sim_{k}(p,p-q)$ and taking the transitive closure
defines $\sim_{k}$.

######  Lemma 5.17.

If $(p,q)\sim_{k}(r,s)$, then $\alpha(p,q)$ is Schofield
semistable/polystable/stable if and only if $\alpha(r,s)$ is Schofield
semistable/polystable/stable.

###### Proof.

This follows from Proposition 5.15 and Corollary 5.14. ∎

######  Lemma 5.18.

Consider the quantity $\Gamma(p,q)=p^{2}+kq^{2}-kpq$. If $(p,q)\sim_{k}(r,s)$,
then $\Gamma(p,q)=\Gamma(r,s)$.

###### Proof.

It is enough to check that $\Gamma(p,q)=\Gamma(kq-p,q)$ when $p<kq$ and that
$\Gamma(p,q)=\Gamma(p,p-q)$ when $q<p$. Both are straightforward computations
and left to the reader. ∎

######  Definition 5.19.

Let $Q$ be a quiver and $\alpha$ a Schur root. We say $\alpha$ is a real
(resp. isotropic imaginary, non-isotropic imaginary) if
$\left<\alpha,\alpha\right>=1$ (resp. $0$, $<0$).

######  Theorem 5.20.

Let $Q$ be a quiver and $\alpha$ a dimension vector. Suppose
$\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{l}$ is the canonical
decomposition of $\alpha$, then the canonical decomposition of $n\alpha$ is

$n\alpha=(n\beta_{1})\oplus(n\beta_{2})\oplus\dots\oplus(n\beta_{l}),$

where

$(n\beta)=\begin{cases}\beta^{\oplus n}&\text{if $\beta$ is real or
isotropic};\\\ n\beta&\text{if $\beta$ is non-isotropic}.\end{cases}$

######  Proposition 5.21.

Let $(p,q)\in\mathbb{Z}_{>1}^{2}$. Then:

  1. (1)

If $q\leq p/2$ and $p\leq kq/2$, then $\alpha(p,q)$ is Schofield stable unless
$k=4$, $p=2q$ and $q>1$.

  2. (2)

If $q>p$ or $p>kq$, then $\alpha(p,q)$ is not Schofield semistable.

  3. (3)

If $q=p$ or $p=kq$, then $\alpha(p,q)$ is Schofield polystable. Further it is
Schofield stable if and only if $(p,q)=(1,1)$.

  4. (4)

In all other cases, we have either $p/2<q<p$ or $kq/2<p<kq$ and $(p,q)$ is
$\sim_{k}$-equivalent to some $(p^{\prime},q^{\prime})$ which falls in the
above three cases.

###### Proof.

The first part is Theorem 1.1. For the second part, observe that if $q>p$,
then $\epsilon_{y_{1}}=(0,1,0,\dots,0)$ appears in the canonical decomposition
of $\alpha$ and $\sigma\cdot\epsilon_{y_{1}}\neq 0$ where
$\sigma=(kq,-p,-p,\dots,-p)$ is the Schofield weight of $\alpha(p,q)$. Thus,
by Lemma 5.2, $\alpha(p,q)$ is not Schofield polystable, and hence not
Schofield semistable by Proposition 5.1. Similarly if $p>kq$, then
$\epsilon_{x}=(1,0,\dots,0)$ appears in the canonical decomposition for
$\alpha(p,q)$ and again $\sigma\cdot\epsilon_{x}\neq 0$. Thus, by a similar
argument, we get that $\alpha(p,q)$ is not Schofield semistable.

For the third part, if $p=q$, then $\alpha(p,q)=(1,1,\dots,1)^{\oplus p}$ is
the canonical decomposition of $\alpha(p,q)$. This follows from the fact that
$(1,1,\dots,1)$ is a real Schur root. It is straightforward to verify that
$(1,1,\dots,1)$ is a real Schur root and we leave that to the reader. Hence if
$p=q=1$, $\alpha(p,q)$ is Schofield stable and if $p=q>1$ then $\alpha(p,q)$
is Schofield polystable, but not Schofield stable. Now, the case $p=kq$. If
$q=1$, then we have the canonical decomposition:

$\alpha(k,1)=(1,1,0\dots,0)\oplus(1,0,1,0,\dots,0)\oplus\dots\oplus(1,0,\dots,0,1).$

which can be argued directly. (for e.g., see the argument in the proof of
Proposition 7.2 in [10]). Then, by Lemma 5.2, we get that $\alpha(k,1)$ is
$\sigma$-polystable and $\sigma$-stable precisely when $k=1$, i.e.,
$k=1,(p,q)=(1,1)$. Now, if $p=kq$ and $q>1$, then by Theorem 5.20, we deduce
the canonical decomposition

$\alpha(kq,q)=(1,1,0\dots,0)^{\oplus q}\oplus(1,0,1,0,\dots,0)^{\oplus
q}\oplus\dots\oplus(1,0,\dots,0,1)^{\oplus q}.$

Thus $\alpha(kq,q)$ is Schofield polystable, but not Schofield stable by Lemma
5.2.

Now, for the last and fourth part. It is clear that in all other cases, we
$p/2<q<p$ or $kq/2<p<kq$. For $(r,s)\in\mathbb{Z}_{>1}^{2}$, let
$\delta(r,s)=krs$ and call $(r,s)$ minimal if it minimizes $\delta$ in its
equivalence class. In the former case (i.e., $p/2<q<p$), since $p>q$, we have
$(p,q)\sim_{k}(p,p-q)$ and $\delta(p,q)=kpq>kp(p-q)=\delta(p,p-q)$ because
$q>p/2$. In the latter case, we have $(p,q)\sim_{k}(kq-p,q)$ and
$\delta(p,q)=kpq>kp(kq-p)=\delta(kq-p,q)$ because $p>kq/2$. Thus we have shown
that $p/2<q<p$ or $kq/2<p<kq$ means that $(p,q)$ is not minimal. Now, if we
let $(p^{\prime},q^{\prime})$ be a minimal element in its equivalence class.
Now, $(p^{\prime},q^{\prime})$ has no choice but to be in the first three
cases since the last case contains only non-minimal elements as we just
argued. ∎

######  Lemma 5.22.

Recall that $\Gamma(p,q)=p^{2}+kq^{2}-kpq$, Let ${\rm g.c.d.}(p,q)=d$.

  * •

If $q\leq p/2$ and $p\leq kq/2$, then $\Gamma(p,q)<0$ unless $k=4$, $p=2q$, in
which case $\Gamma(p,q)=0$ and $d=q$.

  * •

If $q>p$ or $p>kq$, then $\Gamma(p,q)>d^{2}$.

  * •

If $q=p$, then $\Gamma(p,q)=d^{2}$ and $d=p=q$ and if $p=kq$, then
$\Gamma(p,q)=kq^{2}$ and $d=q$.

###### Proof.

This is a straightforward computation and left to the reader. ∎

######  Corollary 5.23.

Let $(p,q)\in\mathbb{Z}_{>1}^{2}$. Let $\Gamma(p,q)=p^{2}+kq^{2}-kpq$, Let
${\rm g.c.d.}(p,q)=d$. Then,

  * •

$\alpha(p,q)$ is Schofield polystable if and only if $\Gamma(p,q)\leq d^{2}$.

  * •

$\alpha(p,q)$ is Schofield stable if and only if either $\Gamma(p,q)<0$ or
$\Gamma(p,q)\in\\{0,1\\}$ with $d=1$.

  * •

$\alpha(p,q)$ is not Schofield semistable if and only if $\Gamma(p,q)>d^{2}$.

###### Proof.

This follows from Proposition 5.21 and Lemma 5.22. ∎

###### Proof of Theorem 1.2.

This follows from Corollary 5.23 and Theorem 4.3. ∎

##  6\. Polytopal characterization of Schofield stability

For a given quiver $Q$, and a dimension vector $\alpha\in\mathbb{Z}_{\geq
0}^{Q_{0}}$, recall the cone of effective weights:

(8) $\Sigma(Q,\alpha)=\\{\sigma\in\mathbb{Z}^{Q_{0}}\ |\
\operatorname{SI}(Q,\alpha)_{m\sigma}\neq 0\text{ for some $m>0$}\\}$

There are several equivalent descriptions of the cone of effective weights. We
write $\beta\hookrightarrow\alpha$ if a general representation of dimension
$\alpha$ has a $\beta$-dimensional subrepresentation.

(9) $\displaystyle\Sigma(Q,\alpha)$
$\displaystyle=\\{\sigma\in\mathbb{Z}^{Q_{0}}\ |\
\operatorname{SI}(Q,\alpha)_{m\sigma}\neq 0\text{ for some $m>0$}\\}$ (10)
$\displaystyle=\\{\sigma\in\mathbb{Z}^{Q_{0}}\ |\
\operatorname{SI}(Q,\alpha)_{\sigma}\neq 0\\}$ (11)
$\displaystyle=\\{\sigma\in\mathbb{Z}^{Q_{0}}\ |\ \alpha\text{ is
$\sigma$-semistable}\\}.$ (12) $\displaystyle=\\{\sigma\in\mathbb{Z}^{Q_{0}}\
|\ \sigma(\alpha)=0\text{ and }\sigma(\beta)\leq 0\ \forall\
\beta\hookrightarrow\alpha\\}.$

The equivalence of Equation (9) and Equation (10) follows from saturation, see
[13, Theorem 3]. Given that, the equivalence of Equation (9) and Equation (10)
with Equation (11) and Equation (12) is essentially due to King [30].

For a weight $\sigma\in\mathbb{Z}^{Q_{0}}$, we also define

$\overline{\Sigma}(Q,\sigma):=\\{\alpha\in\mathbb{Z}_{\geq 0}^{Q_{0}}\ |\
\alpha\text{ is $\sigma$-semistable}\\}.$

Suppose $\sigma\in\mathbb{Z}^{Q_{0}}$ such that $\overline{\Sigma}(Q,\sigma)$
is non-empty. Then there exists a dimension vector $\beta\in\mathbb{Z}_{\geq
0}^{Q_{0}}$ such that $\sigma(\gamma)=-\left<\gamma,\beta\right>$ for all
$\gamma\in\mathbb{R}^{Q_{0}}$, see [13, Theorem 1]. Thus, we have

$\displaystyle\overline{\Sigma}(Q,\sigma)$
$\displaystyle=\\{\alpha\in\mathbb{Z}_{\geq 0}^{Q_{0}}\ |\ \alpha\text{ is
$\sigma$-semistable}\\}$ $\displaystyle=\\{\alpha\in\mathbb{Z}_{\geq
0}^{Q_{0}}\ |\ \alpha\text{ is $(-\left<-,\beta\right>$)-semistable}\\}$
$\displaystyle=\\{\alpha\in\mathbb{Z}_{\geq 0}^{Q_{0}}\ |\ \beta\text{ is
$\left<\alpha,-\right>$-semistable}\\},$

The last equality follows from the reciprocity property, see [13, Corollary
1]. From the above, it is easy to see that $\overline{\Sigma}(Q,\sigma)$ is
just $\Sigma(Q,\beta)$ up to a change of coordinates that we now make precise.
Let $\zeta_{Q}:\mathbb{R}^{Q_{0}}\rightarrow\mathbb{R}^{Q_{0}}$ that is
defined as follows: For $\delta\in\mathbb{R}^{Q_{0}}$, let
$\zeta_{Q}(\delta)\in\mathbb{R}^{Q_{0}}$ be defined by

$\displaystyle\zeta_{Q}(\delta)\cdot\gamma=\sum_{x\in
Q_{0}}\zeta_{Q}(\delta)_{x}\gamma_{x}=\left<\delta,\gamma\right>\text{ for
}\gamma\in\mathbb{R}^{Q_{0}}.$

Then, by the above discussion, we conclude that

(13) $\displaystyle\overline{\Sigma}(Q,\sigma)=\zeta_{Q}(\Sigma(Q,\beta)).$

In particular, this means that $\overline{\Sigma}(Q,\sigma)$ is also (the
integer points of) a convex polyhedral cone and we can write down inequalities
that define it in terms of the subrepresentations of $\beta$. For the star
quiver, we define a polytope that carries the same information as the cone.

######  Definition 6.1 (Polytope from slice of the cone).

Consider the polyhedral cone $\Sigma(\Delta[k],\beta)$ of effective weights
for the star quiver $\Delta[k]$ with dimension vector $\beta$. We may instead
study the polytope

$\Sigma_{k}(\beta):=\\{x\in\Sigma(\Delta[k],\beta):x_{1}=1\\},$

the set of weights with first coordinate equal to $1$. We have
$\Sigma(\Delta[k],\beta)=\mathbb{R}_{\geq 0}\cdot\Sigma_{k}(\beta)$ because
the origin is the only element of $\Sigma(\Delta[k],\beta)$ with vanishing
first coordinate.

###  6.1. A polynomial time algorithm for detecting $\sigma$-semistability for
star quivers

In order to give a polynomial time algorithm for detecting
$\sigma$-semistability for star quivers, we reduce the problem to deciding if
a (generalized) Littlewood-Richardson coefficient is non-zero. We first state
the two main results of this section.

######  Theorem 6.2.

Consider the dimension vector $\alpha=(p,q_{1},\dots,q_{k})$ of the star
quiver $\Delta[k]$ and let
$\sigma=(\sigma_{x},\sigma_{y_{1}},\dots,\sigma_{y_{k}})$. Then, we can decide
if $\alpha$ is $\sigma$-semistable, i.e. if
$\sigma\in\Sigma(\Delta[k],\alpha)$, in poly$(p,k)$-time.

Throughout this section, polynomial time will mean polynomial in $p$ and $k$.

######  Proposition 6.3.

Consider the dimension vector $\alpha=(p,q_{1},\dots,q_{k})$ of the star
quiver $\Delta[k]$ and let
$\sigma=(\sigma_{x},\sigma_{y_{1}},\dots,\sigma_{y_{n}})$ be a weight such
that $\sigma_{x}\leq 0$ and $\sigma_{y_{i}}\geq 0$ for all $1\leq i\leq k$.
Then $\alpha$ is $\sigma$-semistable, i.e.
$\sigma\in\Sigma(\Delta[k],\alpha)$, if and only if the Littlewood-Richardson
coefficient
$c_{(\sigma_{y_{1}})^{q_{1}},\dots(\sigma_{y_{k}})^{q_{k}}}^{(-\sigma_{x})^{p}}$
(as defined below) is non-zero.

First, we need to define Littlewood-Richardson coefficients rigorously.
Consider the algebraic group
$G=\operatorname{GL}_{m}=\operatorname{GL}(\mathbb{C}^{m})$. Let
$\Lambda=\Lambda_{m}=\\{\lambda\in\mathbb{Z}^{m}\ |\
\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{m}\\}$. Irreducible
representations of $\operatorname{GL}_{m}$ are naturally indexed by $\Lambda$,
which is often called the dominant Weyl chamber. For each $\lambda\in\Lambda$,
there is an irreducible representation $S_{\lambda}(\mathbb{C}^{m})$ of
$\operatorname{GL}_{m}$. We call $\lambda$ the highest weight for
$S_{\lambda}(\mathbb{C}^{m})$.

For $\lambda^{(1)},\dots,\lambda^{(k)}\in\Lambda$, we decompose the tensor
product as a direct sum of irreducibles

(14) $S_{\lambda^{(1)}}(\mathbb{C}^{m})\otimes
S_{\lambda^{(2)}}(\mathbb{C}^{m})\otimes\dots\otimes
S_{\lambda^{(k)}}(\mathbb{C}^{m})=\bigoplus_{\mu}S_{\mu}(\mathbb{C}^{m})^{\oplus\left(c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\mu}\right)}.$

The multiplicities $c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\mu}$ are called
(generalized) Littlewood-Richardson coefficients. In literature, often
Littlewood-Richardson coefficients refer to the case when $k=2$, i.e., the
multiplicites obtained by decomposing a tensor product of two irreducibles.

For $\lambda=(\lambda_{1},\dots,\lambda_{m})\in\Lambda$, we define
$\widehat{\lambda}=(-\lambda_{m},-\lambda_{m-1},\dots,-\lambda_{1})\in\Lambda$.
And we note that

(15)
$\displaystyle(S_{\lambda}(\mathbb{C}^{m}))^{*}=S_{\lambda}((\mathbb{C}^{m})^{*})=S_{\widehat{\lambda}}(\mathbb{C}^{m})$

For the product of general linear groups
$\operatorname{GL}(\alpha)=\operatorname{GL}_{p}\times\operatorname{GL}_{q_{1}}\times\dots\times\operatorname{GL}_{q_{k}}$,
the irreducible representations are all of the form

$E_{\lambda^{(x)},\lambda^{(y_{1})},\dots,\lambda^{(y_{k})}}=S_{\lambda^{(x)}}(\mathbb{C}^{p})\otimes\bigotimes_{i}S_{\lambda^{(y_{i})}}(\mathbb{C}^{q_{i}}).$

One observes that $\operatorname{SI}(\Delta[k],\alpha)_{\sigma}$ is precisely
the isotypic component in $\mathbb{C}[{\rm Rep}(\Delta[k],\alpha)]$
corresponding to
$E_{(\sigma_{x})^{p},(\sigma_{y_{1}})^{q_{1}},\dots,(\sigma_{y_{k}})^{q_{k}}}$.
To do this, let us decompose $\mathbb{C}[{\rm Rep}(\Delta[k],\alpha)]$ into
irreducible representations for the action of $\operatorname{GL}(\alpha)$.

We define $\Lambda_{m}^{+}=\Lambda_{m}\cap\mathbb{Z}_{\geq 0}^{m}$. We observe
that ${\rm
Rep}(\Delta[k],\alpha)=\bigoplus_{i=1}^{k}(\mathbb{C}^{p})\otimes(\mathbb{C}^{q_{i}})^{*}$.
Thus, we have

$\displaystyle\mathbb{C}[{\rm Rep}(\Delta[k],\alpha)]$ $\displaystyle={\rm
Sym}({\rm Rep}(\Delta[k],\alpha)^{*})$ $\displaystyle=\bigotimes_{i=1}^{k}{\rm
Sym}((\mathbb{C}^{p})^{*}\otimes\mathbb{C}^{q_{i}}))$
$\displaystyle=\bigoplus_{\lambda^{(y_{i})}\in\Lambda_{y_{i}}^{+}}\left(\bigotimes_{i=1}^{k}S_{\lambda^{(y_{i})}}(\mathbb{C}^{p})^{*}\otimes
S_{\lambda^{(y_{i})}}(\mathbb{C}^{q_{i}})\right)$
$\displaystyle=\bigoplus_{\lambda^{(y_{i})}\in\Lambda_{y_{i}}^{+}}\left(\bigotimes_{i=1}^{k}S_{\lambda^{(y_{i})}}(\mathbb{C}^{p})^{*}\right)\otimes\left(\bigotimes_{i=1}^{k}S_{\lambda^{(y_{i})}}(\mathbb{C}^{q_{i}})\right)$
$\displaystyle=\bigoplus_{\lambda^{(y_{i})}\in\Lambda_{y_{i}}^{+}}\left(\bigoplus_{\mu}S_{\widehat{\mu}}(\mathbb{C}^{p})^{\oplus\left(c_{\lambda^{(y_{1})},\dots,\lambda^{(y_{k})}}^{\mu}\right)}\right)\otimes\left(\bigotimes_{i=1}^{k}S_{\lambda^{(y_{i})}}(\mathbb{C}^{q_{i}})\right)$
$\displaystyle=\bigoplus_{\mu,\lambda^{(y_{i})}\in\Lambda_{y_{i}}^{+}}(E_{\widehat{\mu},\lambda^{(y_{1})},\dots,\lambda^{(y_{k})}})^{\oplus\left(c_{\lambda^{(y_{1})},\dots,\lambda^{(y_{k})}}^{\mu}\right)}$

The first equality follows from the fact that polynomial functions
$\mathbb{C}[W]$ on any vector space $W$ is the same as ${\rm Sym}(W^{*})$. The
second equality is clear. The third equality follows from Cauchy’s identity
and the fourth equality is simply a regrouping of terms. The fifth equality is
a consequence of the definition of Littlewood-Richardson coefficients, i.e.,
Equation 14, and Equation 15. The final equality is simply a rearrangement of
terms.

###### Proof of Proposition 6.3.

From the above discussion, we observe that
$\operatorname{SI}(\Delta[k],\alpha)_{\sigma}$ is non-zero if and only if
$c_{(\sigma_{y_{1}})^{q_{1}},\dots,(\sigma_{y_{k}})^{q_{k}}}^{(-\sigma_{x})^{p}}$
is non-zero because $\widehat{(-\sigma_{x})^{p}}=(\sigma_{x})^{p}$. Since
$\operatorname{SI}(\Delta[k],\alpha)_{\sigma}\neq 0$ precisely when $\alpha$
is $\sigma$-semistable, we have the required conclusion. ∎

Now, we turn to proving Theorem 6.2.

###### Proof of Theorem 6.2.

From Proposition 6.3, we see that it suffices to show that the vanishing of
(generalized) Littlewood-Richardson coefficients can be decided in poly-time.
In the case $k=2$, i.e., $c_{\lambda,\mu}^{\nu}$, a (strongly) polynomial time
algorithm for deciding its vanishing was shown in [33] (another algorithm was
given in [3]). One can simply adapt this to get a poly-time algorithm for
computing $c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}.$

To describe this, we must first describe the algorithm of [34]. The authors
show $c_{\lambda,\mu}^{\nu}$ vanishes if and only if the polyhedron
$P_{\lambda,\mu}^{\nu}:=\\{x\in\mathbb{Q}^{m_{2}}:Ax\leq
b_{\lambda,\mu}^{\nu}\\}$ is empty, where
$b_{\lambda,\mu}^{\nu}\in\mathbb{Q}^{m_{1}}$ is a vector whose entries are
homogeneous linear forms in $\lambda,\mu,\nu$ with coefficients in
$\\{0,-1,1\\}$ and $A\in\operatorname{Mat}_{m_{1},m_{2}}(\\{0,-1,1\\})$ is a
matrix depending only on $n$ and not $\lambda,\mu,\nu$. The numbers
$m_{1},m_{2}$ are both $O(n^{2})$. Thus deciding the vanishing of
$c_{\lambda,\mu}^{\nu}$ can be decided in strongly polynomial time by the
existence of strongly polynomial time algorithms for linear programs with
coefficients in $\\{0,-1,1\\}$ [39].

We claim that a similar statement holds for
$c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\mu},$ namely that there is a vector
$b_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}\in\mathbb{Q}^{m_{1}(k)}$ whose
entries are linear forms in $\lambda^{(1)},\dots,\lambda^{(k)},\nu$ with
coefficients in $\\{0,-1,1\\}$ and
$A_{k}\in\operatorname{Mat}_{m_{1}(k),m_{2}(k)}(\\{0,-1,1\\})$ depending only
on $n$ and $k$ and not $\lambda^{(1)},\dots,\lambda^{(k)},\nu$ such that
$c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\mu}=0$ if and only if the polyhedron
$P_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}:=\\{x\in\mathbb{Q}^{m_{2}(k)}:A_{k}x\leq
b_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}\\}$ is empty. Moreover,
$m_{1}(k),m_{2}(k)=O(kn^{2})$. This will imply that the vanishing of
$c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}$ can also be decided in strongly
polynomial time. We prove the claim by induction on $k\geq 2$; the base case
is already established with $m_{1}=m_{1}(2),m_{2}=m_{2}(2),A_{2}=A$.

Now suppose $k\geq 3$. By saturation,
$c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}>0$ if and only if there exists
$t\in\mathbb{N}$ such that $c_{t\lambda^{(1)},\dots,t\lambda^{(k)}}^{t\nu}>0$.
By definition, $c_{t\lambda^{(1)},\dots,t\lambda^{(k)}}^{t\nu}>0$ if and only
if there some partition $\alpha$ such that
$c_{t\lambda^{(1)},\dots,t\lambda^{(k-1)}}^{\alpha}>0$ and
$c_{t\lambda^{(k)},\alpha}^{t\nu}>0.$ That is,
$c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}>0$ if and only if there exists
$t\in\mathbb{N}$, a non-increasing vector $\alpha\in\mathbb{Z}_{\geq 0}^{n}$,
and $x_{1}\in\mathbb{Q}^{m_{2}(k-1)},x_{2}\in\mathbb{Q}^{m_{2}}$ such that
$A_{k-1}x_{1}\leq b_{t\lambda^{(1)},\dots,t\lambda^{(k-1)}}^{\alpha}$ and
$Ax_{2}\leq b_{t\lambda^{(k)},\alpha}^{t\nu}.$ As the entries of
$b_{\lambda^{(1)},\dots,\lambda^{(k-1)}}^{\alpha},b_{\lambda^{(k)},\alpha}^{\nu}$
are homogeneous linear forms in the partitions, this is true if and only if
there exists $t\in\mathbb{N}$, a non-increasing vector
$\alpha\in\mathbb{Z}_{\geq 0}^{n}$, and
$x_{1}\in\mathbb{Q}^{m_{2}(k-1)},x_{2}\in\mathbb{Q}^{m_{2}}$ such that
$A_{k-1}x_{1}\leq b_{\lambda^{(1)},\dots,\lambda^{(k-1)}}^{\alpha/t}$ and
$Ax_{2}\leq b_{\lambda^{(k)},\alpha/t}^{\nu}.$ This is equivalent to the
existence of a non-increasing vector $\alpha\in\mathbb{Q}_{\geq 0}^{n}$ and
$x_{1}\in\mathbb{Q}^{m_{2}(k-1)},x_{2}\in\mathbb{Q}^{m_{2}}$ such that
$A_{k-1}x_{1}\leq b_{\lambda^{(1)},\dots,\lambda^{(k-1)}}^{\alpha}$ and
$Ax_{2}\leq b_{\lambda^{(k)},\alpha}^{\nu}.$ This is then equivalent to the
feasibility of a larger linear program, namely the nonemptiness of the
polyhedron

$\displaystyle
P^{\prime}=\\{(\alpha,x_{1},x_{2})\in\mathbb{Q}^{n+m_{2}(k-1)+m_{2}}:A_{k-1}x_{1}$
$\displaystyle\leq b_{\lambda^{(1)},\dots,\lambda^{(k-1)}}^{\alpha},$
$\displaystyle Ax_{2}$ $\displaystyle\leq b_{\lambda^{(k)},\alpha}^{\nu},$
$\displaystyle\alpha_{i+1}$ $\displaystyle\leq\alpha_{i}\;\forall i\in[n-1]$
$\displaystyle\alpha_{i}$ $\displaystyle\geq 0;\forall i\in[n]\\}.$

One checks that the matrix $A_{k}$ and vector
$b_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}\\}$ such that

$P^{\prime}=\\{y\in\mathbb{Q}^{m_{2}(k)}:A_{k}y\leq
b_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}\\}$

have the desired properties. Namely, we may $m_{2}(k)=m_{2}(k-1)+m_{2}+n$ and
$m_{1}(k)=m_{1}(k-1)+m_{1}+n^{2}+n$ which, by induction, are both $O(kn^{2})$.
Moreover, because $A_{k-1}$ and $A$ have entries in $\\{0,-1,1\\}$ and
$b_{\lambda^{(1)},\dots,\lambda^{(k-1)}}^{\alpha},b_{\lambda^{(k)},\alpha}^{\nu}$
are homogeneous linear forms with coefficients in $\\{0,-1,1\\}$, $A_{k}$ has
entries in $\\{0,-1,1\\}$ and $b_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}$
consists of homogeneous, linear forms with coefficients in $\\{0,-1,1\\}$.
This completes the proof. ∎

As a corollary, the polyhedra $\Sigma_{k}(\beta),\Sigma(\Delta[k],\beta)$ of
Eq. 8, Definition 6.1 are _well-described_ :

######  Corollary 6.4.

The polyhedra $\Sigma(\Delta[k],\beta),\Sigma_{k}(\beta)$ are _well-described_
, i.e. it is given by a finite collection of rational inequalities, each of
polynomial encoding length. Moreover, there is a polynomial time algorithm to
optimize linear functions over $\Sigma_{k}(\beta)$.

###### Proof.

Consider the Horn polytope $P$, the rational cone generated by partitions
$(\lambda^{(1)},\dots,\lambda^{(k)},\nu)$ such that
$c_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}>0$. By Proposition 6.3, the cone
$\Sigma(\Delta[k],\beta)$ is a preimage of $P$ under a linear map with Boolean
matrix entries in the coordinate bases. $\Sigma_{k}(\beta)$ an intersection of
$\Sigma(\Delta[k],\beta)$ with the coordinate plane $x_{1}=1$, and hence is
well-described if $\Sigma_{k}(\beta)$ is. Therefore it is enough to show that
$P$ is well-described. We now argue that this is the case.

Indeed, for rational $(\lambda^{(1)},\dots,\lambda^{(k)},\nu)$, we saw in the
proof of Theorem 6.2 that there are vectors whose entries are linear forms in
$\lambda^{(1)},\dots,\lambda^{(k)},\nu$ with coefficients in $\\{0,-1,1\\}$
and a Boolean matrix $A_{k}$ such that
$y=(\lambda^{(1)},\dots,\lambda^{(k)},\nu)\in P$ if and only if the polyhedron
$P_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}:=\\{x\in\mathbb{Q}^{m_{2}(k)}:A_{k}x\leq
b_{\lambda^{(1)},\dots,\lambda^{(k)}}^{\nu}\\}$ is nonempty. This implies $P$
is a projection of a well-described polyhedron $Q$, and hence $P$ is well-
described. This can be seen in several ways, e.g. Fourier-Motzkin elimination.

It remains to be seen why there is a polynomial time algorithm to optimize
linear functions over $\Sigma_{k}(\beta)$. By Theorem 6.4.1 in [24], strong
optimization and separation are polynomially equivalent for well-described
polyhedra, so it is enough to show that $\Sigma_{k}(\beta)$ has a polynomial
time separation oracle. By Proposition 6.3, the cone $\Sigma_{k}(\beta)$ is a
preimage of the slice $P^{\prime}:=P\cap\\{\sigma:x_{1}=1\\}$ of the Horn
polytope under a linear map with Boolean matrix entries in the coordinate
bases.

Thus it suffices to show that $P^{\prime}$ has a strong separation oracle.
Recall that $P$ is a coordinate projection of $Q$, and that both are well-
described. Thus $P^{\prime}$ is a coordinate projection of a well-described
coordinate slice $Q^{\prime}$ of $Q$. As the inequalities of $Q$ and hence
$Q^{\prime}$ are given explicitly, $Q^{\prime}$ has a strong separation oracle
and hence a strong optimization oracle. As $P^{\prime}$ is a projection of
$Q^{\prime}$, $P^{\prime}$ has a strong optimization oracle as well. Hence
$P^{\prime}$ also has a strong separation oracle. This completes the proof.∎

###  6.2. The Derksen-Weyman characterization of $\sigma$-stability

The set of $\sigma$-semistable dimension vectors, i.e.
$\overline{\Sigma}(Q,\sigma)$, is (the integer points of) a rational convex
polyhedral cone and membership of dimension vectors in
$\overline{\Sigma}(Q,\sigma)$ can be solved in polynomial time by Theorem 6.2.
On the other hand, $\sigma$-stability of a dimension vector $\alpha$ is more
subtle to understand and detect. The subset of dimension vectors in
$\overline{\Sigma}(Q,\sigma)$ that are $\sigma$-stable usually lacks
convexity. We say a dimension vector $\alpha\in\mathbb{Z}^{Q_{0}}$ is
indivisible if ${\rm g.c.d.}(\alpha_{x}:x\in Q_{0})=1$.

######  Lemma 6.5.

Let $Q$ be a quiver, $\alpha$ a dimension vector and $\sigma$ a weight.
Suppose $\alpha$ generates an extremal ray of $\overline{\Sigma}(Q,\sigma)$
(in particular $\alpha$ is indivisible). Then $\alpha$ is $\sigma$-stable.

###### Proof.

By definition of $\overline{\Sigma}(Q,\sigma)$, $\alpha$ is
$\sigma$-semistable. There is a concept of $\sigma$-stable decomposition,
i.e., for any dimension vector $\beta$ which is $\sigma$-semistable, we write
$\beta=\gamma_{1}\dotplus\gamma_{2}\dotplus\dots\dotplus\gamma_{k}$ if a
general representation of dimension $\beta$ has a Jordan-Hölder filtration (in
the category of $\sigma$-semistable representations) whose factors are of
dimensions $\gamma_{1},\gamma_{2},\dots,\gamma_{k}$ (in some order), see [14,
Section 3]. Each of the $\gamma_{i}$ are $\sigma$-stable, hence
$\sigma$-semistable and so $\gamma_{i}\in\overline{\Sigma}(Q,\sigma)$.

Now, since $\alpha$ generates an extremal ray, it cannot be written non-
trivially as a sum of dimension vectors in $\overline{\Sigma}(Q,\sigma)$.
Thus, its $\sigma$-stable decomposition must be trivial, i.e., $\alpha$ is
$\sigma$-stable. ∎

The locus of $\sigma$-stable dimension vectors in
$\overline{\Sigma}(Q,\sigma)$ apart from the extremal rays is more
complicated, but is captured by the result below due to Derksen and Weyman,
see [14, Theorem 6.4]. Suppose we have a sequence of dimension vectors
$\underline{\delta}=\delta_{1},\dots,\delta_{t}$ (all $\delta_{i}$ distinct),
we define $G(\underline{\delta})$ to be the directed graph with vertex set
$\\{1,2,\dots,t\\}$ and an edge from $i$ to $j$ if
$\left<\delta_{i},\delta_{j}\right><0$.

######  Theorem 6.6 (Derksen-Weyman).

Suppose $\alpha\in\overline{\Sigma}(Q,\sigma)$. Then there exists a linearly
independent sequence $\underline{\delta}=\delta_{1},\dots,\delta_{t}$ of
dimension vectors that generate extremal vectors of
$\overline{\Sigma}(Q,\sigma)$ such that $\alpha$ is a positive rational
combination of $\delta_{1},\dots,\delta_{t}$. The dimension vector $\alpha$ is
$\sigma$-stable if and only if

  1. (1)

$\alpha=\delta_{1}$ and $\delta_{1}$ is a real Schur root.

  2. (2)

or $\left<\delta_{i},\alpha\right>\leq 0$ and
$\left<\alpha,\delta_{i}\right>\leq 0$ for all $i$ and $G(\underline{\delta})$
is path-connected and $\alpha$ is indivisible if it is isotropic.

For fixed $\delta_{1},\dots,\delta_{t}$, the condition for $\alpha$ to be
proportional to some element of the semistable locus is polyhedral. Thus we
have the following.

######  Corollary 6.7.

Let $S$ be the locus of $\sigma$-stable dimension vectors in
$\overline{\Sigma}(Q,\sigma)$. Then $\mathbb{R}_{+}S=\\{as:a\in\mathbb{R},s\in
S\\}$ is a finite union of convex cones.

###  6.3. A polynomial time algorithm for detecting $\sigma$-stability for
star quivers

We turn to proving the following theorem:

######  Theorem 6.8.

Let $\alpha$ be a dimension vector and $\sigma$ be a weight for the quiver
$\Delta[k]$. Then, in polynomial time, we can decide if $\alpha$ is
$\sigma$-stable.

It is well-known that, given an optimization oracle for a polytope, one can
efficiently express interior points as convex combinations of vertices.

######  Theorem 6.9 (6.5.11 in [24]).

For a well-described polytope $(P;n,\Phi)$ given by a strong optimization
oracle, there is a polynomial time algorithm to express a rational interior
point $p$ as a convex combination of vertices $v_{1},\dots,v_{t}$ of $P$.

This theorem applies to our case because the polyhedron $\Sigma(Q,\beta)$ is
given by a strong optimization oracle by Corollary 6.4:

######  Corollary 6.10.

Consider the quiver $\Delta[k]$ and let $\alpha$ be a dimension vector and
$\sigma$ a weight such that $\alpha\in\overline{\Sigma}(Q,\sigma)$. Then, in
polynomial time, we can find linearly independent integral vectors
$\delta_{1},\dots,\delta_{t}$ that generate extremal vectors of
$\overline{\Sigma}(Q,\sigma)$ such that $\alpha$ is a positive rational
combination of $\delta_{1},\dots,\delta_{t}$.

###### Proof of Theorem 6.8.

We first check if $\alpha$ is $\sigma$-semistable (can be done efficiently by
Theorem 6.2). If $\alpha$ is $\sigma$-semistable, then we can find efficiently
$\delta_{1},\dots,\delta_{k}$ satisfying the hypothesis in Theorem 6.6. Now,
it is straightforward and efficient to check whether condition $(1)$ or
condition $(2)$ is satisfied, so we can decide whether $\alpha$ is
$\sigma$-stable efficiently. ∎

Now we have proved all we need for Theorem 1.3.

###### Proof of Theorem 1.3.

The first statement follows from Corollary 2.7, the second from the discussion
at the beginning of this section, and the third from Corollary 6.7 and Theorem
6.8.∎

###  6.4. Randomized algorithm for generic stability of star quivers

Here we include an algorithm to decide the generic stability of star quivers,
and hence generic unique existence of iPCA. The algorithm takes advantage of
the fact that semistability and polystability are generically equivalent for
star quivers. First we decide semistability, and hence polystability, by
checking whether the likelihood function is bounded below. Next we distinguish
polystability from stability by checking if there is a stabilizer subgroup of
positive dimension. For the latter calculation, we need only check that there
is no nontrivial element of the Lie algebra stabilizing the element in
question. Because the Lie algebra action is linear, we can check this by
evaluating the rank of the Lie algebra action as a linear map. Let

$\pi:\mathfrak{gl}(\alpha)=\operatorname{Mat}_{p,p}\oplus\bigoplus_{i=1}^{k}\operatorname{Mat}_{q_{i},q_{i}}\to{\rm
Rep}(\Delta[k],\alpha))=\bigoplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$

denote the Lie algebra action of $\mathfrak{gl}(\alpha)$ on some element
$B\in\bigoplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$. We can compute the Lie
algebra action as follows:

(16)
$\pi(H_{0},H_{1},\dots,H_{k})B=(H_{0}B_{1}-B_{1}H_{1},\dots,H_{0}B_{k}-B_{k}H_{k}).$

Any element of the form $H_{0}=tI_{p},H_{i}=tI_{q_{i}}$ will be in the
stabilizer, but any other elements are non-trivial.

Input:

A dimension vector $\alpha=(p,q_{1},\dots,q_{k})$.

Output:

Yes if $\alpha$ is generically stable, No otherwise.

Algorithm:

  1. (1)

For each $i\in\\{1,\dots,k\\}$, sample a $p\times q_{i}$ matrices $B_{i}$ with
i.i.d. complex Gaussian entries. Think of $B=(B_{1},\dots B_{k})$ as a
representation of the start quiver $\Delta[k]$.

  2. (2)

Check if $B=(B_{1},\dots,B_{k})$ is semistable using e.g. [21]. If not, output
No. Otherwise continue to the next step.

  3. (3)

Compute the Lie algebra action $\pi$ of $\mathfrak{gl}(\alpha)$ on $B$ as in
Eq. 16. If the dimension of the kernel of $\pi$ is at most $1$, output Yes.
Otherwise output No.

Algorithm 1 Randomized algorithm to decide generic stability of star quivers.

##  Appendix A From complex iPCA to real iPCA

A complex (affine) variety $X$ with an $\mathbb{R}$-structure (i.e., an
$\mathbb{R}$-subalgebra $\mathbb{R}[X]\subseteq\mathbb{C}[X]$ such that
$\mathbb{R}[X]\otimes_{\mathbb{R}}\mathbb{C}=\mathbb{C}[X]$333For any complex
(affine) variety $Y$, we denote by $\mathbb{C}[Y]$ its coordinate ring) is
called an (affine) $\mathbb{R}$-variety. As a technical point, we identify a
variety $X$ with its complex points $X_{\mathbb{C}}$ (which can be viewed as
algebra morphisms $\mathbb{C}[X]\rightarrow\mathbb{C}$). A morphism
$f:X\rightarrow Y$ of (affine) varieties is equivalent to a map on the
coordinate rings $f^{*}:\mathbb{C}[Y]\rightarrow\mathbb{C}[X]$. The morphism
$f$ is said to be defined over $\mathbb{R}$ if
$f^{*}(\mathbb{R}[Y])\subseteq\mathbb{R}[X]$. The real points $X_{\mathbb{R}}$
as the points in $X_{\mathbb{C}}$ (i.e., the algebra morphisms
$\mathbb{C}[X]\rightarrow\mathbb{C}$) that are defined over $\mathbb{R}$. A
complex algebraic group $G$ is called an $\mathbb{R}$-group if it is an
(affine) $\mathbb{R}$-variety such that the multiplication map and inverse map
are defined over $\mathbb{R}$, and its real points $G_{\mathbb{R}}$ is an
algebraic group over $\mathbb{R}$.

######  Remark A.1.

We only deal with affine varieties in this paper, so we will drop the prefix
affine from here on.

For a representation $V$ of an algebraic group $G$ (over $K=\mathbb{R}$ or
$\mathbb{C}$), we say that $V$ is generically $G$-semistable (resp.
$G$-polystable, $G$-stable) if there is a Zariski open dense subset of $V$
that consists of $G$-semistable points (resp. $G$-polystable points,
$G$-stable points). The following is a very important result.

######  Proposition A.2.

Let $G$ be a connected reductive $\mathbb{R}$-group. Let $V$ be a rational
representation of $G$ that is defined over $\mathbb{R}$. Then $V$ is
generically $G$-semistable (resp. $G$-polystable, $G$-stable) if and only if
$V_{\mathbb{R}}$ is generically $G_{\mathbb{R}}$-semistable (resp.
$G_{\mathbb{R}}$-polystable, $G_{\mathbb{R}}$-stable).

###### Proof.

See [10, Proposition 2.23]. ∎

Thus, as far as (generic) semistability/polystability/stability is concerned,
there is no difference whether you consider the (actions corresponding to the)
real model $\mathcal{M}_{\alpha,\mathbb{R}}$ or the complex model
$\mathcal{M}_{\alpha,\mathbb{C}}$. However, there is still one subtle issue,
namely that stability is not equivalent to existence of a unique MLE for real
Gaussian groups (see Theorem 2.3). However, we leverage Lemma 2.4 instead.
First, a lemma:

######  Lemma A.3.

Let $\alpha=(p,q_{1},\dots,q_{r})$. Suppose a generic $V\in{\rm
Rep}(\Delta[k],\alpha)_{\mathbb{R}}$ is indecomposable (over $\mathbb{R}$),
then a generic $V\in{\rm Rep}(\Delta[k],\alpha)_{\mathbb{C}}$ is
indecomposable (over $\mathbb{C}$).

###### Proof.

We will prove this by contradiction. Suppose a generic $V\in{\rm
Rep}(\Delta[k],\alpha)_{\mathbb{C}}$ is not indecomposable. Then, its
canonical decomposition must be non-trivial, say it is

$\alpha=\beta_{1}\oplus\beta_{2}\oplus\dots\oplus\beta_{k}.$

This means that the natural map
$\psi:\operatorname{GL}(\alpha)_{\mathbb{C}}\times\prod_{i=1}^{k}{\rm
Rep}(\Delta[k],\beta_{i})_{\mathbb{C}}=:X\rightarrow{\rm
Rep}(\Delta[k],\alpha)_{\mathbb{C}}=:Y$ is a dominant morphism of varieties.
Observe that both $X$ and $Y$ are varieties that are defined over $\mathbb{R}$
in a natural way and that the $\mathbb{R}$-points $X_{\mathbb{R}}$ and
$Y_{\mathbb{R}}$ are Zariski-dense in $X$ and $Y$ respectively. Since
$X_{\mathbb{R}}$ is Zariski-dense and $\psi$ is dominant, we conclude that
$\psi(X_{\mathbb{R}})$ is Zariski-dense in $Y$ and hence Zariski-dense in
$Y_{\mathbb{R}}$. Now, $\psi:X_{\mathbb{R}}\rightarrow Y_{\mathbb{R}}$ is a
morphism of real algebraic varieties and hence the image
$\psi(X_{\mathbb{R}})$ is a semi-algebraic set. Since it is dense, it must be
a full dimensional semi-algebraic set.

Observe that $\psi(X_{\mathbb{R}})$ consists of representations that are
decomposable (over $\mathbb{R}$). Thus, we have a full dimensional semi-
algebraic set, i.e., $\psi(X_{\mathbb{R}})$ which consists of representations
that are decomposable. Thus, we cannot have a Zariski open dense subset of
$Y_{\mathbb{R}}$ consisting of indecomposable representations. ∎

######  Lemma A.4.

Let $V\in{\rm Rep}(Q,\alpha)_{\mathbb{R}}$. If $V$ is decomposable over
$\mathbb{R}$, then the image of the stabilizer
$H=\rho_{\alpha}((\operatorname{GL}(\alpha)_{\sigma}(\mathbb{R}))_{V})$ is not
compact.

###### Proof.

Suppose $V=U\oplus W$ is a decomposition of $V$ over $\mathbb{R}$. In other
words, for each $x\in Q_{0}$, we have a splitting
$V(x)=\mathbb{R}^{\alpha_{x}}=U(x)\oplus W(x)$ and for each arrow $a\in
Q_{1}$, $V(a)$ maps $U(ta)\rightarrow U(ha)$ and maps $W(ta)\rightarrow
W(ha)$. Suppose $\dim(U)=\beta$ and $\dim(W)=\gamma$.

Let $\lambda=2$ and let $\mu\in\mathbb{R}_{>0}$ be such that
$\lambda^{\sigma\cdot\beta}\mu^{\sigma\cdot\gamma}=1$. We need to argue that
such a $\mu$ exists. Let $\nu_{1}=\sigma\cdot\beta$ and
$\nu_{2}=\sigma\cdot\gamma$. Since $\sigma\cdot\alpha=0$, either both
$\nu_{1}$ and $\nu_{2}$ are zero or both non-zero. If both are zero, then any
choice of $\mu$ works. If both are non-zero, then there is a unique choice of
$\mu$.

Consider the linear transformation $g_{x}\in\operatorname{GL}_{\alpha_{x}}$
that acts on $U(x)$ by the scalar $\lambda$ and on $W(x)$ by the scalar $\mu$.
It is easy to check that $g=(g_{x})_{x\in
Q_{0}}\in\operatorname{GL}(\alpha)_{\sigma}$ because $\prod_{x\in
Q_{0}}\det(g_{x})^{\sigma_{x}}=\lambda^{\sigma\cdot\beta}\mu^{\sigma\cdot\gamma}=1$.

It is straightforward to see that $\rho_{\alpha}(g)$ has eigenvalues with
absolute value $\neq 1$. Thus the sequence
$\\{\rho_{\alpha}(g^{m})\\}_{m\in\mathbb{Z}_{>0}}$ is a sequence in $H$ with
no convergent subsequence. Thus $H$ cannot be compact. ∎

Before we prove the theorem, we will need the following lemma. For an
algebraic group $G$ (defined over $K=\mathbb{R}$ or $\mathbb{C}$), we denote
by $G^{\circ}$ its identity component, i.e., the connected component (in the
Euclidean topology) of $G$ that contains the identity element. The following
lemma was proven for $K=\mathbb{C}$ in [10, Lemma 2.17], but it is also true
for $K=\mathbb{R}$ as we will now show.

######  Lemma A.5.

Let $K=\mathbb{R}$ or $\mathbb{C}$ denote the ground field. Let $G$ be an
algebraic group and $V$ a rational representation. Let $G^{\circ}$ denote the
identity component of $G$. Then for $v\in V$, $v$ is
$G$-semistable/polystable/stable if and only if $v$ is
$G^{\circ}$-semistable/polystable/stable.

###### Proof.

First, we note that $G^{\circ}$ is a subgroup of finite index in $G$,
equivalently that $G$ has finitely many components. For $K=\mathbb{C}$, this
is well known because connected components in the Euclidean topology agree
with the connected components in the Zariski topology, the latter of which is
always finite by Noetherianity. For $K=\mathbb{R}$, this follows from Whitney
[43, Theorem 3] which proves that any real algebraic variety has finitely many
topological components.

There is one subtle issue in the case of $K=\mathbb{R}$. We note that
$G^{\circ}$ may not be an algebraic group over $\mathbb{R}$! However, it is
still a Lie group and all the arguments used here should be taken in the
framework of Lie theory. We only defined stability notions for algebraic
groups, but their definitions extend easily to any group action, in particular
for actions of Lie groups.

Suppose $v$ is $G$-semistable, then it is clearly $G^{\circ}$-semistable.
Conversely, suppose $v$ is not $G$-semistable, then there is a sequence
$\\{g_{m}\\}$ such that $\lim_{m\to\infty}g_{m}v=0$. Since $G$ has finitely
many components, we can extract a subsequence $\\{h_{m}\\}$ with the same
property but is contained in one connected component, say $jG^{\circ}$ for
some $j\in G$. Then, we see that $\\{j^{-1}h_{m}\\}$ is a sequence in
$G^{\circ}$ such that $\lim_{m\to\infty}j^{-1}h_{m}v=0$. Hence $v$ is
$G^{\circ}$-unstable.

The arguments for polystability and stability are the same as in [10, Lemma
2.17], but we recall them here for completeness. For polystability, we observe
that the $G$-orbit is a disjoint finite union of $G^{\circ}$-orbits each of
which forms a connected component of the $G$-orbit. Hence the $G$-orbit of $v$
is closed if and only if the $G^{\circ}$ orbit of $v$ is closed.

For stability, we need to observe that (by the orbit-stabilizer theorem)
$\dim(G_{v})=\dim((G^{\circ})_{v})$ because their orbits have the same
dimensions. Further suppose the action of $G$ on $V$ is given by
$\rho:G\rightarrow\operatorname{GL}(V)$. Then, we observe that the kernel of
$\rho$ (denoted $\Delta$) and the kernel of $\rho|_{G^{\circ}}$ (i.e.,
$\Delta\cap G^{\circ}$) have the same dimension because they both have the
same Lie algebra. Thus, $\dim(G_{v})=\dim(\Delta)$ if and only if
$\dim((G^{\circ})_{v})=\dim(\Delta\cap G^{\circ})$. ∎

Now, we have all the ingredients need to prove Theorem 4.3.

###### Proof of Theorem 4.3.

We observe that $\operatorname{GL}(\alpha)_{\sigma}$ is a reductive
$\mathbb{R}$-group and ${\rm
Rep}(\Delta[k],\alpha))=\oplus_{i=1}^{k}\operatorname{Mat}_{p,q_{i}}$ is an
$\mathbb{R}$-variety in a natural way. Since
$\operatorname{GL}(\alpha)_{\sigma}$ is not a connected group, which prevents
us from applying Proposition A.2 directly, which is precisely why we prove
Lemma A.5.

Let $\mu=(\mu(x),\mu(y_{1}),\dots,\mu(y_{k}))$ be the unique indivisible
integral weight that is a positive multiple of $\sigma$. In other words,
$\sigma=d\mu$ for some $d\in\mathbb{Z}_{\geq 1}$ and ${\rm
g.c.d.}(\mu(x),\mu(y_{1}),\dots,\mu(y_{k}))=1$. First, we observe that
$\operatorname{GL}(\alpha)_{\mu}$ is connected, see [10, Remark 2.19].
Consider the map
$\mu:\operatorname{GL}(\alpha)_{\sigma}\rightarrow\mathbb{C}^{\times}$ given
by
$\mu((g_{x},g_{y_{1}},\dots,g_{y_{k}}))=\det(g_{x})^{\mu(x)}\cdot\prod_{i=1}^{k}\det(g_{y_{i}})^{\mu(y_{i})}$.
Then the kernel of $\mu$ is $\operatorname{GL}(\alpha)_{\mu}$ and the image is
finite as it is contained in the set of $d^{th}$ roots of unity. This means
that the identity component of $\operatorname{GL}(\alpha)_{\sigma}$ is
$\operatorname{GL}(\alpha)_{\mu}$. Thus
$\operatorname{GL}(\alpha)_{\sigma}$-semistability/polystability/stability is
the same as
$\operatorname{GL}(\alpha)_{\mu}$-semistability/polystability/stability by
Lemma A.5. By Proposition A.2, we deduce that generic
semistability/polystability/stability for the action of
$\operatorname{GL}(\alpha)_{\mu}$ on ${\rm Rep}(\Delta[k],\alpha))$ is
equivalent to generic semistability/polystability/stability for the action of
$(\operatorname{GL}(\alpha)_{\mu})_{\mathbb{R}}$ on $({\rm
Rep}(\Delta[k],\alpha)))_{\mathbb{R}}$.

A similar argument as above will show that
$(\operatorname{GL}(\alpha)_{\mu})_{\mathbb{R}}$ is a subgroup of finite index
in $(\operatorname{GL}(\alpha)_{\sigma})_{\mathbb{R}}$. Hence their identity
components are the same, so generic semistability/polystability/stability for
the action of $(\operatorname{GL}(\alpha)_{\mu})_{\mathbb{R}}$ on $({\rm
Rep}(\Delta[k],\alpha)))_{\mathbb{R}}$ is equivalent to generic
semistability/polystability/stability for the action of
$(\operatorname{GL}(\alpha)_{\sigma})_{\mathbb{R}}$ on $({\rm
Rep}(\Delta[k],\alpha)))_{\mathbb{R}}$, again by Lemma A.5.

Thus, $\alpha$ being $\sigma$-semistable/polystable/stable is the same as
generic
semistability/polystability/stability for the action of
$(\operatorname{GL}(\alpha)_{\sigma})_{\mathbb{R}}$ on $({\rm
Rep}(\Delta[k],\alpha)))_{\mathbb{R}}$. Thus Proposition 2.6 and Corollary 2.7
immediately give the first two statements and the backwards direction of the
last statement. So, only the forward direction of the last statement remains,
which we prove by contradiction.

Suppose $\alpha$ is not $\sigma$-stable. If $\alpha$ is not
$\sigma$-polystable, then we do not have generic existence of MLE does not
exist so clearly we cannot have generic existence of a unique MLE. So, we can
assume $\alpha$ is not $\sigma$-stable, but $\sigma$-polystable. So we do not
have generic indecomposability in ${\rm Rep}(Q,\alpha)_{\mathbb{C}}$. By Lemma
A.3, we do not have generic indecomposability in ${\rm
Rep}(Q,\alpha)_{\mathbb{R}}$. By Lemma A.4, we do not have generically have a
compact stabilizer. Thus, by Lemma 2.4, we do have generically have the
existence of a unique MLE. ∎

######  Remark A.6.

In the last paragraph of the proof above, we are careful to use phrases like
“we do not have generic property P” instead of “we generically have the
property of not P.” There is a subtle difference in the two. The former says
that the set of points satisfying P does not contain a Zariski-open dense
subset and the latter says that the set of points satisfying P is contained in
a lower dimensional subvariety.

### Acknowledgements

CF thanks Ankur Moitra for mentioning iPCA and Tiffany M. Tang for helpful
conversations. The authors thank Michael Walter for correcting a subtle point
about the time complexity of our algorithm.

## References

  * [1] Carlos Améndola, Kathlén Kohn, Philipp Reichenbach, and Anna Seigal.  Invariant theory and scaling algorithms for maximum likelihood estimation.  arXiv preprint arXiv:2003.13662, 2020.
  * [2] Peter Bürgisser, Cole Franks, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson.  Efficient algorithms for tensor scaling, quantum marginals, and moment polytopes.  In 59th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2018, pages 883–897. IEEE Computer Soc., Los Alamitos, CA, 2018.
  * [3] Peter Bürgisser and Christian Ikenmeyer.  Deciding positivity of littlewood–richardson coefficients.  SIAM Journal on Discrete Mathematics, 27(4):1639–1681, 2013.
  * [4] Calin Chindris, Brett Collins, and Daniel Kline.  Membership in moment cones, quiver semi-invariants, and generic semi-stability for bipartite quivers.  arXiv preprint arXiv:2211.01990, 2022.
  * [5] Harm Derksen and Gregor Kemper.  Computational invariant theory, volume 130 of Encyclopaedia of Mathematical Sciences.  Springer, Heidelberg, enlarged edition, 2015.  With two appendices by Vladimir L. Popov, and an addendum by Norbert A’Campo and Popov, Invariant Theory and Algebraic Transformation Groups, VIII.
  * [6] Harm Derksen and Visu Makam.  Generating invariant rings of quivers in arbitrary characteristic.  J. Algebra, 489:435–445, 2017.
  * [7] Harm Derksen and Visu Makam.  Polynomial degree bounds for matrix semi-invariants.  Adv. Math., 310:44–63, 2017.
  * [8] Harm Derksen and Visu Makam.  Degree bounds for semi-invariant rings of quivers.  J. Pure Appl. Algebra, 222(10):3282–3292, 2018.
  * [9] Harm Derksen and Visu Makam.  An exponential lower bound for the degrees of invariants of cubic forms and tensor actions.  Adv. Math., 368:107136, 25, 2020.
  * [10] Harm Derksen and Visu Makam.  Maximum likelihood estimation for matrix normal models via quiver representations.  arXiv preprint arXiv:2007.10206, 2020.
  * [11] Harm Derksen, Visu Makam, and Michael Walter.  Maximum likelihood estimation for tensor normal models via castling transforms.  arXiv preprint arXiv:2011.03849, 2020.
  * [12] Harm Derksen, Aidan Schofield, and Jerzy Weyman.  On the number of subrepresentations of a general quiver representation.  Journal of the London Mathematical Society, 76(1):135–147, 2007\.
  * [13] Harm Derksen and Jerzy Weyman.  Semi-invariants of quivers and saturation for littlewood-richardson coefficients.  Journal of the American Mathematical Society, 13(3):467–479, 2000\.
  * [14] Harm Derksen and Jerzy Weyman.  The combinatorics of quiver representations.  In Annales de l’Institut Fourier, volume 61, pages 1061–1131, 2011\.
  * [15] Harm Derksen and Jerzy Weyman.  An introduction to quiver representations, volume 184.  American Mathematical Soc., 2017.
  * [16] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky.  Quivers with potentials and their representations ii: applications to cluster algebras.  Journal of the American Mathematical Society, 23(3):749–790, 2010\.
  * [17] Mátyás Domokos and Alexander N Zubkov.  Semi-invariants of quivers as determinants.  Transformation groups, 6(1):9–24, 2001.
  * [18] Michael A Forbes and Amir Shpilka.  Explicit noether normalization for simultaneous conjugation via polynomial identity testing.  In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 527–542. Springer, 2013.
  * [19] Cole Franks, Rafael Oliveira, Akshay Ramachandran, and Michael Walter.  Near optimal sample complexity for matrix and tensor normal models via geodesic convexity.  arXiv preprint arXiv:2110.07583, 2021.
  * [20] William Cole Franks and Ankur Moitra.  Rigorous guarantees for tyler’s m-estimator via quantum expansion.  In Conference on Learning Theory, pages 1601–1632. PMLR, 2020.
  * [21] Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson.  A deterministic polynomial time algorithm for non-commutative rational identity testing.  In 57th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2016, pages 109–117. IEEE Computer Soc., Los Alamitos, CA, 2016.
  * [22] Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson.  Algorithmic and optimization aspects of brascamp-lieb inequalities, via operator scaling.  Geometric and Functional Analysis, 28(1):100–145, 2018.
  * [23] Victor Ginzburg.  Lectures on nakajima’s quiver varieties.  arXiv preprint arXiv:0905.0686, 2009.
  * [24] Martin Grötschel, László Lovász, and Alexander Schrijver.  Geometric algorithms and combinatorial optimization, volume 2 of Algorithms and Combinatorics.  Springer-Verlag, Berlin, second edition, 1993.
  * [25] Pavel Hrubes and Avi Wigderson.  Non-commutative arithmetic circuits with division.  In Moni Naor, editor, Innovations in Theoretical Computer Science, ITCS’14, Princeton, NJ, USA, January 12-14, 2014, pages 49–66. ACM, 2014.
  * [26] Gábor Ivanyos, Youming Qiao, and K. V. Subrahmanyam.  Constructive non-commutative rank computation is in deterministic polynomial time.  Comput. Complexity, 27(4):561–593, 2018.
  * [27] Victor G Kac.  Infinite root systems, representations of graphs and invariant theory.  Inventiones mathematicae, 56(1):57–92, 1980.
  * [28] Victor G Kac.  Infinite root systems, representations of graphs and invariant theory, ii.  Journal of algebra, 78(1):141–162, 1982.
  * [29] Bernhard Keller.  Cluster algebras, quiver representations and triangulated categories.  arXiv preprint arXiv:0807.1960, 2008.
  * [30] Alastair D King.  Moduli of representations of finite dimensional algebras.  The Quarterly Journal of Mathematics, 45(4):515–530, 1994.
  * [31] Allen Knutson and Terence Tao.  The honeycomb model of $GL_{n}(\mathbb{C})$ tensor products i: Proof of the saturation conjecture.  Journal of the American Mathematical Society, 12(4):1055–1090, 1999\.
  * [32] Ketan D. Mulmuley.  Geometric complexity theory V: Efficient algorithms for Noether normalization.  J. Amer. Math. Soc., 30(1):225–309, 2017.