# Shift orbits for elementary representations of Kronecker quivers

Daniel Bissinger

###### Abstract.

Let $r\in\mathbb{N}_{\geq 3}$. We denote by $K_{r}$ the wild $r$-Kronecker
quiver with $r$ arrows $\gamma_{i}\colon 1\longrightarrow 2$ and consider the
action of the group $G_{r}\subseteq\operatorname{Aut}(\mathbb{Z}^{2})$
generated by
$\delta\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z}^{2},(x,y)\mapsto(y,x)$
and
$\sigma_{r}\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z}^{2},(x,y)\mapsto(rx-y,x)$
on the set of regular dimension vectors

$\mathcal{R}=\\{(x,y)\in\mathbb{N}^{2}\mid x^{2}+y^{2}-rxy<1\\}.$

A fundamental domain of this action is given by
$\mathcal{F}_{r}\coloneqq\\{(x,y)\in\mathbb{N}^{2}\mid\frac{2}{r}x\leq y\leq
x\\}$. We show that $(x,y)\in\mathcal{F}_{r}$ is the dimension vector of an
elementary representation if and only if

$y\leq\min\\{\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r,\lceil\frac{x}{r}\rceil-\frac{x}{\lceil\frac{x}{r}\rceil}+r,r-1\\},$

where we interpret
$\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r$ as $\infty$
for $1\leq x<r$. In this case we also identify the set of elementary
representations as a dense open subset of the irreducible variety of
representations with dimension vector $(x,y)$. A complete combinatorial
description of elementary representations for $r=3$ has been given by Ringel.
We show that such a compact description is out of reach when we consider
$r\geq 4$, altough the representation theory of $K_{3}$ is as difficult as the
representation theory of $K_{r}$ for $r\geq 4$.

††2020 Mathematics Subject Classification: 16G20††Keywords: Kronecker quivers,
Steiner bundles, Elementary representations, Generic representations

## Introduction

Let $\mathbbm{k}$ be an algebraically closed field of arbitrary characteristic
and $Q$ be a finite, connected and wild quiver without oriented cycles. It is
well known that the theory of finite dimensional representations over $Q$ is
undecidable (see [2, 4.4], [13]), which makes a full classification of the
category $\operatorname{rep}(Q)$ of finite dimensional representations a
hopeless task.

The indecomposable representations of $Q$ fall into three classes: There are
countable many (isomorphism classes of) so-called preinjective and
preprojective indecomposable representations that are well-understood. All
other indecomposable representations are called regular. A (not necessarily
indecomposable) representation is called regular if all of its indecomposable
direct summands are regular and we denote by
$\operatorname{reg}(Q)\subseteq\operatorname{rep}(Q)$ the full subcategory
containing all regular representations. This subcategory contains the large
majority of representations and is responsible for the wild behaviour of the
category $\operatorname{rep}(Q)$.

Since regular representations are closed under extensions, there is a uniquely
determined smallest class of regular representation
$\mathcal{E}\subseteq\operatorname{reg}(Q)$ closed under isomorphisms, whose
extension-closure is $\operatorname{reg}(Q)$. In particular, every
representation $M$ possesses a (in general not uniquely determined) finite
filtration

$0=M_{0}\subset M_{1}\subset M_{2}\subset\cdots\subset M_{l-1}\subset M_{l}=M$

with $M_{i}/M_{i-1}\in\mathcal{E}$ for all $i\in\\{1,\ldots,l\\}$. The
representations in $\mathcal{E}$ are called elementary and are the simple
objects in the category of regular representations. The definition of
elementary representations is due to Crawley-Boevey and is a natural
generalization of quasi-simple representations living in regular tubes of tame
hereditary algebras.

Among other things, elementary representations are of interest because they
are closely related (see [11, 3.1]) to the graph of domination (see [10, 15.2,
15.3] for a precise definition), whose sinks are given by the extensively
studied wild Kronecker algebras corresponding to generalized Kronecker quivers

$\textstyle{K_{r}=}$$\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\gamma_{2}}$$\scriptstyle{\gamma_{r}}$$\scriptstyle{\vdots}$$\textstyle{2}$$\textstyle{,r\in\mathbb{N}_{\geq
3}.}$

Since a representation in $\operatorname{rep}(Q)$ is elementary if and only if
its Auslander-Reiten translate $\tau_{Q}(E)$ is elementary and the Coxeter
transformation describes the $\tau_{Q}$-orbits on the level of the
Grothendieck group, it is natural to consider Coxeter-orbits that belong to
elementary representations.
These orbits have been studied systematically in [12, 11] and it has been
shown that there are only finitely many Coxeter-orbits of dimension vectors of
elementary representations. The explicit number $e(Q)\in\mathbb{N}$ of
Coxeter-orbits of elementary representations is known in a few cases (see for
example [12, 4.2.1]). But even for generalized Kronecker quivers this was
unknown until recently: In [16] the equality $e(K_{3})=4$ has been proven.

In this article we tackle the general case and arrive at a criterion thats
allows us to decide wether or not a dimension vector $(x,y)$ is the dimension
vector of an elementary representation. In particular, we can decide wether a
Coxeter-orbit belongs to the dimension vector of an elementary representation.
As noted in [16], it suffices to identify the elements in

$\mathcal{F}_{r}\coloneqq\\{(x,y)\in\mathbb{N}^{2}\mid\frac{2}{r}x\leq y\leq
x\\}$

that are the dimension vector of an elementary representation to obtain such a
criterion. We follow this approach and arrive at:

######  Theorem A.

An element $(x,y)\in\mathcal{F}_{r}$ is the dimension vector of an elementary
representation if and only if

$(\ast)\quad
y\leq\min\\{\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r,\lceil\frac{x}{r}\rceil-\frac{x}{\lceil\frac{x}{r}\rceil}+r,r-1\\},$

where we interpret
$\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r$ as $\infty$
for $1\leq x<r$.

In the following we outline the structure of this article and point out
differences to [16] in the proof of Theorem A. A crucial step in [16], to show
that an elementary representation $E$ with dimension vector
$\operatorname{\underline{dim}}E\in\mathcal{F}_{3}$ has to satisfy
$\operatorname{\underline{dim}}E\in\\{(1,1),(2,2)\\}$, is an elegant
application of the Projective Dimension Theorem (see [7, I.7.2]). The
Projective Dimension Theorem is used to prove that every
$K_{3}$-representation $M$ with dimension $(x,y)$ and and $2\leq y\leq x+1$
has a subrepresentation with dimension vector $(1,2)$.

In the case $r\geq 4$ this geometric tool no longer yields strong enough
restrictions on dimension vectors in $\mathcal{F}_{r}$ that are the dimension
vector associated to an elementary representation: For $r=4$ the approach does
not rule out the dimension vectors $(3,3)\in\mathcal{F}_{4}$ altough it can
not belong to an elementary representation by Theorem A.

Our approach rests on the consideration of the full subcategories
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\subseteq\operatorname{rep}(K_{r})$
($1\leq d<r$), of so-called relative $d$-projective Kronecker representations,
each being equivalent to the category of Steiner bundles on the Grassmannian
$\operatorname{Gr}_{d}(A_{r})$ (see [3, 3.2.3]), where
$A_{r}=\bigoplus^{r}_{i=1}\mathbbm{k}\gamma_{i}$ denotes the arrow space of
the path algebra $\mathbbm{k}K_{r}$. Restrictions on the minimal rank of non-
trivial Steiner bundles, first established in [1] for fields of characteristic
zero, and the connection between relative projective representations and
representations with the so-called equal socle property (this definition
originated from modular representation theory of finite group schemes) allow
us to prove that an elementary representation $E$ with dimension vector
$(x,y)\in\mathcal{F}_{r}$ has to satisfy $(\ast)$. For
$(x,y)\in\mathcal{F}_{r}$ satisfying $(\ast)$, we show that the elements
$f\in\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{x})\coloneqq\operatorname{Hom}_{\mathbbm{k}}(\mathbbm{k}^{x},\mathbbm{k}^{y})^{r}$
with $(\mathbbm{k}^{x},\mathbbm{k}^{y},f)\in\operatorname{rep}(K_{r})$
elementary form an open set $\mathcal{E}(x,y)$ in the affine variety
$\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{x})$. We do so by
showing that being relative $d$-projective and having the equal socle property
is an open property. Moreover, we prove that this set has to be non-empty by
dimension reasons, showing that the assumptions in Theorem A are sufficient.

An important tool in the proof of Theorem A is a new description by Reineke
(see [14, 3.4]) concerning general subrepresentations of Kronecker
representations. We characterize the category of representations with the
equal $d$-socle property as those representations that do not have
subpresentations with dimension vector $(1,a)$ for all
$a\in\\{0,\ldots,r-d\\}$. This allows us to apply Reineke’s Theorem and
generalize Ringel’s approach.

In the last section of this article we study the internal structure of
elementary representations for $K_{r}$ with $r\geq 3$. For the tame Kronecker
quiver $K_{2}$, the quasi-simple representations are well-known and completely
determined in terms of their coefficient quiver, i.e. there exists a non-zero
element $\alpha\in A_{2}$ such that the coefficient quiver has the following
form:

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\bullet.}$

For $r=3$, the elementary representations with dimension vector in
$\mathcal{E}_{3}=\\{(1,1),(2,2)\\}$ can also be described combinatorially in
terms of the coefficient quiver (see [16]). More precisely: There exists a
basis $\alpha,\beta,\gamma$ of the arrow space $A_{3}$ such that the
coefficient quiver has one of the following two forms:

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet.}$

Rephrasing this in the terms of the natural action of the general linear group
$\operatorname{GL}(A_{r})$ on $\operatorname{rep}(K_{r})$, this just means
that a representation $E\in\operatorname{rep}(K_{3})$ with dimension vector in
$\mathcal{F}_{3}$ is elementary if and only if $M$ is isomorphic to an element
in the $\operatorname{GL}(A_{r})$-orbit of
$E_{1}\coloneqq(\mathbbm{k},\mathbbm{k},(\operatorname{id}_{\mathbbm{k}},0,0))$
or
$E_{2}\coloneqq(\mathbbm{k}^{2},\mathbbm{k}^{2},(\operatorname{id}_{\mathbbm{k}^{2}},\beta,\gamma))$
with $\beta(a,b)=(0,a)$ and $\gamma(a,b)=(b,0)$ for all
$(a,b)\in\mathbbm{k}^{2}$. Since the action of $\operatorname{GL}(A_{r})$ on
$\operatorname{rep}(K_{r})$ commutes with the Auslander-Reiten translation
$\tau_{K_{r}}$, we therefore can compute every elementary representation from
$E_{1}$ and $E_{2}$. We show that the situation is quite different for $r\geq
4$.

######  Theorem B.

Let $r\geq 4$. Then there are infinitely many, pairwise non-isomorphic
elementary representations with the same dimension vector
$(x,y)\in\mathcal{F}_{r}$ that all are in different
$\operatorname{GL}(A_{r})$-orbits.

It is well known that $\mathbbm{k}K_{s}$ is wild algebra if and only $s\geq
3$. In particular, the representation theory of $K_{3}$ is as difficult as the
representation theory of $K_{r}$ for $r\geq 4$. Moreover, in all cases, known
to the author, proofs for $K_{3}$ can be easily generalized to $K_{r}$ by
substituting $r$ for 3. However, the above theorem tells us that the problem
of classifying elementary representations gets much more difficult, when we
consider $r\geq 4$ arrows.

## Acknowledgement

I would like to thank Rolf Farnsteiner for helpful comments and suggestions on
an earlier version of this article, that helped to improve the exposition of
this article. Furthermore I thank Claus Michael Ringel for sharing his
insights on elementary Kronecker representations and answering questions
regarding the first draft of [16].

##  1\. Preliminaries

Throughout $\mathbbm{k}$ denotes an algebraically closed field of arbitrary
characteristic and all $\mathbbm{k}$-vector spaces under consideration are of
finite dimension.

###  1.1. Wild quivers

We assume that the reader is familiar with basic results on the
representations theory of wild quivers. In the following, we will give a brief
introduction, recalling the main definitions that we will use throughout this
work. For a well written survey on the subject, where all the details and
proofs may be found, we refer to [10].
Let $Q$ be a finite, connected and wild quiver without oriented cycles and
vertex set $Q_{0}=\\{1,\ldots,n\\}$. We denote by $\operatorname{rep}(Q)$ the
category of finite dimensional representations over $Q$ and let $\mathbbm{k}Q$
be the corresponding path algebra. The category $\operatorname{rep}(Q)$ and
the category of finite dimensional (left) $\mathbbm{k}Q$-modules
$\operatorname{mod}\mathbbm{k}Q$ are equivalent which allows us to switch
freely between representations and modules.
Let $M\in\operatorname{mod}\mathbbm{k}Q$. Then
$\operatorname{Ext}^{1}_{\mathbbm{k}Q}(M,\mathbbm{k}Q)$ is a right
$\mathbbm{k}Q$-module, so that
$\tau_{\mathbbm{k}Q}(M)\coloneqq\operatorname{Ext}^{1}_{Q}(M,\mathbbm{k}Q)^{\ast}$
is a left $\mathbbm{k}Q$-module. There results an endofunctor, the Auslander-
Reiten translation,

$\tau_{\mathbbm{k}Q}\colon\operatorname{mod}\mathbbm{k}Q\longrightarrow\operatorname{mod}\mathbbm{k}Q$

which is left exact, since $\mathbbm{k}Q$ is hereditary. We denote the induced
functor on $\operatorname{rep}(Q)$ by
$\tau_{Q}\colon\operatorname{rep}(Q)\longrightarrow\operatorname{rep}(Q)$.
Similarly, we obtain the functor
$\tau^{-1}_{Q}\colon\operatorname{rep}(Q)\longrightarrow\operatorname{rep}(Q)$
induced by
$\tau^{-1}_{\mathbbm{k}Q}\colon\operatorname{mod}\mathbbm{k}Q\longrightarrow\operatorname{mod}\mathbbm{k}Q;M\mapsto\operatorname{Ext}^{1}_{\mathbbm{k}Q}(M^{\ast},\mathbbm{k}Q)$.

An indecomposable representation $M\in\operatorname{rep}(Q)$ is called
preprojective (preinjective), provided $\tau^{n}_{Q}(M)=0$ (resp.
$\tau^{-n}_{Q}(M)=0$) for some $n\in\mathbb{N}$. All other indecomposable
representations are called regular. Since $Q$ is a wild quiver, and therefore
not of Dynkin type, the three classes preprojective, preinjective and regular
are mutually exclusive.
Given a representation $M\in\operatorname{rep}(Q)$, we let
$\operatorname{\underline{dim}}M\coloneqq(\dim_{\mathbbm{k}}M_{i})_{i\in
Q_{0}}\in\mathbb{Z}^{n}$ be its dimension vector. This gives rise to an
isomorphism

$\operatorname{\underline{dim}}\colon
K_{0}(\operatorname{rep}(Q))\longrightarrow\mathbb{Z}^{n},$

which identifies the Grothendieck group $K_{0}(\operatorname{rep}(Q))$ of
$\operatorname{rep}(Q)$ with $\mathbb{Z}^{n}$. Given $i\in Q_{0}$ we denote by
$S(i)$ the simple representation corresponding to $i$ and by $P(i)$ and $I(i)$
its projective cover and injective hull, respectively. The sets
$\\{\operatorname{\underline{dim}}P(i)\mid i\in\mathbb{Q}_{0}\\}$,
$\\{\operatorname{\underline{dim}}I(i)\mid i\in\mathbb{Q}_{0}\\}$ are
$\mathbb{Z}$-bases of $\mathbb{Z}^{n}$. The Coxeter transformation $\Phi_{Q}$
is the $\mathbb{Z}$-linear map
$\Phi_{Q}\colon\mathbb{Z}^{n}\longrightarrow\mathbb{Z}^{n}$ with

$\Phi_{Q}(\operatorname{\underline{dim}}P(i))=-\operatorname{\underline{dim}}I(i)$

for all $i\in Q_{0}$. We have

$\operatorname{\underline{dim}}\tau_{Q}(M)=\Phi_{Q}(\operatorname{\underline{dim}}M)\
\text{and}\
\operatorname{\underline{dim}}\tau^{-1}_{Q}(N)=\Phi^{-1}_{Q}(\operatorname{\underline{dim}}M)$

for $M,N$ indecomposable with $M\not\cong P(i),I\not\cong I(i)$ for all $i\in
Q_{0}$. An arbitary non-zero representation $M\in\operatorname{rep}(Q)$ is
called preprojective, preinjective or regular, provided all its indecomposable
direct summands are preprojective, preinjective or regular, respectively. By
definition the zero representation is preprojective, preinjective and regular.

###  1.2. Wild Kronecker quivers

We specialize our considerations to the family of wild Kronecker quivers.
Throughout we let $r\in\mathbb{N}_{\geq 3}$. The (generalized) Kronecker
quiver with $r$ arrows, denoted by $K_{r}$, is the bipartite quiver with two
vertices $1,2$ and arrows $\gamma_{i}\colon 1\longrightarrow 2$ ($1\leq i\leq
r$). A representation $M$ over $K_{r}$ is a tuple
$M=(M_{1},M_{2},(M(\gamma_{i}))_{1\leq i\leq r})$ consisting of finite
dimensional vector spaces $M_{1},M_{2}$ and $\mathbbm{k}$-linear maps
$M(\gamma_{i})\colon M_{1}\longrightarrow M_{2}$ for each $1\leq i\leq r$. A
morphism $f\colon M\longrightarrow N$ of representations is a pair
$(f_{1},f_{2})$ of $\mathbbm{k}$-linear maps such that, for each
$i\in\\{1,\ldots,r\\}$, the diagram

$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{M(\gamma_{i})}$$\scriptstyle{f_{1}}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{2}}$$\textstyle{N_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N(\gamma_{i})}$$\textstyle{N_{2}}$

commutes. The simple representations corresponding to the vertices $1$ and $2$
are denoted by $S_{1}=S(1)$ and $S_{2}=S(2)$, respectively. We let
$A_{r}\coloneqq\bigoplus^{r}_{i=1}\mathbbm{k}\gamma_{i}$ be the arrow space of
$K_{r}$ and realize the path algebra of $K_{r}$ as

$\mathbbm{k}K_{r}=\begin{pmatrix}\mathbbm{k}&0\\\
A_{r}&\mathbbm{k}\end{pmatrix}.$

We let

$\langle-,-\rangle_{r}\colon\mathbb{Z}^{2}\times\mathbb{Z}^{2}\longrightarrow\mathbb{Z},(x,y)\mapsto
x_{1}y_{1}+x_{2}y_{2}-rx_{1}y_{2}$

be the bilinear form given by $K_{r}$, with corresponding Tits quadratic form
$q_{r}\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z},x\mapsto\langle
x,x\rangle_{r}$.

###  1.3. Shift functors

We denote by
$\sigma_{K_{r}},\sigma^{-1}_{K_{r}}\colon\operatorname{rep}(K_{r})\longrightarrow\operatorname{rep}(K_{r})$
the shift functors. These functors correspond to the BGP-reflection functors
but take into account that the opposite quiver of $K_{r}$ is isomorphic to
$K_{r}$, i.e. $D_{K_{r}}\circ\sigma_{K_{r}}\cong\sigma_{K_{r}}^{-1}\circ
D_{K_{r}}$, where
$D_{K_{r}}\colon\operatorname{rep}(K_{r})\longrightarrow\operatorname{rep}(K_{r})$
denotes the standard duality.
For a representation $M\in\operatorname{rep}(K_{r})$ we consider the
$\mathbbm{k}$-linear map

$f_{M}\colon(M_{1})^{r}\longrightarrow
M_{2},(m_{i})\mapsto\sum^{r}_{i=1}M(\gamma_{i})(m_{i}).$

Then $\sigma_{K_{r}}(M)$ is by definition the representation

$(\sigma_{K_{r}}(M)_{1},\sigma_{K_{r}}(M)_{2},(\sigma_{K_{r}}(M)(\gamma_{i}))_{1\leq
i\leq r})=(\ker f_{M},M_{1},(\pi_{i}|_{\ker f_{M}})_{1\leq i\leq r}),$

where $\pi_{i}\colon(M_{1})^{r}\longrightarrow M_{1}$ is the projection onto
the $i$-th component for each $i\in\\{1,\ldots,r\\}$. Recall that
$\sigma_{K_{r}}$ induces an equivalence

$\sigma_{K_{r}}\colon\operatorname{rep}_{2}(K_{r})\longrightarrow\operatorname{rep}_{1}(K_{r})$

between the full subcategories $\operatorname{rep}_{i}(K_{r})$ of
$\operatorname{rep}(K_{r})$, whose objects do not have have any direct
summands isomorphic to $S_{i}$. By the same token, $\sigma^{-1}_{K_{r}}$ is a
quasi-inverse of $\sigma_{K_{r}}$. The map

$\sigma_{r}\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z}^{2};(x,y)\mapsto(rx-y,x)$

is invertible and satisfies

$\operatorname{\underline{dim}}\sigma_{K_{r}}(M)=\sigma_{r}(\operatorname{\underline{dim}}M)\
\text{and}\
\operatorname{\underline{dim}}\sigma^{-1}_{K_{r}}(N)=\sigma^{-1}_{r}(\operatorname{\underline{dim}}N)$

for all $M\in\operatorname{rep}_{2}(K_{r})$ and
$N\in\operatorname{rep}_{1}(K_{r})$. Moreover, we have
$\sigma_{K_{r}}\circ\sigma_{K_{r}}\cong\tau_{K_{r}}$ and
$\sigma^{2}_{r}=\Phi_{r}\coloneqq\Phi_{K_{r}}$.

###  1.4. Indecomposable representations and Kac’s Theorem

The preprojective and preinjective indecomposable Kronecker representations
are well-understood: We define $P_{0}\coloneqq S_{2}$ and
$P_{i}\coloneqq\sigma^{-1}_{K_{r}}(P_{i-1})$ for all $i\geq 1$. The
representations $P_{i}$ form a complete list of representatives of the
isomorphism classes of indecomposable preprojective Kronecker representations.
By the same token, a complete list of representatives of the isomorphism
classes of indecomposable preinjective Kronecker representations is given by
$I_{i}\coloneqq D_{K_{r}}(P_{i})$, $i\in\mathbb{N}_{0}$. Since $\sigma_{r}$
and $\sigma_{r}^{-1}$ leave the Tits form invariant and
$q_{r}(1,0)=1=q_{r}(0,1)$, this shows that
$q_{K_{r}}(\operatorname{\underline{dim}}N)=1$ for $N$ indecomposable and
preprojective or preinjective. We let
$L_{r}\coloneqq\frac{r+\sqrt{r^{2}-4}}{2}$ and note that $L_{r}$ and
$\frac{1}{L_{r}}$ are the roots of the polynomial $f_{r}\coloneqq
X^{2}-rX+1\in\mathbb{Z}[X]$. Therefore they satisfy the equation
$\frac{1}{L_{r}}=r-L_{r}$. Moreover, we have $r-1<L_{r}<r$ since $r\geq 3$. We
let
$\delta\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z}^{2};(a,b)\mapsto(b,a)$ be
twist function on $\mathbb{Z}^{2}$. Then we have

$\operatorname{\underline{dim}}P_{i}=(a_{i},a_{i+1})=\delta(\operatorname{\underline{dim}}I_{i}),\
\text{where for all}\ i\in\mathbb{N}_{0}\
a_{i}\coloneqq\frac{(L_{r})^{i}-(\frac{1}{L_{r}})^{i}}{\sqrt{r^{2}-4}}.$

We recall a simplified version of Kac’s Theorem (see [9, Thm.B] and [15,
Thm.3]) and an immediate consequence thereof that suffice for our purposes.

######  Theorem 1.1.

(Kac’s Theorem for $K_{r}$) Let $\delta\in\mathbb{N}^{2}_{0}\setminus\\{0\\}$.

  1. (1)

If $\delta=\operatorname{\underline{dim}}M$ for some indecomposable
$M\in\operatorname{rep}(K_{r})$, then $q_{r}(\delta)\leq 1$.

  2. (2)

If $q_{r}(\delta)=1$, then there is a, up to isomorphism, unique
indecomposable representation $M\in\operatorname{rep}(K_{r})$ such that
$\operatorname{\underline{dim}}M=\delta$. The representation $M$ is
preprojective or preinjective and preprojective if and only if
$\delta_{1}\leq\delta_{2}$.

  3. (3)

If $q_{r}(\delta)\leq 0$, then there are infinitely many pairwise non-
isomorphic indecomposable representations with dimension vector $\delta$, each
being regular.

######  Corollary 1.2.

Let $M\in\operatorname{rep}(K_{r})$ be indecomposable. The following
statements hold.

  1. (1)

$M$ is preprojective if and only if
$\dim_{\mathbbm{k}}M_{1}<\frac{1}{L_{r}}\dim_{\mathbbm{k}}M_{2}$.

  2. (2)

$M$ is regular if and only if
$\frac{1}{L_{r}}\dim_{\mathbbm{k}}M_{2}<\dim_{\mathbbm{k}}M_{1}<L_{r}\dim_{\mathbbm{k}}M_{2}$.

  3. (3)

$M$ is preinjective if and only if
$L_{r}\dim_{\mathbbm{k}}M_{2}<\dim_{\mathbbm{k}}M_{1}$.

##  2\. Geometric considerations and restrictions on dimension vectors

Throughout this section $d$ denotes a natural number with $1\leq d<r$. For
$(x,y)\in\mathbb{N}^{2}_{0}$ we write

$\Delta_{(x,y)}(d)\coloneqq y-dx\ \text{and}\ \nabla_{(x,y)}(d)\coloneqq
dy-x.$

For a representation $M\in\operatorname{rep}(K_{r})$, or vector spaces
$M_{1},M_{2}\in\operatorname{mod}\mathbbm{k}$, we define

$\Delta_{M}(d)\coloneqq\Delta_{(M_{1},M_{2})}(d)\coloneqq\Delta_{(\dim_{\mathbbm{k}}M_{1},\dim_{\mathbbm{k}}M_{2})}(d)\
\text{and}$
$\nabla_{M}(d)\coloneqq\nabla_{(M_{1},M_{2})}(d)\coloneqq\nabla_{(\dim_{\mathbbm{k}}M_{1},\dim_{\mathbbm{k}}M_{2})}(d).$

###  2.1. Relative projective representations and vector bundles

Let $M\in\operatorname{rep}(K_{r})$ be a representation with structure map

$\psi_{M}\colon A_{r}\otimes_{\mathbbm{k}}M_{1}\longrightarrow M_{2}\ ;\
\sum^{r}_{i=1}\gamma_{i}\otimes m\mapsto\sum^{r}_{i=1}M(\gamma_{i})(m).$

We say that $M\in\operatorname{rep}(K_{r})$ is relative $d$-projective,
provided that $\psi_{M}|_{\mathfrak{v}\otimes M_{1}}$ is injective for each
$\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$, where
$\operatorname{Gr}_{d}(A_{r})$ denotes the Grassmannian of $d$-dimensional
subspaces of $A_{r}$.

######  Remark 2.1.

The terminology "relative $d$-projective" is motivated by the fact that
$\psi_{M}|_{\mathfrak{v}\otimes M_{1}}$ is injective if and only the
restriction of the $\mathbbm{k}K_{r}$-module $M$ to the subalgebra

$\mathbbm{k}K_{d}\cong\mathbbm{k}\mathfrak{v}\coloneqq\begin{pmatrix}\mathbbm{k}&0\\\
\mathfrak{v}&\mathbbm{k}\end{pmatrix}\subseteq\mathbbm{k}K_{r}$

is projective (cf. [3, 2.1.5]).

The full subcategory of relative $d$-projective representations is denoted by
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$. This category is a
torsion-free class (see [3, 2.2.1]) closed under $\sigma_{K_{r}}^{-1}$ and
gives rise to special vector bundles (locally free coherent sheaves) on
$\operatorname{Gr}_{d}(A_{r})$. In the following we recall results and
definitions from [3] and [1].
Let $\operatorname{Vect}(\operatorname{Gr}_{d}(A_{r}))$ be the category of
vector bundles on $\operatorname{Gr}_{d}(A_{r})$ with structure sheaf
$\mathcal{O}_{\operatorname{Gr}_{d}(A_{r})}$. Moreover, let
$\mathcal{U}_{(r,d)}$ be the universal vector bundle of
$\operatorname{Gr}_{d}(A_{r})$. A locally free sheaf
$\mathcal{F}\in\operatorname{Coh}(\operatorname{Gr}_{d}(A_{r}))$ is called
Steiner bundle, provided there exist vector spaces $V_{1},V_{2}$ and a short
exact sequence

$0\longrightarrow V_{1}\otimes_{\mathbbm{k}}\mathcal{U}_{(d,r)}\longrightarrow
V_{2}\otimes_{\mathbbm{k}}\mathcal{O}_{\operatorname{Gr}_{d}(A_{r})}\longrightarrow\mathcal{F}\longrightarrow
0.$

We denote by $\operatorname{StVect}(\operatorname{Gr}(A_{r}))$ the full
subcategory of Steiner bundles on $\operatorname{Gr}_{d}(A_{r})$. The
following result is proven in [3, 2.3.2, 3.3.2, 3.3.3]. The proof of (2)
elaborates on [1, 2.4], where the result was first shown for algebraically
closed fields of characteristic zero.

######  Theorem 2.2.

The following statements hold.

  1. (1)

There exists a fully faithful and exact functor

$\widetilde{\Theta}_{d}\colon\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\longrightarrow\operatorname{Vect}(\operatorname{Gr}_{d}(A_{r}))$

with essential image $\operatorname{StVect}(\operatorname{Gr}_{d}(A_{r}))$.
Moreover, there is a short exact sequence

$0\longrightarrow M_{1}\otimes_{\mathbbm{k}}\mathcal{U}_{(r,d)}\longrightarrow
M_{2}\otimes_{\mathbbm{k}}\mathcal{O}_{\operatorname{Gr}_{d}(A_{r})}\longrightarrow\widetilde{\Theta}_{d}(M)\longrightarrow
0$

for each $M\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$.

  2. (2)

For each Steiner bundle

$0\longrightarrow V_{1}\otimes_{\mathbbm{k}}\mathcal{U}_{(r,d)}\longrightarrow
V_{2}\otimes_{\mathbbm{k}}\mathcal{O}_{\operatorname{Gr}_{d}(A_{r})}\longrightarrow\mathcal{F}\longrightarrow
0$

we have
$\operatorname{rk}(\mathcal{F})\geq\min\\{d(r-d),(\dim_{\mathbbm{k}}V_{1})(r-d)\\}$.

  3. (3)

Let $M\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$, then
$\Delta_{M}(d)\geq\min\\{d(r-d),\dim_{\mathbbm{k}}M_{1}(r-d)\\}$.

We record direct consequences of 2.2 that will be needed later on when we
study elementary representations.

######  Corollary 2.3.

Let $M\in\operatorname{rep}_{\operatorname{inj}}(K_{r},d)\coloneqq
D_{K_{r}}(\operatorname{rep}_{\operatorname{proj}}(K_{r},d))$, then

$-\nabla_{M}(d)\geq\min\\{d(r-d),\dim_{\mathbbm{k}}M_{2}(r-d)\\}.$

###### Proof.

Since $M\in\operatorname{rep}_{\operatorname{inj}}(K_{r},d)$, we have
$D_{K_{r}}(M)\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$ and
therefore

$\displaystyle-\nabla_{M}(d)$
$\displaystyle=\dim_{\mathbbm{k}}M_{1}-d\dim_{\mathbbm{k}}M_{2}=\Delta_{D_{K_{r}}(M)}(d)$
$\displaystyle\geq\min\\{d(r-d),\dim_{\mathbbm{k}}(D_{K_{r}}(M))_{1}(r-d)\\}=\min\\{d(r-d),\dim_{\mathbbm{k}}M_{2}(r-d)\\}.$

∎

######  Corollary 2.4.

The following statements hold.

  1. (1)

Let $M\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$ with
$\dim_{\mathbbm{k}}M_{1}\leq d$, then $M$ is projective.

  2. (2)

Let $M\in\operatorname{rep}_{\operatorname{inj}}(K_{r},d)$ with
$\dim_{\mathbbm{k}}M_{2}\leq d$, then $M$ is injective.

###### Proof.

  1. (1)

Let $N$ be an indecomposable direct summand of $M$ not isomorphic to $P_{0}$.
Then $N_{1}\neq 0$, $N\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$
and $\dim_{\mathbbm{k}}N_{1}\leq d$. We have

$\dim_{\mathbbm{k}}N_{2}-d\dim_{\mathbbm{k}}N_{1}=\Delta_{N}(d)\geq\min\\{d(r-d),\dim_{\mathbbm{k}}N_{1}(r-d)\\}=\dim_{\mathbbm{k}}N_{1}(r-d)$

and conclude $\dim_{\mathbbm{k}}N_{2}\geq r\dim_{\mathbbm{k}}N_{1}$. Since
$\dim_{\mathbbm{k}}N_{1}\neq 0$, we also have a projective resolution

$0\longrightarrow
P_{0}^{r\dim_{\mathbbm{k}}N_{1}-\dim_{\mathbbm{k}}N_{2}}\longrightarrow
P_{1}^{\dim_{\mathbbm{k}}N_{1}}\longrightarrow N\longrightarrow 0$

and conclude $r\dim_{\mathbbm{k}}N_{1}-\dim_{\mathbbm{k}}N_{2}=0$ as well as
$N\cong P_{1}^{\dim_{\mathbbm{k}}N_{1}}$.

  2. (2)

This follows from duality.

∎

###  2.2. The equal socle property and connections to relative projective
representations

Constant rank type modules have been defined and studied in [6] in the context
of elementary abelian $p$-groups over fields of characteristic $p$ as a
generalization of constant Jordan type modules. Inspired by these
considerations, representations with the equal socle type have been introduced
in [4] for Kronecker representations over fields of arbitrary characteristic.
It is the aim of this section characterize dimension vectors that admit
representations with the equal socle property. This description plays a
crucial role in Section 3, when we determine the dimension vectors admitting
an elementary representation.
We obtain these restrictions with the help of a recent result of Reineke in
the framework of generic representations for Kronecker representations. Let
$M\in\operatorname{rep}(K_{r})$ and
$\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$. Given
$a=\sum^{r}_{i=1}\alpha_{i}\gamma_{i}\in A_{r}$ we denote by $a_{M}\colon
M_{1}\longrightarrow M_{2}$ the $\mathbbm{k}$-linear map

$a_{M}\colon M_{1}\longrightarrow M_{2}\ ;\ m\mapsto
a.m\coloneqq\sum^{r}_{i=1}\alpha_{i}M(\gamma_{i})(m).$

######  Definition.

(cf. [4, 2.3]). A representation $M\in\operatorname{rep}(K_{r})$ has the equal
$d$-socle property, provided $\\{0\\}=\bigcap_{a\in\mathfrak{v}}\ker a_{M}$
for all $\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$.

We note that $\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$ and
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$ are closed under
subrepresentations and direct sums. Relative projective representations and
representations with the equal socle property are closely related:

######  Lemma 2.5.

Let $N\in\operatorname{rep}(K_{r})$ and $1\leq d<r$. The following statements
are equivalent.

  1. (1)

$N\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$.

  2. (2)

$\sigma_{K_{r}}(N)\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$.

###### Proof.

We define $M\coloneqq\sigma_{K_{r}}(N)$. Clearly,
$P_{0}\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$ and
$\sigma_{K_{r}}(P_{0})=\\{0\\}\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$.
Since the involved categories are closed under direct sums and summands, we
may assume that $N$ does not have $P_{0}$ as a direct summand.

(1) $\implies$ (2). We assume that
$N\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$. By definition we
find $\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$ such that

$\psi_{N}|_{\mathfrak{v}\otimes_{\mathbbm{k}}N_{1}}\colon\mathfrak{v}\otimes_{\mathbbm{k}}N_{1}\longrightarrow
N_{2};a\otimes n\mapsto a.n$

is not injective. Let $(a_{1},\ldots,a_{d})$ be a basis of $\mathfrak{v}$ and
$x=\sum^{d}_{j=1}a_{j}\otimes n_{j}$ be a non-zero element in
$\ker\psi_{N}|_{\mathfrak{v}\otimes N_{1}}$. We write
$a_{j}=\sum^{r}_{i=1}\beta_{ij}\gamma_{i}$ for $1\leq j\leq d$ and set
$n_{i}^{\prime}\coloneqq\sum^{d}_{j=1}\beta_{ij}n_{j}$ for $1\leq i\leq r$. By
definition we have

$x=\sum^{d}_{j=1}\sum^{r}_{i=1}\beta_{ij}\gamma_{i}\otimes
n_{j}=\sum^{r}_{i=1}\gamma_{i}\otimes\sum^{d}_{j=1}{\beta_{ij}n_{j}}=\sum^{r}_{i=1}\gamma_{i}\otimes
n_{i}^{\prime}.$

Recall that

$M_{2}=N_{1},M_{1}=\ker(N_{1}^{r}\longrightarrow N_{2};(y_{i})_{1\leq i\leq
r}\mapsto\sum^{r}_{i=1}\gamma_{i}.y_{i})$

and $\gamma_{j}.((y_{i})_{1\leq i\leq r})=M(\gamma_{j})((y_{i})_{1\leq i\leq
r})=y_{j}$ for $j\in\\{1,\ldots,r\\}$. We have

$0=\psi_{N}|_{\mathfrak{v}\otimes
N_{1}}(x)=\sum^{r}_{i=1}\gamma_{i}.n_{i}^{\prime},$

which shows $m\coloneqq(n_{i}^{\prime})\in M_{1}\setminus\\{0\\}$. Let
$A\coloneqq\\{\delta\in\mathbbm{k}^{r}\mid\sum^{r}_{i=1}\delta_{i}n_{i}^{\prime}=0\\}$.
Since
$\sum^{r}_{i=1}\mathbbm{k}n_{i}^{\prime}\subseteq\sum^{d}_{i=1}\mathbbm{k}n_{i}$,
we have $\dim_{\mathbbm{k}}A\geq r-d$. We fix a subspace $B\subseteq A$ of
dimension $r-d$. Let
$\mathfrak{u}\coloneqq\\{\sum^{r}_{i=1}\delta_{i}\gamma_{i}\mid\delta\in
B\\}\in\operatorname{Gr}_{r-d}(A_{r})$. Let
$a=\sum^{r}_{i=1}\delta_{i}\gamma_{i}\in\mathfrak{u}$, then
$a.m=\sum^{r}_{i=1}\delta_{i}\gamma_{i}.m=\sum^{r}_{i=1}\delta_{i}n_{i}^{\prime}=0$.
Hence $0\neq m\in\bigcap_{a\in\mathfrak{u}}\ker a_{M}$ and
$M\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$.

(2) $\implies$ (1). Assume that
$M\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$. We find
$\mathfrak{u}\in\operatorname{Gr}_{r-d}(A_{r})$ and $0\neq
m\in\bigcap_{a\in\mathfrak{u}}\ker a_{M}\setminus\\{0\\}$. By definition we
have $m=(n_{1},\ldots,n_{r})\in N_{1}^{r}\setminus\\{0\\}$ and
$0=\sum^{r}_{i=1}\gamma_{i}.n_{i}$. Let
$a=\sum^{r}_{i=1}\lambda_{i}\gamma_{i}\in\mathfrak{u}$, then
$0=a_{M}(m)=\sum^{r}_{i=1}\lambda_{i}M(\gamma_{i})(m)=\sum^{r}_{i=1}\lambda_{i}n_{i}$.
Hence $\\{\delta\in\mathbbm{k}^{r}\mid\sum^{r}_{i=1}\delta_{i}n_{i}=0\\}$ is a
vector space of dimension at least $r-d$ and $\sum^{r}_{i=1}\mathbbm{k}n_{i}$
a vector space of dimension at most $d$. Let $(x_{1},\ldots,x_{m})$ be a basis
of $\sum^{r}_{i=1}\mathbbm{k}n_{i}$. We write
$n_{i}=\sum^{m}_{j=1}\lambda_{ij}x_{j}$ for $1\leq i\leq r$ and let
$b_{l}\coloneqq\sum^{r}_{j=1}\lambda_{jl}\gamma_{j}$ for $1\leq l\leq m$. Let
$\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$ such that
$\sum^{m}_{l=1}\mathbbm{k}b_{l}\subseteq\mathfrak{v}$. We have
$0\neq\sum^{m}_{l=1}b_{l}\otimes
x_{l}\in\mathfrak{v}\otimes_{\mathbbm{k}}N_{1}$ and

$\displaystyle\psi_{N}|_{\mathfrak{v}\otimes N_{1}}(\sum^{m}_{l=1}b_{l}\otimes
x_{l})$
$\displaystyle=\sum^{m}_{l=1}(\sum^{r}_{j=1}\lambda_{jl}\gamma_{j}).x_{l}=\sum^{r}_{j=1}\gamma_{j}.\sum^{m}_{l=1}\lambda_{jl}x_{l}=\sum^{r}_{j=1}\gamma_{j}.n_{j}=0.$

Hence $N\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$. ∎

###  2.3. Generic representations and applications

Let $(V_{1},V_{2})$ be a pair vector spaces. We denote by
$\operatorname{rep}(K_{r};V_{1},V_{2})\coloneqq\operatorname{Hom}_{\mathbbm{k}}(V_{1},V_{2})^{r}$
the affine variety of representations of $K_{r}$ on $(V_{1},V_{2})$. Given
$\mathcal{S}\subseteq\operatorname{rep}(K_{r})$ and
$\mathcal{T}\subseteq\operatorname{rep}(K_{r};V_{1},V_{2})$ we define

$\mathcal{S}\cap\mathcal{T}\coloneqq\mathcal{T}\cap\mathcal{S}\coloneqq\\{g\in\mathcal{T}\mid(V_{1},V_{2},g)\in\mathcal{S}\\}\subseteq\operatorname{rep}(K_{r};V_{1},V_{2}).$

Let
$\operatorname{\mathbf{d}}\coloneqq(\dim_{\mathbbm{k}}V_{1},\dim_{\mathbbm{k}}V_{2})$.
For
$\operatorname{\mathbf{e}}\leq\operatorname{\mathbf{d}}\in\mathbb{N}^{2}_{0}$
(componentwise) we let
$\operatorname{rep}(K_{r};V_{1},V_{2})_{\operatorname{\mathbf{e}}}$ be the
Zariski-closed subset (cf. [17, 3.1]) of
$\operatorname{rep}(K_{r};V_{1},V_{2})$ consisting of all representations
admitting a subrepresentation of dimension vector $\operatorname{\mathbf{e}}$.
We write $\operatorname{\mathbf{e}}\hookrightarrow\operatorname{\mathbf{d}}$
if
$\operatorname{rep}(K_{r};V_{1},V_{2})_{\operatorname{\mathbf{e}}}=\operatorname{rep}(K_{r};V_{1},V_{2})$.
Otherwise we write
$\operatorname{\mathbf{e}}\not\hookrightarrow\operatorname{\mathbf{d}}$.
Schofield gave in [17] a criterion (in a way more general setting) in
characteristic zero to decide whether
$\operatorname{\mathbf{e}}\hookrightarrow\operatorname{\mathbf{d}}$ holds.
Crawley-Boevey extended this criterion in [5] to positive characteristic. The
statemenent in the Kronecker setting reads as follows:

######  Theorem 2.6 (Crawley-Boevey, Schofield).

We have
$\operatorname{\mathbf{e}}\operatorname{\hookrightarrow}\operatorname{\mathbf{d}}$
if and only if
$\langle\operatorname{\mathbf{f}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}\rangle_{r}\geq
0$ for all
$\operatorname{\mathbf{f}}\operatorname{\hookrightarrow}\operatorname{\mathbf{e}}$.

For imaginary roots the statement can be simplified:

######  Proposition 2.7.

(see [14, 3.4]) Assume that $q_{r}(\operatorname{\mathbf{d}})\leq 0$. The
following statements are equivalent for
$\operatorname{\mathbf{e}}\in\mathbb{N}^{2}_{0}$ with
$\operatorname{\mathbf{e}}\leq\operatorname{\mathbf{d}}$.

  1. (1)

$\operatorname{\mathbf{e}}\hookrightarrow\operatorname{\mathbf{d}}$.

  2. (2)

$\langle\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}\rangle_{r}\geq
0$.

We adapt the proof of Reineke to show:

######  Proposition 2.8.

The following statements are equivalent for
$\operatorname{\mathbf{e}}\in\mathbb{N}^{2}_{0}$ with
$\operatorname{\mathbf{e}}\leq\operatorname{\mathbf{d}}$ and
$q_{r}(\operatorname{\mathbf{e}})\leq 1$.

  1. (1)

$\operatorname{\mathbf{e}}\operatorname{\hookrightarrow}\operatorname{\mathbf{d}}$.

  2. (2)

$\langle\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}\rangle_{r}\geq
0$.

###### Proof.

(1) $\implies$ (2). Apply 2.6 for
$\operatorname{\mathbf{f}}=\operatorname{\mathbf{e}}$.

(2) $\implies$ (1). Let
$\operatorname{\mathbf{f}}\operatorname{\hookrightarrow}\operatorname{\mathbf{e}}$.
In view of 2.6 it suffices to show that
$\langle\operatorname{\mathbf{f}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}\rangle_{r}\geq
0$. Since $q_{r}(\operatorname{\mathbf{e}})\leq 1$ holds,
$\operatorname{\mathbf{e}}$ is a Schur root (see for example [3, 1.2.2]).
Hence [17, 6.1] implies
$0<\langle\operatorname{\mathbf{f}},\operatorname{\mathbf{e}}\rangle_{r}-\langle\operatorname{\mathbf{e}},\operatorname{\mathbf{f}}\rangle_{r}=r(e_{1}f_{2}-e_{2}f_{1})$.
In particular, $e_{1}\neq 0$ and $f_{2}>\frac{e_{2}f_{1}}{e_{1}}$. We conclude
with $d_{2}-e_{2}\geq 0$

$\displaystyle\langle\operatorname{\mathbf{f}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}\rangle_{r}$
$\displaystyle=f_{1}(d_{1}-e_{1}-r(d_{2}-e_{2}))+f_{2}(d_{2}-e_{2})\geq
f_{1}(d_{1}-e_{1}-r(d_{2}-f_{2}))+\frac{f_{1}e_{2}}{e_{1}}(d_{2}-e_{2})$
$\displaystyle=\frac{f_{1}}{e_{1}}\langle\operatorname{\mathbf{e}},\operatorname{\mathbf{d}}-\operatorname{\mathbf{e}}\rangle_{r}\geq
0.$

∎

In order to use Proposition 2.8, we give a characterization of
$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$ and
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$ in terms of abscence of
subrepresentations:

######  Proposition 2.9.

Let $M\in\operatorname{rep}(K_{r})$.

  1. (1)

The following statements are equivalent.

    1. (i)

$M\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$.

    2. (ii)

There exists $a\in\\{0,\ldots,r-d\\}$ and a subrepresentation $X\subseteq M$
with dimension vector $(1,a)$.

  2. (2)

The following statements are equivalent.

    1. (i)

$M\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$.

    2. (ii)

There exist $a\in\\{1,\ldots,d\\}$, $a^{\prime}\in\\{0,\ldots,ar-1\\}$ and a
subrepresentation $X\subseteq M$ with dimension vector $(a,a^{\prime})$.

###### Proof.

  1. (1)

(i) $\implies$ (ii). Let $\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$ and
$0\neq x\in\bigcap_{a\in\mathfrak{v}}\ker a_{M}$. We denote by $X$ the
representation generated by $x$. Let
$\mathfrak{u}\in\operatorname{Gr}_{r-d}(A_{r})$ such that
$\mathfrak{u}\oplus\mathfrak{v}=A_{r}$. Then
$X_{2}=\operatorname{im}\psi_{M}(\mathfrak{v}\otimes_{\mathbbm{k}}X_{1})+\operatorname{im}\psi_{M}(\mathfrak{u}\otimes_{\mathbbm{k}}X_{1})=\operatorname{im}\psi_{M}(\mathfrak{u}\otimes_{\mathbbm{k}}X_{1})$.
Since $\dim_{\mathbbm{k}}X_{1}=1$, we obtain
$\psi_{M}(\mathfrak{u}\otimes_{\mathbbm{k}}X_{1})\leq\dim_{\mathbbm{k}}\mathfrak{u}=r-d$.

(ii) $\implies$ (ii). Let $x\in X_{1}\setminus\\{0\\}$. Then $x$ generates an
indecomposable representation $\langle x\rangle\subseteq X$ with
$\operatorname{\underline{dim}}\langle x\rangle=(1,u)$ for some $0\leq u\leq
r-d$ and

$\psi_{M}|_{A_{r}\otimes_{\mathbbm{k}}\mathbbm{k}x}\colon
A_{r}\otimes_{\mathbbm{k}}\mathbbm{k}x\longrightarrow(\langle
x\rangle)_{2};a\otimes m\mapsto a.m$

is surjective. We have
$\dim_{\mathbbm{k}}\ker\psi_{M}|_{A_{r}\otimes_{\mathbbm{k}}\mathbbm{k}x}=r-u\geq
r-(r-d)=d$. Hence we find $\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$ such
that $\mathfrak{v}.x=\\{0\\}$, $0\neq x\in\bigcap_{a\in\mathfrak{v}}a_{M}$ and
$M\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$.

  2. (2)

(i) $\implies$ (ii). By definition we find
$\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$ such that
$\psi_{M}|_{\mathfrak{v}\otimes_{\mathbbm{k}}M_{1}}\colon\mathfrak{v}\otimes_{\mathbbm{k}}M_{1}\longrightarrow
M_{2}$ is not injective. We fix a basis $(v_{1},\ldots,v_{d})$ of
$\mathfrak{v}$ and an element $0\neq x=\sum^{d}_{i=1}v_{i}\otimes m_{i}$ in
the kernel of $\psi_{M}|_{\mathfrak{v}\otimes_{\mathbbm{k}}M_{1}}$. We
consider the module $X\subseteq M$ generated by $\\{m_{1},\ldots,m_{d}\\}$,
then $1\leq\dim_{\mathbbm{k}}X_{1}\leq d$. Let
$\mathfrak{u}\in\operatorname{Gr}_{r-d}(A_{r})$ be a direct complement of
$\mathfrak{v}$ in $A_{r}$. We have

$\displaystyle\dim_{\mathbbm{k}}X_{2}$
$\displaystyle\leq\dim_{\mathbbm{k}}\psi_{M}(\mathfrak{v}\otimes_{\mathbbm{k}}X_{1})+\dim_{\mathbbm{k}}\psi_{M}(\mathfrak{u}\otimes_{\mathbbm{k}}X_{1})$
$\displaystyle\leq d\dim_{\mathbbm{k}}X_{1}-1+(r-d)\dim_{\mathbbm{k}}X_{1}$
$\displaystyle=r\dim_{\mathbbm{k}}X_{1}-1.$

(ii) $\implies$ (i). We write $X=Y\oplus P_{0}^{\ell}$ such that $P_{0}$ is
not a direct summand of $Y$. Then

$\psi_{M}|_{A_{r}\otimes_{\mathbbm{k}}Y_{1}}\colon
A_{r}\otimes_{\mathbbm{k}}Y_{1}\longrightarrow Y_{2}$

is surjective. We have $\dim_{\mathbbm{k}}Y_{1}=a$,
$\dim_{\mathbbm{k}}Y_{2}\leq ar-1$ and obtain
$\dim_{\mathbbm{k}}\ker(\psi_{M}|_{A_{r}\otimes_{\mathbbm{k}}Y_{1}})\geq
ra-(ar-1)=1$. Let $(v_{1},\ldots,v_{a})$ be a basis of $Y_{1}$. We find $0\neq
x=\sum^{a}_{i=1}y_{i}\otimes
v_{i}\in\ker(\psi_{M}|_{A_{r}\otimes_{\mathbbm{k}}Y_{1}})$ and
$\mathfrak{v}\in\operatorname{Gr}_{d}(A_{r})$ containing $y_{1},\ldots,y_{a}$.
Therefore $0\neq x\in\ker\psi|_{\mathfrak{v}\otimes_{\mathbbm{k}}M_{1}}$.

∎

######  Remark 2.10.

Note that the subrepresentations $X$ in (1) and (2) are not in
$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$ and
$\operatorname{rep}_{\operatorname{proj}}(K_{r},r-d)$, respectively. In
particular, they are not preprojective.

######  Theorem 2.11.

Let $V_{1},V_{2}$ be vector spaces such that $V_{1}\oplus V_{2}\neq 0$. The
following statements hold.

  1. (1)

The set
$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$
is open in $\operatorname{rep}(K_{r};V_{1},V_{2})$.

  2. (2)

The following statements are equivalent.

    1. (i)

$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})\neq\varnothing$.

    2. (ii)

$V_{1}=0$ or $\nabla_{(V_{1},V_{2})}(d)\geq d(r-d)$.

###### Proof.

  1. (1)

By Proposition 2.9 we have

$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})=\operatorname{rep}(K_{r};V_{1},V_{2})\setminus\bigcup_{i=0}^{r-d}\operatorname{rep}(K_{r};V_{1},V_{2})_{(1,i)}.$

  2. (2)

(i) $\implies$ (ii) Assume that
$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})\neq\varnothing$.
We assume that $V_{1}\neq 0$. Then $\dim_{\mathbbm{k}}V_{2}>r-d$ by
Proposition 2.9. Another application of Proposition 2.9 implies
$(1,r-d)\not\hookrightarrow(\dim_{\mathbbm{k}}V_{1},\dim_{\mathbbm{k}}V_{2})$.
We have $q_{r}(1,r-d)\leq 1$ and conclude with Proposition 2.8

$\displaystyle 0$
$\displaystyle>\langle(1,r-d),(\dim_{\mathbbm{k}}V_{1},\dim_{\mathbbm{k}}V_{2})-(1,r-d)\rangle_{r}$
$\displaystyle=\dim_{\mathbbm{k}}V_{1}-d\dim_{\mathbbm{k}}V_{2}-(1-d(r-d))$
$\displaystyle=-\nabla_{(V_{1},V_{2})}(d)+d(r-d)-1.$

(ii) $\implies$ (i). If $V_{1}=0$, the statement is clear. Hence we assume
$\dim_{\mathbbm{k}}V_{1}\neq 0$ and $\nabla_{(V_{1},V_{2})}(d)\geq d(r-d)$. We
have $d\dim_{\mathbbm{k}}V_{2}\geq\nabla_{(V_{1},V_{2})}(d)\geq d(r-d)$ and
conclude $\dim_{\mathbbm{k}}V_{2}\geq r-d$. Hence
$(1,r-d)\leq\operatorname{\underline{dim}}(V_{1},V_{2})$ with

$\langle(1,r-d),\operatorname{\underline{dim}}(V_{1},V_{2})-(1,r-d)\rangle_{r}=-\nabla_{(V_{1},V_{2})}(d)+d(r-d)-1\leq-1.$

Since $q_{r}(1,r-d)\leq 1$, we conclude with Proposition 2.8 that
$\operatorname{rep}(K_{r};V_{1},V_{2})\setminus\operatorname{rep}(K_{r};V_{1},V_{2})_{(1,r-d)}$
is non-empty. Note that $\dim_{\mathbbm{k}}V_{2}\geq r-d$ and $V_{1}\neq 0$
imply
$\operatorname{rep}(K_{r};V_{1},V_{2})_{(1,r-d)}=\bigcup_{i=0}^{r-d}\operatorname{rep}(K_{r};V_{1},V_{2})_{(1,i)}$.
Now we apply Proposition 2.9.

∎

Recall that a representation $M\in\operatorname{rep}(K_{r})$ is called brick
if $\operatorname{End}_{K_{r}}(M)\cong\mathbbm{k}$. Clearly, bricks are
indecomposable. Given $M\in\operatorname{rep}(K_{r})$ indecomposable, Kac’s
Theorem implies $q_{r}(\operatorname{\underline{dim}}M)\leq 1$. Therefore the
following result describes all dimension vectors that can be realized by
indecomposable elements in $\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$.

######  Corollary 2.12.

Let $V_{1},V_{2}$ be a pair of vector spaces such that $V_{1}\oplus V_{2}\neq
0$ and $q_{r}(\operatorname{\underline{dim}}(V_{1},V_{2}))\leq 1$. The
following statements are equivalent.

  1. (i)

$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})\neq\varnothing$.

  2. (ii)

$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$
is a dense open subset $\operatorname{rep}(K_{r};V_{1},V_{2})$.

  3. (iii)

There exists a brick $N\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$
with dimension vector
$\operatorname{\underline{dim}}N=\operatorname{\underline{dim}}(V_{1},V_{2})$.

  4. (iv)

$\nabla_{(V_{1},V_{2})}(d)\geq d(r-d)$ or
$\operatorname{\underline{dim}}(V_{1},V_{2})=(0,1)$.

###### Proof.

(i) $\implies$ (ii). This is clear since
$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$
is open in $\operatorname{rep}(K_{r};V_{1},V_{2})$ by 2.11 and
$\operatorname{rep}(K_{r};V_{1},V_{2})$ is irreducible.
(ii) $\implies$ (iii). Since
$q_{r}(\operatorname{\underline{dim}}(V_{1},V_{2}))\leq 1$ and $V_{1}\oplus
V_{2}\neq 0$, we know from [3, 1.2.2] that the open set

$\mathcal{B}{(V_{1},V_{2})}\coloneqq\\{g\in\operatorname{rep}(K_{r};V_{1},V_{2})\mid(V_{1},V_{2},g)\
\text{is a brick}\\}$

is dense in $\operatorname{rep}(K_{r};V_{1},V_{2})$. Hence
$\mathcal{B}{(V_{1},V_{2})}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$
lies also dense $\operatorname{rep}(K_{r};V_{1},V_{2})$ and is in particular
non-empty.
(iii) $\implies$ (iv). Follows from 2.11 and the fact that a representation
with dimension vector $(0,\dim_{\mathbbm{k}}V_{2})$ is isomorphic to
$P_{0}^{\dim_{\mathbbm{k}}V_{2}}$.
(iv) $\implies$ (i) Follows from 2.11. ∎

Now we have the tools to give an alternative proof of 2.2(2).

######  Corollary 2.13.

(cf. [3, 2.3.2, 3.3.2], [1, 2.4]) Let $V_{1},V_{2}$ be vector spaces such that
$V_{1}\oplus V_{2}\neq 0$. The following statements hold.

  1. (1)

The set
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$
is open in $\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$.

  2. (2)

The following statements are equivalent.

    1. (i)

$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})\neq\varnothing$.

    2. (ii)

$\Delta_{(V_{1},V_{2})}(d)\geq\min\\{d(r-d),\dim_{\mathbbm{k}}V_{1}(r-d)\\}$.

  3. (3)

Let $\mathcal{F}$ be a Steiner bundle on $\operatorname{Gr}_{d}(A_{r})$ with
resolution

$0\longrightarrow V_{1}\otimes_{\mathbbm{k}}\mathcal{U}_{(r,d)}\longrightarrow
V_{2}\otimes_{\mathbbm{k}}\mathcal{O}_{\operatorname{Gr}_{d}(A_{r})}\longrightarrow\mathcal{F}\longrightarrow
0,$

then
$\operatorname{rk}(\mathcal{F})\geq\min\\{d(r-d),\dim_{\mathbbm{k}}V_{1}(r-d)\\}$.

###### Proof.

  1. (1)

In view of Proposition 2.9 the set

$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})=\operatorname{rep}(K_{r};V_{1},V_{2})\setminus\bigcup_{\operatorname{\mathbf{e}}\in\mathfrak{M}}\operatorname{rep}(K_{r};V_{1},V_{2})_{\operatorname{\mathbf{e}}}$

for $\mathfrak{M}:=\\{(a,a^{\prime})\mid
a\in\\{0,\ldots,d\\},a^{\prime}\in\\{0,\ldots,ad-1\\}\\}$ is open.

  2. (2)

(i) $\implies$ (ii). Let
$M\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$. We write
$M=P_{0}^{a}\oplus P_{1}^{b}\oplus N$ such that $P_{0},P_{1}\nmid N$. If
$N\neq 0$, we have
$\operatorname{\underline{dim}}\sigma_{K_{r}}(M)=\sigma_{K_{r}}(N)\oplus
P_{0}^{b}$ and 2.5 implies
$\sigma_{K_{r}}(N)\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$. We
have

$\displaystyle\operatorname{\underline{dim}}\sigma_{K_{r}}(N)$
$\displaystyle=\sigma_{r}(\dim_{\mathbbm{k}}M_{1}-b,\dim_{\mathbbm{k}}M_{2}-a-rb)$
$\displaystyle=(r(\dim_{\mathbbm{k}}M_{1}-b)-\dim_{\mathbbm{k}}M_{2}+a+rb,\dim_{\mathbbm{k}}M_{1}-b)$
$\displaystyle=(r\dim_{\mathbbm{k}}M_{1}-\dim_{\mathbbm{k}}M_{2}+a,\dim_{\mathbbm{k}}M_{1}-b).$

Since $N$ is not projective, we have $\sigma_{K_{r}}(N)_{1}\neq 0$ and
conclude with 2.11

$\displaystyle d(r-d)$ $\displaystyle\leq\nabla_{\sigma_{K_{r}}(N)}(r-d)$
$\displaystyle=(r-d)(\dim_{\mathbbm{k}}M_{1}-b)-(r\dim_{\mathbbm{k}}M_{1}-\dim_{\mathbbm{k}}M_{2}+a)$
$\displaystyle=\dim_{\mathbbm{k}}M_{2}-d\dim_{\mathbbm{k}}M_{1}-b(r-d)-a=\Delta_{M}(d)-b(r-d)-a.$

Hence

$d(r-d)\leq d(r-d)+b(r-d)+a\leq\Delta_{M}(d).$

Now assume that $N=0$, i.e. $M$ is projective. Then
$\Delta_{M}(d)=b(r-d)+a\geq b(r-d)=\dim_{\mathbbm{k}}M_{1}(r-d)$.

(ii) $\implies$ (i). At first we consider the case
$\Delta_{(V_{1},V_{2})}(d)\geq\dim_{\mathbbm{k}}V_{1}(r-d)$. Then we have
$\dim_{\mathbbm{k}}V_{2}\geq r\dim_{\mathbbm{k}}V_{1}$, i.e.
$\Delta_{(V_{1},V_{2})}(r)\geq 0$. Since
$(\dim_{\mathbbm{k}}V_{1})P_{1}\oplus\Delta_{(V_{1},V_{2})}(r)P_{0}\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$
has dimension vector $\operatorname{\underline{dim}}(V_{1},V_{2})$, we
conclude
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})\neq\varnothing$.

Now we consider the case $\Delta_{(V_{1},V_{2})}(d)\geq d(r-d)$. By the first
case, we may assume that $\dim_{\mathbbm{k}}V_{2}<r\dim_{\mathbbm{k}}V_{1}$
holds. We consider
$(r\dim_{\mathbbm{k}}V_{1}-\dim_{\mathbbm{k}}V_{2},\dim_{\mathbbm{k}}V_{1})\in\mathbb{N}_{0}^{2}\setminus\\{(0,0)\\}$.
Then
$\nabla_{(r\dim_{\mathbbm{k}}V_{1}-\dim_{\mathbbm{k}}V_{2},\dim_{\mathbbm{k}}V_{1})}(r-d)=(r-d)\dim_{\mathbbm{k}}V_{1}-(r\dim_{\mathbbm{k}}V_{1}-\dim_{\mathbbm{k}}V_{2})=\Delta_{(V_{1},V_{2})}(d)\geq
d(r-d)$. We apply 2.11 and find
$M\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$ with dimension vector
$(r\dim_{\mathbbm{k}}V_{1}-\dim_{\mathbbm{k}}V_{2},\dim_{\mathbbm{k}}V_{1})$.
Since $\operatorname{rep}_{\operatorname{esp}}(K_{r},r-d)$ does not contain
$I_{0}=S_{1}$, we conclude
$\operatorname{\underline{dim}}\sigma_{K_{r}}^{-1}(M)=\operatorname{\underline{dim}}(V_{1},V_{2})$
and 2.5 implies
$\sigma_{K_{r}}^{-1}(M)\in\operatorname{rep}_{\operatorname{proj}}(K_{r},d)$.

  3. (3)

This follows from 2.2(1) in conjunction with (2).

∎

We record two more consequences that we will need in the next section for the
study of elementary representations.

######  Corollary 2.14.

Let $V_{1},V_{2}$ be vector spaces such that $V_{1}\oplus V_{2}\neq 0$. The
set
$\operatorname{rep}_{\operatorname{inj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$
is open in $\operatorname{rep}(K_{r};V_{1},V_{2})$ and non-empty if
$-\nabla_{(V_{1},V_{2})}(d)\geq d(r-d)$.

###### Proof.

Note that
$\operatorname{rep}_{\operatorname{inj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$
is open in $\operatorname{rep}(K_{r};V_{1},V_{2})$, since the duality
$D_{K_{r}}\colon\operatorname{rep}(K_{r})\longrightarrow\operatorname{rep}(K_{r})$
induces an isomorphism of varieties

$\operatorname{rep}(K_{r};V_{2},V_{1})\longrightarrow\operatorname{rep}(K_{r};V_{1}^{\ast},V_{2}^{\ast})\cong\operatorname{rep}(K_{r};V_{1},V_{2})$

that takes
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{2},V_{1})$
to
$\operatorname{rep}_{\operatorname{inj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$.
If $-\nabla_{(V_{1},V_{2})}(d)\geq d(r-d)$, we
$\Delta_{(V_{2},V_{1})}(d)=\dim_{\mathbbm{k}}V_{1}-d\dim_{\mathbbm{k}}V_{2}=-\nabla_{(V_{1},V_{2})}(d)\geq
d(r-d)$. Hence Corollary 2.13 implies that
$\operatorname{rep}_{\operatorname{proj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{2},V_{1})$
is non-empty. By duality

$\operatorname{rep}_{\operatorname{inj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1}^{\ast},V_{2}^{\ast})\cong\operatorname{rep}_{\operatorname{inj}}(K_{r},d)\cap\operatorname{rep}(K_{r};V_{1},V_{2})$

is non-empty. ∎

######  Corollary 2.15.

Let $M\in\operatorname{rep}(K_{r})$ be a representation with
$(1,r-d)\leq\operatorname{\underline{dim}}(M_{1},M_{2})$. We assume that one
of the following conditions holds:

  1. (i)

$\nabla_{M}(d)<d(r-d)$, or

  2. (ii)

$M\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$.

Then there exists a non-preprojective subpresentation $U_{r-d}$ of $M$ with
dimension vector $(1,r-d)$.

###### Proof.

In case (i) we conclude with $\nabla_{M}(d)<d(r-d)$ and 2.11 that
$M\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$, since
$M_{1}\neq\\{0\\}$. In case (ii) we apply Proposition 2.9 and find a
subrepresentation $U\subseteq M$ with dimension vector $(1,a)$ for some
$a\in\\{0,\ldots,r-d\\}$. Since $\dim_{\mathbbm{k}}M_{2}\geq r-d$, we can
extend $U$ to a subrepresentation $U_{r-d}$ with dimension vector $(1,r-d)$.
The only preprojective indecomposable representation $U$ with dimension vector
$\operatorname{\underline{dim}}U\leq(1,r-d)$ is $U=P_{0}$ with dimension
vector $\dim P_{0}=(0,1)$. Hence $U_{r-d}$ is not preprojective. ∎

##  3\. Elementary representations

###  3.1. General results

Let $Q$ be a connected and wild quiver.

######  Definition.

A non-zero regular representation $E\in\operatorname{rep}(Q)$ is called
elementary, provided there is no short exact sequence

$0\longrightarrow A\longrightarrow E\longrightarrow B\longrightarrow 0$

with $A,B\in\operatorname{rep}(Q)$ regular and non-zero.

By definition the elementary representations are the simple objects in the
full subcategory of regular representations and the analogue of quasi-simple
regular representations in the context of tame quivers. Elementary
representations for wild quivers were first systematically studied in [11] and
[12]. There, the authors showed that, parallel to the tame situation, there
exist only finitely many Coxeter-orbits of dimension vectors of elementary
representations. A very useful characterization of elementary representations,
established more recently in [16, Appendix A], is the following:

######  Proposition 3.1.

Let $E\in\operatorname{rep}(Q)$ be a non-zero regular representation. The
following statements are equivalent.

  1. (1)

$E$ is elementary.

  2. (2)

Given any subrepresentation $U$ of $E$, $U$ is preprojective or the quotient
$E/U$ is preinjective.

Now we return to the case $Q=K_{r}$ for $r\geq 3$. It is well known (see for
example [15, 3.4]) that the region

$\mathcal{C}_{r}\coloneqq\\{(x,y)\in\mathbb{N}^{2}\mid\frac{1}{r-1}x\leq
y<(r-1)x\\}$

is a fundamental domain for the action of the Coxeter transformation
$\Phi_{r}=\begin{pmatrix}r^{2}-1&-r\\\
r&-1\end{pmatrix}\in\operatorname{GL}_{2}(\mathbb{Z})$111We identify
$\Phi_{K_{r}}\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z}^{2}$ and the
Coxeter-matrix $\Phi_{r}$ with its natural action on $\mathbb{Z}^{2}$ by left
multiplication. on the set

$\mathcal{R}_{r}\coloneqq\\{(x,y)\in\mathbb{N}^{2}\mid x^{2}+y^{2}-rxy<1\\}$

of dimension vectors of regular representations in
$\operatorname{rep}(K_{r})$. Ultimately, we are interested in the set

$\widetilde{\mathcal{E}}_{r}\coloneqq\\{(x,y)\in\mathcal{C}_{r}\mid\exists
E\in\operatorname{rep}(K_{r})\
\text{elementary},\operatorname{\underline{dim}}E=(x,y)\\}.$

By [16, Section 2] the set

$\mathcal{F}_{r}\coloneqq\\{(x,y)\in\mathbb{N}^{2}\mid\frac{2}{r}x\leq y\leq
x\\}\subseteq\mathcal{C}_{r}$

is a fundamental domain for the action of the group
$G_{r}\subseteq\operatorname{GL}_{2}(\mathbb{Z})$ generated by $\sigma_{r}$
and the twist function
$\delta\colon\mathbb{Z}^{2}\longrightarrow\mathbb{Z}^{2};(x,y)\mapsto(y,x)$ on
$\mathcal{R}$. In fact, the statement was only proven for $r=3$ but the
arguments extend to the general case.

$\mathcal{C}_{3}$$\mathcal{F}_{3}$ Figure 1. Illustration of $\mathcal{C}_{3}$
and $\mathcal{F}_{3}$.

We define

$\mathcal{E}_{r}\coloneqq\widetilde{\mathcal{E}}_{r}\cap\mathcal{F}_{r}=\\{(x,y)\in\mathcal{F}_{r}\mid\exists
E\in\operatorname{rep}(K_{r})\
\text{elementary},\operatorname{\underline{dim}}E=(x,y)\\}.$

Given $M\in\operatorname{rep}(K_{r})$ regular indecomposable, we have
$\operatorname{\underline{dim}}\sigma_{K_{r}}(M)=\sigma_{r}(\operatorname{\underline{dim}}M)$
and
$\operatorname{\underline{dim}}D_{K_{r}}(M)=\delta(\operatorname{\underline{dim}}M)$.
Since $M$ is elementary if and only if its dual (respectively its
$\sigma_{K_{r}}$-shift$)$ is elementary and
$\sigma_{r}\circ\sigma_{r}=\Phi_{r}$, the determination of $\mathcal{E}_{r}$
only necessitates the knowledge of $\widetilde{\mathcal{E}}_{r}$. The set
$\mathcal{E}_{3}$ has been determined in [16] and is given by

$\mathcal{E}_{3}=\\{(1,1),(2,2)\\}.$

In the following we determine the set $\mathcal{E}_{r}$ for arbitrary $r\geq
3$. We start our considerations with the following simple observations, that
will be needed later on.

######  Lemma 3.2.

Let $M\in\operatorname{rep}(K_{r})$ and $\dim_{\mathbbm{k}}M_{2}\leq 2(r-1)$.

  1. (1)

If $M$ is preinjective, then $M\in\operatorname{add}(I_{0}\oplus I_{1}\oplus
I_{2})$.

  2. (2)

If $\operatorname{\underline{dim}}M\in\mathcal{F}_{r}$ and $U\subseteq M$ such
$M/U$ is preinjective, then $M/U\in\operatorname{add}(I_{0}\oplus I_{1})$ and
$M/U\cong-\nabla_{M/U}(r)I_{0}\oplus\dim_{\mathbbm{k}}(M/U)_{1}I_{1}$.

  3. (3)

If $\operatorname{\underline{dim}}M\in\mathcal{F}_{r}$ and $U\subseteq M$ such
that

$r(\dim_{\mathbbm{k}}(M/U)_{2})>\dim_{\mathbbm{k}}(M/U)_{1},$

then $M/U$ is not preinjective.

  4. (4)

A representation $N\in\operatorname{rep}(K_{r})$ with
$\dim_{\mathbbm{k}}N_{1}<L_{r}\dim_{\mathbbm{k}}N_{2}$ is not preinjective.

  5. (5)

A representation $N\in\operatorname{rep}(K_{r})$ with
$2\leq\dim_{\mathbbm{k}}N_{2}$ and $\dim_{\mathbbm{k}}N_{1}<2r$ is not
preinjective.

###### Proof.

  1. (1)

We have
$\dim_{\mathbbm{k}}(I_{l})_{2}\geq\dim_{\mathbbm{k}}(I_{3})_{2}=r^{2}-1$ for
all $l\geq 3$. Moreover, we have $r^{2}-1>2(r-1)\geq\dim_{\mathbbm{k}}M_{2}$
since $r\geq 2$.

  2. (2)

We apply (1) to $M/U$ and know that $M/U\in\operatorname{add}(I_{0}\oplus
I_{1}\oplus I_{2})$. Moreover, we have

$\dim_{\mathbbm{k}}(M/U)_{1}\leq\dim_{\mathbbm{k}}M_{1}\leq\frac{r}{2}\dim_{\mathbbm{k}}M_{2}\leq\frac{r}{2}2(r-1)=r^{2}-r<r^{2}-1=\dim_{\mathbbm{k}}(I_{2})_{1}.$

Hence $M/U\in\operatorname{add}(I_{0}\oplus I_{1})$ and therefore
$M/U\cong-\nabla_{M/U}(r)I_{0}\oplus\dim_{\mathbbm{k}}(M/U)_{2}I_{1}$.

  3. (3)

This is a direct consequence of (2) since
$\operatorname{\underline{dim}}I_{1}=(r,1)$ and
$\operatorname{\underline{dim}}I_{0}=(1,0)$.

  4. (4)

This follows immediately by applying Corollary 1.2 to the direct summands of
$N$.

  5. (5)

We have
$\dim_{\mathbbm{k}}(I_{l})_{1}\geq\dim_{\mathbbm{k}}(I_{2})_{1}=r^{2}-1>2r$
for all $l\geq 2$. Assume that $N$ is preinjective. Then
$N\in\operatorname{add}(I_{0}\oplus I_{1})$. Since
$\operatorname{\underline{dim}}I_{0}=(1,0)$ and
$\operatorname{\underline{dim}}I_{1}=(r,1)$, we conclude with
$\dim_{\mathbbm{k}}N_{2}\geq 2$ that $\dim_{\mathbbm{k}}N_{1}\geq 2r$, a
contradiction.

∎

###  3.2. Restricting y

######  Proposition 3.3.

Let $(x,y)\in\mathcal{F}_{r}$ with $y\geq r$ and $E$ be a representation with
dimension vector $\operatorname{\underline{dim}}E=(x,y)$. Then $E$ is not
elementary.

###### Proof.

Since elementary representations are bricks (see [11, 1.4]), we can assume
that $E$ is a regular indecomposable representation. From now on we proceed in
steps. Since
$\nabla_{E}(1)=\dim_{\mathbbm{k}}E_{2}-\dim_{\mathbbm{k}}E_{1}\leq 0<1(r-1)$
and $\dim_{\mathbbm{k}}E_{2}=y\geq r-1$, we can apply Corollary 2.15 and find
a non-preprojective subrepresentation $U_{r-1}\subseteq E$ with dimension
vector $(1,r-1)$.

At first we assume that $y\geq 2(r-1)$. Then quotient $E/U_{r-1}$ has
dimension vector $(x-1,y-r+1)$. We claim that this dimension vector can not
belong to a preinjective representation. Indeed, since $r-1<L_{r}$, we have
$r-2L_{r}<0$ and conclude

$\displaystyle(x-1)-(y-r+1)L_{r}$ $\displaystyle\stackrel{{\scriptstyle
x\leq\frac{r}{2}y}}{{\leq}}(\frac{r}{2}-L_{r})y+L_{r}(r-1)-1=(\frac{r-2L_{r}}{2})y+L_{r}(r-1)-1$
$\displaystyle\stackrel{{\scriptstyle y\geq
2(r-1)}}{{\leq}}(\frac{r-2L_{r}}{2})2(r-1)+L_{r}(r-1)-1$
$\displaystyle=(r-2L_{r})(r-1)+L_{r}(r-1)-1=(r-1)(r-L_{r})-1.$

Recall that $L_{r}$ is a root of the polynomial
$f=X^{2}-rX+1\in\mathbb{R}[X]$. Hence

$(x-1)-(y-r+1)L_{r}\leq(r-L_{r})(r-1)-1=(r-L_{r})(r-1)+L_{r}(L_{r}-r)=(r-L_{r})(r-1-L_{r})<0,$

since $r-1<L_{r}<r$. Now 3.2(4) implies that $E/U_{r-1}$ is not preinjective.
We conclude with Proposition 3.1 that $E$ is not elementary.

Therefore we can assume from now on that $r\leq y<2(r-1)$. Given
$d\in\mathbb{N}$ we define

$\nabla(d):=r(r-d).$

We begin with the case $\nabla(1)\leq ry-x$. We have
$\operatorname{\underline{dim}}E/U_{r-1}=(x-1,y-(r-1))$ and $y-(r-1)\neq 0$.
Therefore

$r\dim_{\mathbbm{k}}(E/U_{r-1})_{2}=r(y-(r-1))=ry-\nabla(1)\geq
x>x-1=\dim_{\mathbbm{k}}(E/U_{r-1})_{1}.$

Since $\dim_{\mathbbm{k}}E_{2}=y<2(r-1)$, we can apply 3.2(3) and conclude
that $E/U_{r-1}$ is not preinjective. Now Proposition 3.1 implies that $E$ is
not elementary.

Now we assume that $ry-x<\nabla(1)$. Since $ry-x\geq r\frac{2}{r}x-x=x\geq
y\geq r=\nabla(r-1)$, we find a natural number $2\leq d\leq r-1$ such that

$\nabla(d)\leq ry-x<\nabla(d-1).$

We consider two cases:

  * •

$d\in\\{2,\ldots,r-2\\}$, then $d-(r-1)\leq-1$ and

$\displaystyle\nabla_{(x,y)}(d)-d(r-d)$
$\displaystyle=\nabla_{(x,y)}(r)-y(r-d)-d(r-d)<\nabla(d-1)-(y+d)(r-d)$
$\displaystyle\leq\nabla(d-1)-(r+d)(r-d)=r(r-d+1)-r^{2}+d^{2}$
$\displaystyle=-rd+d^{2}+r=r+d(d-r)$
$\displaystyle=(d-1)(d-(r-1))+1\leq(d-1)\cdot(-1)+1\leq-1+1=0,$

By Corollary 2.15 there is a subrepresentation $U_{r-d}\subseteq E$ with
dimension vector $(1,r-d)$ that is not preprojective. We have
$\operatorname{\underline{dim}}E/U_{r-d}=(x-1,y-(r-d))$ and the choice of $d$
gives us

$r\dim_{\mathbbm{k}}(E/U_{r-d})_{2}=r(y-(r-d))=ry-\nabla(d)\geq
x>x-1=\dim_{\mathbbm{k}}(E/U_{r-d})_{1}.$

Therefore $E/U_{r-d}$ is not preinjective by 3.2(3) and Proposition 3.1
implies that $E$ is not elementary.

  * •

$d=r-1$, i.e. $\nabla(r-1)\leq ry-x<\nabla(r-2)$. We get

$x>r(y-2)\geq r(\frac{2}{r}x-2)=2x-2r\Leftrightarrow x<2r$

and conclude with $r(y-2)<x<2r$ that $y<4$. Since $3\leq r\leq y<4$, we
conclude $r=3$. Hence the statement follows since
$\mathcal{E}_{r}=\\{(1,1),(2,2)\\}$ by [16].

∎

###  3.3. Existence of elementary representations

For $x,y\in\mathbb{N}_{0}$, we define

$\mathcal{E}(x,y)\coloneqq\\{g\in\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})\mid(\mathbbm{k}^{x},\mathbbm{k}^{y},g)\
\text{elementary}\\},$

and

$\mathcal{B}(x,y)\coloneqq\\{g\in\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})\mid(\mathbbm{k}^{x},\mathbbm{k}^{y},g)\
\text{brick}\\}.$

Since elementary representations are bricks (see [11, 1.4]) we have
$\mathcal{E}(x,y)\subseteq\mathcal{B}(x,y)$.

We assume from now on that $(x,y)\in\mathcal{F}_{r}$. We recall from [3,
1.2.2] that $\mathcal{B}(x,y)$ is a dense subset of
$\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})$ since
$q_{r}(x,y)\leq 1$.
In following we determine under which assumptions on $(x,y)$ the set
$\mathcal{E}(x,y)$ is non-empty. Since Proposition 3.3 implies that
$\mathcal{E}(x,y)\neq\varnothing$ can only happen for $y<r$, we assume from
now on that $y<r$. Before we tackle the general case, we consider an example
that illustrates the strategy of proof.

######  Example.

We have $(6,3)\in\mathcal{F}_{4}$ with $3<4=r$ and claim that
$(6,3)\in\mathcal{E}_{4}$. We have $\nabla_{(6,3)}(3)=3\geq 3=3(4-3)$.
Moreover, we have $-\nabla_{(6,3)}(1)=3\geq 1(4-1)$. Hence 2.11 and Corollary
2.14 imply that
$\operatorname{rep}_{\operatorname{esp}}(K_{r},3)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},1)\cap\mathcal{B}(6,3)$
is non-empty. We fix a representation $E$ in the above set. Let $0\neq
U\subseteq E$ a non-preprojective representation. We now show that $E/U$ is
preinjective. Since $U$ is not projective, we find $0\neq u\in U_{1}$. We
consider the subrepresentation $\langle u\rangle$ generated by $u$. Then
$\operatorname{\underline{dim}}\langle u\rangle=(1,z)$ for some
$z\in\\{0,1,2,3\\}$. Since $\operatorname{rep}_{\operatorname{esp}}(K_{r},3)$
is closed under subrepresentations, we have $\langle
u\rangle\in\operatorname{rep}_{\operatorname{esp}}(K_{r},3)$ and conclude with
2.11 that $3z-1=\nabla_{(1,z)}(3)\geq 3(4-3)=3$. Hence $z\geq 2$. Therefore
$\operatorname{\underline{dim}}E/\langle u\rangle=(5,b)$ with $b\in\\{0,1\\}$.
Since $E\in\operatorname{rep}_{\operatorname{inj}}(K_{r},1)$ and
$\operatorname{rep}_{\operatorname{inj}}(K_{r},1)$ is closed under images
(since $\operatorname{rep}_{\operatorname{proj}}(K_{r},1)$ is closed under
subrepresentation), we have $E/\langle
u\rangle\in\operatorname{rep}_{\operatorname{inj}}(K_{r},1)$. Now we apply
Corollary 2.4 to conclude that $E/\langle u\rangle$ is preinjective. Finally,
the presence of the canonical epimorphism $E/\langle u\rangle\longrightarrow
E/U$ implies that $E/U$ is injective.

Now we consider the general case and start with the following simple
observation.

######  Lemma 3.4.

Let $y=1$, then $\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\neq\varnothing$.

###### Proof.

Let $M\in\mathcal{B}(x,y)$, then $M$ is indecomposable and regular. Let
$U\subseteq M$ be a proper subrepresentation. Then
$0\neq\dim_{\mathbbm{k}}U_{2}$ and therefore $M/U\in\operatorname{add}(I_{0})$
is injective. In particular, $M$ is elementary. This shows
$\varnothing\neq\mathcal{B}(x,y)=\mathcal{E}(x,y)$. ∎

We assume from now on that $1<y<r$ and set
$b\coloneqq\lceil\frac{x}{r}\rceil\in\mathbb{N}$ which is the uniquely
determined natural number such that

$(b-1)r<x\leq br.$

######  Remark 3.5.

.

  1. (1)

We have $1\leq b<y<r$: Assume that $\lceil\frac{x}{r}\rceil=b\geq y$. Then
$\frac{x}{r}>y-1$ and therefore

$\frac{r}{2}y\geq x=\frac{x}{r}r>ry-r.$

Hence $2>y$, a contradiction since we assume $2\leq y$.

  2. (2)

We extend to definition of $\operatorname{rep}_{\operatorname{inj}}(K_{r},d)$
to $d\in\\{0,\ldots,r-1\\}$ be setting
$\operatorname{rep}_{\operatorname{inj}}(K_{r},0)\coloneqq\operatorname{rep}(K_{r})$.

######  Proposition 3.6.

Let $(x,y)\in\mathcal{F}_{r}$ with $1<y<r$ and
$b\coloneqq\lceil\frac{x}{r}\rceil$. The following statements hold.

  1. (1)

If $\mathcal{E}(x,y)$ is non-empty, then

$(b-1)(y+r-(b-1))\leq x\leq b(r-y+b)$

and
$\mathcal{E}(x,y)\subseteq\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+b)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$.

  2. (2)

If

$(b-1)(y+r-(b-1))\leq x\leq b(r-y+b),$

then $\mathcal{E}(x,y)$ is a non-empty open set given by

$\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+b)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1).$

###### Proof.

  1. (1)

Let $E$ be an elementary representation with dimension vector
$\operatorname{\underline{dim}}E=(x,y)$. We denote by $F\coloneqq
D_{K_{r}}(E)$ the dual representation with dimension vector $(y,x)$. We
proceed in steps.

    1. (i)

We have $(b-1)(y+r-(b-1))\leq x$ and
$E\in\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$: We assume that
$x<(b-1)(y+r-(b-1))$ or
$E\not\in\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$. In both cases we
conclude $b\neq 1$ and therefore $b-1\in\\{1,\ldots,r-1\\}$. If
$x<(b-1)(y+r-(b-1))$, we have

$\Delta_{F}(b-1)=x-(b-1)y<(b-1)(r-(b-1))\ \text{and}\ x\leq\frac{r}{2}y<ry\
\text{gives}$ $\displaystyle\Delta_{F}(b-1)$
$\displaystyle=x-(b-1)y<ry-(b-1)y=y(r-(b-1))$
$\displaystyle=\dim_{\mathbbm{k}}F_{1}(r-(b-1)).$

Hence 2.2(3) implies
$F\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},b-1)$. If
$E\not\in\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$, we immediately
get $F\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},b-1)$ from the
definition.

The book-keeping: In both cases we arrive at
$F\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},b-1)$ with $b-1\neq
0$.

In view of Proposition 2.9 we find $a\in\\{1,\ldots,b-1\\}$ and
subrepresentation
$Y\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},b-1)$ of $F$ with
$\operatorname{\underline{dim}}Y=(a,a^{\prime})$ and $a^{\prime}\leq
ar-1\leq(b-1)r-1$. The inequality $(b-1)r-1<x$ ensures that we can extend $Y$
with a semisimple projective direct summand to a subrepresentation $Y$ of $F$
with dimension vector $(a,(b-1)r-1)$ that satisfies
$Y\not\in\operatorname{rep}_{\operatorname{proj}}(K_{r},b-1)$. In particular,
$Y$ is not preprojective by Proposition 2.9.

Since $F$ is elementary, we can apply Proposition 3.1 to conclude that
$(y-a,x-(b-1)r+1)=\operatorname{\underline{dim}}F/X$ belongs to a preinjective
representation. But this is impossible since $x-(b-1)r+1\geq 1$, $y-a<r$ and
$\dim_{\mathbbm{k}}(I_{l})_{1}\geq r$ for all $l\geq 1$ and
$\operatorname{\underline{dim}}I_{0}=(1,0)$. Hence $(b-1)(y+r-(b-1))\leq x$
and $E\in\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$.

    2. (ii)

We have $x\leq b(r-y+b)$ and
$E\in\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+b)$: We assume that
$x>b(r-y+b)$. We set $d\coloneqq r-(y-b)$ and note that
$d\in\\{1,\ldots,r-1\\}$ by 3.5. We get

$\displaystyle\nabla_{E}(d)-d(r-d)$ $\displaystyle=d(y-r+d)-x$
$\displaystyle=b(r-y+b)-x<0.$

Hence $E\not\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$ by 2.11.
Since $r-d=y-b\leq y$, we conclude with Corollary 2.15 that there exists a
non-preprojective subrepresentation $U_{r-d}\subseteq E$ with dimension vector
$(1,r-d)$. Once again we apply Proposition 3.1 and conclude that $E/U_{r-d}$
with dimension vector $(x-1,y-(r-d))=(x-1,b)$ is preinjective. We apply 3.2(3)
and conclude $br\leq x-1$. But this is a contradiction to the definition of
$b$ since $x\leq br$.
We note that this also shows
$E\in\operatorname{rep}_{\operatorname{esp}}(K_{r},d)=\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+b)$.

  2. (2)

We set $d\coloneqq r-(y-b)\in\\{1,\ldots,r-1\\}$ and have

$\nabla_{E}(d)-d(r-d)=d(y-r+d)-x=b(r-y+b)-x\geq 0,$

and

$\quad-\nabla_{E}(b-1)=x-(b-1)y\geq(b-1)(r-(b-1)).$

We can apply 2.11 and Corollary 2.14 (for $b\neq 1$) to conclude that
$\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},d)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$
is non-empty (for $b=1$ we have
$\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)=\operatorname{rep}(K_{r})$).
We fix a representation $E$ in this space and show now that $E$ is elementary.
Let $U\subseteq E$ be a non-preprojective representation, then we find $u\in
U\setminus\\{0\\}$. Recall from Section 2.2 that
$\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$ is closed under
subrepresentation. Therefore the subrepresentation $\langle u\rangle$
generated by $u$ is in $\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$ and
$\operatorname{\underline{dim}}\langle u\rangle=(1,z)$ for some
$z\in\\{1,\ldots,y\\}$. We conclude with 2.11 that

$dz-1=\nabla_{\langle u\rangle}(d)\geq d(r-d)\Leftrightarrow d(z-(r-d))\geq
1.$

In particular, $z\geq r-d+1=y-(b-1)$. In other words, $E/\langle u\rangle$
satisfies $\operatorname{\underline{dim}}E/\langle u\rangle=(x-1,a)$ with
$0\leq a\leq b-1$. If $b=1$, we conclude that $a=0$ and therefore
$\operatorname{\underline{dim}}E/\langle u\rangle$ is injective and the the
presence of the canonical epimorphism $E/\langle u\rangle\longrightarrow E/U$
implies that $E/U$ is injective. If $b\neq 1$ we have
$E\in\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$ with $b-1\neq 0$.
Since $\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$ is closed under
images (since $\operatorname{rep}_{\operatorname{proj}}(K_{r},b-1)$ is closed
under subrepresentation), we have $\operatorname{\underline{dim}}E/\langle
u\rangle\in\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)$ and can apply
Corollary 2.4 to conclude that $E/\langle u\rangle$ is injective and presence
of the canonical epimorphism $E/\langle u\rangle\longrightarrow E/U$ implies
that $E/U$ is preinjective. Hence $E$ is elementary by Proposition 3.1.

The book-keeping: We have shown that

$\varnothing\neq\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+b)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},b-1)\subseteq\mathcal{E}(x,y).$

Now we apply (1) to finish the proof.

∎

######  Remark 3.7.

We extend the definition of $\operatorname{rep}_{\operatorname{esp}}(K_{r},d)$
to $\\{1,\ldots,r\\}$ by setting
$\operatorname{rep}_{\operatorname{esp}}(K_{r},r)\coloneqq\operatorname{rep}(K_{r})$.

######  Theorem 3.8.

Let $(x,y)\in\mathcal{F}_{r}$.

  1. (1)

$\mathcal{E}(x,y)\neq\varnothing$ implies $y<r$.

  2. (2)

For $y<r$ the following statements are equivalent.

    1. (i)

$\mathcal{E}(x,y)\neq\varnothing$.

    2. (ii)

$(\lceil\frac{x}{r}\rceil-1)(y+r-(\lceil\frac{x}{r}\rceil-1))\leq
x\leq\lceil\frac{x}{r}\rceil(r-y+\lceil\frac{x}{r}\rceil)$.

If one the equivalent statements holds, we have

$\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+\lceil\frac{x}{r}\rceil)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},\lceil\frac{x}{r}\rceil-1).$

###### Proof.

  1. (1)

This is the statement of Proposition 3.3.

  2. (2)

At first we assume that $y=1$. Then $x\leq\frac{r}{2}y<r$ and
$\lceil\frac{x}{r}\rceil=1$. So in this case the inequalities in (ii) are
always satisfied and by 3.4 we have
$\mathcal{E}(x,y)=\mathcal{B}(x,y)\neq\varnothing$ as well as

$\displaystyle\mathcal{E}(x,y)$
$\displaystyle=\mathcal{B}(x,y)=\mathcal{B}(x,y)\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},0)$
$\displaystyle=\mathcal{B}(x,y)\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+\lceil\frac{x}{r}\rceil)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},\lceil\frac{x}{r}\rceil-1).$

Now we assume that $1<y<r$. Then the equivalence of (i) and (ii) is precisely
the statement of Proposition 3.6.

∎

######  Corollary 3.9.

Let $(x,y)\in\mathcal{F}_{r}$ such that $y\leq x<r$. The following statements
are equivalent:

  1. (1)

$\mathcal{E}(x,y)\neq\varnothing$.

  2. (2)

$x+y\leq r+1$.

In this case we have

$\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+1).$

###### Proof.

We have $\lceil\frac{x}{r}\rceil=1$. Hence $(x,y)\in\mathcal{F}$ satisfies the
inequality of the above Theorem if and only if $x\leq r-y+1$. Moreover, we
have in this case

$\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+1)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},0)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+1).$

∎

######  Corollary 3.10.

Let $(x,y)\in\mathcal{F}_{r}$. The following statements are equivalent.

  1. (1)

$\mathcal{E}(x,y)\neq\varnothing$.

  2. (2)

$\lfloor\frac{x}{r}\rfloor(y+r-\lfloor\frac{x}{r}\rfloor)\leq
x\leq\lceil\frac{x}{r}\rceil(r-y+\lceil\frac{x}{r}\rceil)$ and $y<r$.

  3. (3)

$y\leq\min\\{\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r,\lceil\frac{x}{r}\rceil-\frac{x}{\lceil\frac{x}{r}\rceil}+r,r-1\\}$,
where we interpret
$\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r$ as $\infty$
for $1\leq x<r$.

If one of the equivalent statements holds, we have

$\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+\lceil\frac{x}{r}\rceil)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},\lfloor\frac{x}{r}\rfloor).$

###### Proof.

Assume that $\frac{x}{r}\in\mathbb{N}$. In this case we have
$\frac{x}{r}=\lceil\frac{x}{r}\rceil$ and

$x<ry\Leftrightarrow x>x-\frac{x}{r}y+(\frac{x}{r})^{2}\Leftrightarrow
x>\lceil\frac{x}{r}\rceil(r-y+\lceil\frac{x}{r}\rceil).$

Now 3.8 implies $\mathcal{E}(x,y)=\varnothing$. Hence we can assume
$\frac{x}{r}\not\in\mathbb{N}$. Then
$\lceil\frac{x}{r}\rceil-1=\lfloor\frac{x}{r}\rfloor$ and 3.8 implies the
equivalence of (1) and (2).
The equivalence of (2) and (3) follows from direct computation and Corollary
3.9. ∎

######  Proposition 3.11.

We have

$\mathcal{E}_{r}=\\{(x,y)\in\mathbb{N}_{\leq\frac{r(r-1)}{2}}\times\mathbb{N}_{\leq
r-1}\mid\frac{2x}{r}\leq
y\leq\min\\{\lfloor\frac{x}{r}\rfloor+\frac{x}{\lfloor\frac{x}{r}\rfloor}-r,\lceil\frac{x}{r}\rceil-\frac{x}{\lceil\frac{x}{r}\rceil}+r,x\\}\\}.$

###### Proof.

Recall that $(x,y)\in\mathcal{F}_{r}$ with $\mathcal{E}(x,y)$ implies
$\frac{2x}{r}\leq y\leq x$ and $y\leq r-1$. In particular,
$x\leq\frac{r(r-1)}{2}$. ∎

######  Example.

In the following we discuss the cases $r=3,4$ in detail to illustrate how to
apply our formulas.

  1. (1)

The case $r=3$. We have $1\leq x\leq\frac{r(r-1)}{2}=3$ and $1\leq y\leq
r-1=2$. We consider the inequalties

$\displaystyle\underline{x=1}:\frac{2}{3}\leq
y\leq\min\\{\infty,3,x=1,r-1=2\\}=1,$
$\displaystyle\underline{x=2}:\frac{4}{3}\leq y\leq\min\\{\infty,2,2,2\\}=2,$
$\displaystyle\underline{x=3}:2\leq y\leq\min\\{1,1,3,2\\}=1.$

This shows $\mathcal{E}_{3}=\\{(1,1),(2,2)\\}$. Moreover, we have

$\mathcal{E}(1,1)=\mathcal{B}{(1,1)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{3},3)\cap\operatorname{rep}_{\operatorname{inj}}(K_{3},0)=\mathcal{B}{(1,1)}\
\text{and}$
$\mathcal{E}(2,2)=\mathcal{B}{(2,2)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{3},2)\cap\operatorname{rep}_{\operatorname{inj}}(K_{3},0)=\mathcal{B}{(1,1)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{3},2).$

The following figure on the left-hand side shows the elementary dimensions
vector in $\mathcal{E}_{3}$ and the figure on the right hand side shows
$\widetilde{\mathcal{E}}_{3}$. The dashed red line is the restriction $y\leq
r-1=2$.

$\bullet$$\bullet$$\mathcal{F}_{3}$

$\mathcal{C}_{3}$$\bullet$$\bullet$$\bullet$$\bullet$

  2. (2)

The case $r=4$. We have $1\leq x\leq\frac{r(r-1)}{2}=6$ and $y\leq r-1=3$. We
consider the inequalties

$\displaystyle\underline{x=1}:\frac{1}{2}\leq y\leq\min\\{\infty,4,1,3\\}=1,$
$\displaystyle\underline{x=2}:1\leq y\leq\min\\{\infty,3,2,3\\}=2$
$\displaystyle\underline{x=3}:\frac{3}{2}\leq y\leq\min\\{\infty,2,3,3\\}=2,$
$\displaystyle\underline{x=4}:2\leq y\leq\min\\{1,1,4,3\\}=1$
$\displaystyle\underline{x=5}:\frac{5}{2}\leq
y\leq\min\\{2,\frac{7}{2},5,3\\}=2,$ $\displaystyle\underline{x=6}:3\leq
y\leq\min\\{3,3,6,3\\}=3.$

Hence $\mathcal{E}_{4}=\\{(1,1),(2,1),(2,2),(3,2),(6,3)\\}$. Moreover, we have

$\displaystyle\mathcal{E}(1,1)$
$\displaystyle=\mathcal{B}{(1,1)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},4)\cap\operatorname{rep}_{\operatorname{inj}}(K_{4},0)=\mathcal{B}(1,1),$
$\displaystyle\mathcal{E}(2,1)$
$\displaystyle=\mathcal{B}{(2,1)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},4)\cap\operatorname{rep}_{\operatorname{inj}}(K_{4},0)=\mathcal{B}(2,1),$
$\displaystyle\mathcal{E}(2,2)$
$\displaystyle=\mathcal{B}{(2,2)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},3)\cap\operatorname{rep}_{\operatorname{inj}}(K_{4},0)=\mathcal{B}{(2,2)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},3),$
$\displaystyle\mathcal{E}(3,2)$
$\displaystyle=\mathcal{B}{(3,2)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},3)\cap\operatorname{rep}_{\operatorname{inj}}(K_{4},0)=\mathcal{B}{(3,2)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},3),$
$\displaystyle\mathcal{E}(6,3)$
$\displaystyle=\mathcal{B}{(6,3)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{4},3)\cap\operatorname{rep}_{\operatorname{inj}}(K_{4},1).$

The figure on the left-hand side shows the elementary dimensions vector in
$\mathcal{E}_{4}$ and the figure on the right hand side shows
$\widetilde{\mathcal{E}}_{4}$.

$\mathcal{F}_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\mathcal{C}_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

##  4\. Orbits of elementary representations

It has been shown in [16] that elementary representations $E$ with dimension
vector in $\mathcal{E}_{3}=\\{(1,1),(2,2)\\}$ can be described combinatorially
in terms of their coefficient quiver. More precisely: There exists a basis
$\alpha,\beta,\gamma$ of the arrow space $A_{3}$ such that the coefficient
quiver of $E$ has one of the following two forms:

$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\gamma}$$\textstyle{\bullet}$$\textstyle{\bullet}$$\textstyle{\bullet.}$

In the following we rephrase this result in terms of an algebraic group acting
on the variety of representations. Let $V_{1},V_{2}$ be vector spaces. We
consider the canonical action of the general linear group
$\operatorname{GL}(A_{r})$ on $\operatorname{rep}(K_{r};V_{1},V_{2})$: Given
$g\in\operatorname{GL}(A_{r})$ and
$f\in\operatorname{rep}(K_{r};V_{1},V_{2})$, we write
$g^{-1}(\gamma_{i})=\sum^{r}_{j=1}\lambda_{ij}^{(g)}\gamma_{j}$ with
$\lambda_{ij}^{(g)}\in\mathbbm{k}$ for all $i\in\\{1,\ldots,r\\}$ and let
$f^{(g)}\in\operatorname{rep}(K_{r};V_{1},V_{2})$ be the tuple with entries

$(f^{(g)})_{i}=\sum^{r}_{j=1}\lambda_{ij}^{(g)}f_{j},1\leq i\leq r.$

The algebraic group

$G_{(V_{1},V_{2})}\coloneqq\operatorname{GL}(A_{r})\times\operatorname{GL}(V_{2})\times\operatorname{GL}(V_{1})$

acts on the space of representations $\operatorname{rep}(K_{r};V_{1},V_{2})$
via

$\displaystyle G_{(V_{1},V_{2})}\times\operatorname{rep}(K_{r};V_{1},V_{2})$
$\displaystyle\longrightarrow\operatorname{rep}(K_{r};V_{1},V_{2})$
$\displaystyle((g,h_{2},h_{1}),f)$ $\displaystyle\mapsto((h_{2}\circ
f_{i}\circ h_{1}^{-1})_{1\leq i\leq r})^{(g)}=(h_{2}\circ(f^{(g)})_{i}\circ
h_{1}^{-1})_{1\leq i\leq r}.$

Note that $\dim
G_{(V_{1},V_{2})}=r^{2}+(\dim_{\mathbbm{k}}V_{1})^{2}+(\dim_{\mathbbm{k}}V_{2})^{2}$.
Moreover, we have an action of $\operatorname{GL}(A_{r})$ on
$\operatorname{rep}(K_{r})$

$\operatorname{GL}(A_{r})\times\operatorname{rep}(K_{r})\longrightarrow\operatorname{rep}(K_{r});(g,N)\mapsto
N^{(g)}\coloneqq(N_{1},N_{2},(N(\gamma_{i}))_{1\leq i\leq r}^{(g)})$

and an induced action on the isomorphism classes of Kronecker representations
$[N]^{(g)}\coloneqq[N^{(g)}]$.

Now let $M_{1},M_{2}$ be vector spaces and
$\varnothing\neq\mathcal{O}\subseteq\operatorname{rep}(K_{r};M_{1},M_{2})$ be
a $G_{(M_{1},M_{2})}$-invariant subset. We let
$[\mathcal{O}]\coloneqq\\{[N]\mid N\in\operatorname{rep}(K_{r}),\exists
f\in\mathcal{O}:N\cong(M_{1},M_{2},f)\\}$. By definition we have a one-to-one
correspondence between $\mathcal{O}/G_{(M_{1},M_{2})}$ and
$[\mathcal{O}]/\operatorname{GL}(A_{r})$. For $(x,y)\in\mathbb{N}^{2}$ we let

$G_{(x,y)}\coloneqq\operatorname{GL}(A_{r})\times\operatorname{GL}(\mathbbm{k}^{x})\times\operatorname{GL}(\mathbbm{k}^{y}).$

Since regular representations are $\operatorname{GL}(A_{r})$-invariant, the
set
$\mathcal{E}(x,y)\subseteq\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})$
is $G_{(x,y)}$-invariant. Since $\operatorname{GL}(A_{r})$ acts transitive on
bases of $A_{r}$, we can rephrase the aforementioned results as follows.

######  Theorem 4.1.

(see [16, Theorem]) The following statements hold.

  1. (1)

We have $\mathcal{E}_{3}=\\{(1,1),(2,2)\\}$.

  2. (2)

The sets $\mathcal{E}(1,1)$, $\mathcal{E}(2,2)$ are orbits under the action of
$G_{(1,1)}$ and $G_{(2,2)}$ on
$\operatorname{rep}(K_{3};\mathbbm{k},\mathbbm{k})$ and
$\operatorname{rep}(K_{3};\mathbbm{k}^{2},\mathbbm{k}^{2})$, respectively.

  3. (3)

Let $M\in\operatorname{rep}(K_{3})$ be a representation with dimension vector
$(1,1)$. The representation is elementary if and only if there is
$g\in\operatorname{GL}(A_{3})$ such that
$M^{(g)}\cong(\mathbbm{k},\mathbbm{k},(\operatorname{id}_{\mathbbm{k}},0,0))$.

  4. (4)

Let $M\in\operatorname{rep}(K_{3})$ be a representation with dimension vector
$(2,2)$. The representation is elementary if and only if there is
$g\in\operatorname{GL}(A_{3})$ such that
$M^{(g)}\cong(\mathbbm{k}^{2},\mathbbm{k}^{2},(\operatorname{id}_{\mathbbm{k}^{2}},\beta,\gamma))$
with $\beta(a,b)=(0,a)$ and $\gamma(a,b)=(b,0)$ for all
$(a,b)\in\mathbbm{k}^{2}$.

In following we show that we can not hope for such a nice classification in
case $r\geq 4$.

######  Lemma 4.2.

Let
$\varnothing\neq\mathcal{O}\subseteq\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})$
be a non-empty open and $G_{(x,y)}$-invariant subset of
$\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})$ such that
$q_{r}(x,y)<-r^{2}$. Then $\mathcal{O}/G_{(x,y)}$ is not finite.

###### Proof.

We set $G\coloneqq G_{(x,y)}$. We assume that $\mathcal{O}/G$ is finite and
fix $T_{1},\ldots,T_{n}\in\mathcal{O}$ such that
$\mathcal{O}=\bigcup^{n}_{i=1}G.T_{i}$. Hence

$\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})=\overline{\mathcal{O}}=\bigcup^{n}_{i=1}\overline{G.T_{i}}.$

Because $\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})$ is
irreducible, we find $i\in\\{1,\ldots,n\\}$ such that
$\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})=\overline{G.T_{i}}$.
Since orbits are open in their closure ([8, 8.3]), we conclude with [7, 1.10]
that

$\dim G\geq\dim
G.T_{i}=\dim\overline{G.T_{i}}=\dim\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})=rxy.$

In particular, we have

$\displaystyle 0\leq\dim G-rxy$
$\displaystyle=r^{2}+x^{2}+y^{2}-rxy=r^{2}+q_{r}(x,y),$

in contradiction to the assumption. ∎

######  Corollary 4.3.

Let $(x,y)\in\mathcal{E}_{r}$ such that $q_{r}(x,y)<-r^{2}$. Then
$\mathcal{E}(x,y)/G_{(x,y)}$ is not finite.

###### Proof.

Since $(x,y)\in\mathcal{E}_{r}$, we can apply 3.8 and conclude that

$\mathcal{E}(x,y)=\mathcal{B}{(x,y)}\cap\operatorname{rep}_{\operatorname{esp}}(K_{r},r-y+\lceil\frac{x}{r}\rceil)\cap\operatorname{rep}_{\operatorname{inj}}(K_{r},\lceil\frac{x}{r}\rceil-1)$

is a non-empty open subset of
$\operatorname{rep}(K_{r};\mathbbm{k}^{x},\mathbbm{k}^{y})$. Moreover,
$\mathcal{E}(x,y)$ is $G_{(x,y)}$-invariant, since regular representations are
$\operatorname{GL}(A_{r})$-invariant. Now we apply 4.2. ∎

######  Theorem 4.4.

Let $r\geq 4$. Then there are infinitely pairwise non-isomorphic elementary
representations with dimension vector $(r+2,3)$ that all are in different
$\operatorname{GL}(A_{r})$-orbits.

###### Proof.

We set $x=r+2$ and $y=3<r$. Then $(x,y)\in\mathcal{F}_{r}$ and

$\lfloor\frac{x}{r}\rfloor(y+r-\lfloor\frac{x}{r}\rfloor)=r-2\leq x\leq
2(r-1)=\lceil\frac{x}{r}\rceil(r-y+\lceil\frac{x}{r}\rceil).$

Now Corollary 3.10 implies that $(r+2,3)\in\mathcal{E}_{r}$. Moreover, we have
$q_{K_{r}}(r+2,3)=-2r^{2}-2r+13<-r^{2}$ and can apply Corollary 4.3 ∎

######  Corollary 4.5.

Let $(x,y)\in\mathbb{N}^{2}$ such that $q_{r}(x,y)<-r^{2}$. The number of
different $\operatorname{GL}(A_{r})$-orbits of isomorphism-classes of
elementary representation with dimension vector $(x,y)$ is either $0$ or
infinite.

###### Proof.

We can assume that there is $E\in\operatorname{rep}(K_{r})$ elementary with
dimension vector $(x,y)$. By applying $D_{K_{r}}$ and powers of
$\sigma_{K_{r}}$ to $E$ we find an elementary representation $F$ with
dimension vector $\operatorname{\underline{dim}}F\in\mathcal{E}_{r}$. Since
$\sigma_{K_{r}}$ and $D_{K_{r}}$ do not change the quadratic form, we have

$q_{r}(\operatorname{\underline{dim}}F)=q_{r}(x,y)<-r^{2}.$

Now Corollary 4.3 implies that we get infinitely many orbits. Since
$D_{K_{r}}$ and $\sigma_{r}$ respect $\operatorname{GL}(A_{r})$-orbits (see
for example [3, 6.1.3]), the statements follows. ∎

######  Remark 4.6.

Let $E\in\operatorname{rep}(K_{3})$ be elementary. Then [16] implies that
$q_{3}(\operatorname{\underline{dim}}E)\in\\{-1,-4\\}$. Hence
$q_{3}(\operatorname{\underline{dim}}E)\geq-9=-r^{2}$.

##  5\. Examples

The following figures illustrate our findings for $r\in\\{3,4,5,6,7\\}$. The
dashed red line is the restriction $y\leq r-1$. We would like to remark that
simulations for $5\leq r\leq 500$ indicate that a sharp upper bound for $y$ is
$\lceil\frac{r}{2}\rceil$.

$\bullet$$\bullet$$\mathcal{F}_{3}$

$\mathcal{F}_{4}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\mathcal{F}_{5}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

$\mathcal{F}_{6}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\mathcal{F}_{7}$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$$\bullet$

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