\oplus\!\cE_{n+1,d}^b$.

* Let $(c,s) \in \NN_0\!\times\!\NN$ such that $q_r(c,s\!+\!dc)\!>\!1$, then there exist uniquely determined elements $n,a,b \in \NN_0$ such that a generic Steiner bundle $\cF \in \StVect(\Gr_d(A_r))$ of rank $s$ and with first Chern class $c \sigma_1$ is isomorphic to $\cE_{n,d}^a\!\oplus\! \cE_{n+1,d}^b$.

(1) Let $n \in \NN_0$ and consider the preprojective indecomposable representation $P_n(r)$. In view of $\udim P_n(r)\! =\! (a_n(r),a_{n+1}(r))$, an application of Theorem <ref>,
Proposition <ref> and Corollary <ref> shows that $\cE_{n,d}\!:=\! \TilTheta_d(P_n(r))$ is an exceptional Steiner bundle of rank $\Delta_{P_n(r)}(d)\! =\! a_{n+1}(r)\! -\!da_n(r)$ and first Chern class
$c_1(\cE_{n,d})\! =\! \dim_\KK P_n(r)_1\sigma_1\! =\! a_n(r)\sigma_1$.

(2) This follows immediately from Proposition <ref>, the definition of $\cE_{n,d}$ and the above-mentioned classification of the indecomposable preprojective $K_r$-representations.

(3) In view of (1), exceptional Steiner bundles enjoy the stated properties. Conversely, let $\cF$ be an indecomposable Steiner bundle such that $\rk(\cF)\! =\!a_{n+1}(r)\!-\!da_n(r)$ and $c_1(\cF)\!=\!a_n(r)\sigma_1$.
According to Theorem <ref>, there is an indecomposable representation $M \in \repp(K_r,d)$ such that $\cF \cong \TilTheta_d(M)$ and $\udim M\!=\!(a_n(r),a_{n+1}(r))$. Since $q_r(\udim M)\! =\!
q_r(a_n(r),a_{n+1}(r))\!=\!1$, Theorem <ref> implies that there is a unique indecomposable representation with dimension vector $(a_n(r),a_{n+1}(r))$. Hence $M \cong P_n(r)$ and $\cF \cong \cE_{n,d}$ is an
exceptional Steiner bundle.

(4) Let $\cF \in \StVect(\Gr_d(A_r))$ be a non-zero Steiner bundle that satisfies $\Ext^1_{\Gr_d(A_r)}(\cF,\cF)\!=\!(0)$. Application of Theorem <ref> implies the existence of $(0)\!\ne\!M \in \repp(K_r,d)$ such that
$\cF \cong \TilTheta_d(M)$. Let $N$ be an indecomposable direct summand of $M$. By Proposition <ref> we have $N \in \repp(K_r,d)$. As $\Ext^1_{K_r}(N,N)\!=\!(0)$, Theorem <ref> provides
$i \in \NN_0$ such that $N\cong P_i(r)$. We fix $n \in \NN_0$ minimal subject to $P_n(r)$ being a direct summand of $M$. Let $m\!>\!n$  and assume that $P_m(r)$ is a direct summand of $M$. We set
$l\!:=\!m\!-\!n\!-\!1\!\geq\!0$ and conclude with the Auslander-Reiten formula <cit.>

\begin{align*}
0 &= \dim_\KK \Ext^1_{K_r}(P_m(r),P_n(r)) = \dim_\KK \Hom_{K_r}(P_{n+2}(r),P_m(r))\\
&= \dim_\KK \Hom_{K_r}(\sigma_{K_r}^{n+1}(P_{n+2}(r)),\sigma_{K_r}^{n+1}(P_m(r))) = \dim_\KK \Hom_{K_r}(P_1(r),P_{l}(r)) = \dim_\KK P_{l}(r)_1.
\end{align*}

Hence $l\!=\!0$ and $m\!=\!n\!+\!1$. The result now follows by applying $\TilTheta_d$.

The other implication follows in a similar fashion using the fact that $\Ext^1_{K_r}(P_i(r),P_j(r))\!=\!(0)$ for all $i,j \in \NN$ with $|i\!-\!j|\!\leq\!1$.

(5) We show the statement on the level of representations and fix vector spaces $V_1,V_2$ of dimension $c$ and $s\!+\!dc$, respectively. Note that we have $\dim_\KK V_1\!<\! \dim_\KK V_2$.
Now <cit.> [Using the notation of <cit.> we have $\beta_0\!=\!\udim I_0(r)$ and $\beta_{-i}\!=\!\udim I_i(r)$, $\beta_i\!=\!\udim P_{i-1}(r)$ for all $i \in \NN$. Moreover, since $
\Ext^1_{K_r}(P_i(r),P_j(r))\!=\!(0)$ for all $i,j \in \NN_0$ with $|i\!-\!j|\!\leq\!1$, <cit.> and Theorem <ref> guarantee that <cit.> holds in arbitrary characteristic.] ensures the
existence of uniquely determined elements $a,b,n \in \NN_0$ such that a generic representation in $N \in \rep(K_r;V_1,V_2)$ is of the form $N \cong a P_n(r)\!\oplus\!b P_{n+1}(r) \in \repp(K_r,d)$. By Theorem
<ref> and Corollary <ref> we have $\rk(\TilTheta_d(N))\!=\!\Delta_{(V_1,V_2)}(d)\!=\!s$ and $c_1(\TilTheta_d(N))\!=\!c \sigma_{1}$.

(1) The numerical data in (3) do not imply the indecomposability of $\cF$: Let $E \in \rep(K_3)$ be indecomposable of dimension vector $\udim E\!=\!(1,1)$. Since $\udim \tau_{K_3}^{-1}(E)\!=\!(2,5)$, it
follows from <cit.> that $\tau_{K_r}^{-1}(E) \in \EKP(K_3)\!=\!\repp(K_r,1)$ and hence so is $M\!:=\!\tau^{-1}_{K_3}(E)\!\oplus\!P_1(3)$. We have $\udim M\!=\!(3,8)\!=\!\udim P_2(3)$, but $\cF\!:=\!
\TilTheta_1(M)$ is not  indecomposable.

(2) In view of Corollary <ref>, the functor $\TilTheta_d|_{\repp(K_r,d)}$ is exact. For $i\!\ge\!2$, the almost split sequence

\[(0) \lra P_{i-2}(r) \lra rP_{i-1}(r)\lra P_i(r) \lra (0)\]

thus induces an exact sequence

\[ (0) \lra \cE_{i,d}^\vee \lra r \cE_{i-1,d}^\vee \lra \cE_{i-2,d}^\vee \lra (0)\]

of vector bundles. For $d\!=\!1$, this corresponds to the construction given in <cit.> (with an extra term given by $\cE_{-1,d}\!:=\!\cO(-1)$). As pointed out in <cit.>, the sequences are thus
related to the mutations studied in [35].

§.§ Decomposing $\StVect(\Gr_d(A_r))$ via the Auslander-Reiten quiver

Let $\Gamma(K_r)$ be the Auslander-Reiten quiver of $\rep(K_r)$. Its vertices are the isoclasses of the indecomposable representations, while its arrows correspond to  the so-called
irreducible morphisms.[By abuse of notation, we will henceforth usually not distinguish between indecomposable representations and the vertices of $\Gamma(K_r)$ associated to them.]  Let $\cC \subseteq
\Gamma(K_r)$ be a component. Given $M \in \cC$, we let $(\rightarrow\! M)$ and $(M\!\rightarrow)$ be the sets of predecessors and successors of $M$, respectively. We refer the reader to [4] for basic  facts
on Auslander-Reiten theory and continue by recording some special features of $\Gamma(K_r)$.

Let $r\!\ge\!3$, $d \in \{1,\ldots, r\!-\!1\}$. The Auslander-Reiten quiver $\Gamma(K_r)$ of $\rep(K_r)$ is a disjoint union of components. There is a preprojective component $\cP$ (containing all preprojective modules), a
preinjective component $\cI$ (containing all preinjective modules), and all other components are called regular. Owing to Proposition <ref> the preprojective component $\cP$ is contained in
$\repp(K_r,d)$ and <cit.> implies that $\EKP(K_r)$ contains no preinjective modules. Recall that $\{P_n(r) \ ; \ n\!\ge\!0\}$ is the set of vertices of $\cP$, so that Proposition <ref> ensures that
$\TilTheta_d(\cP)$ is just the set $\{\cE_{n,d} \ ; \ n\!\ge\!0\}$ of exceptional bundles.

Let $\cC \subseteq \Gamma(K_r)$ be a regular AR-component. By Ringel's Theorem <cit.>, which is fundamental for our methods, $\cC$ is a stable translation quiver of isomorphism type
$\ZZ[A_\infty]$. A representation $M \in \cC$ is called quasi-simple, provided $M$ has precisely one immediate predecessor. The quasi-simple representations are therefore located at the “bottom”
of the component.

For each $M \in \cC$, there is a unique directed path

\[ M_{[1]} \rightarrow M_{[2]} \rightarrow \cdots \rightarrow M_{[n-1]} \rightarrow M_{[n]}\!=\!M\]

in $\cC$ such that

(a) for every $i \in \{1,\ldots,n\!-\!1\}$ the irreducible morphism $M_{[i]} \lra M_{[i+1]}$ is injective, and

(b) for every $i \in \{1,\ldots,n\}$ the factor module $M_{[i]}/M_{[i-1]} \cong \tau^{-i+1}_{K_r}(M_{[1]})$ ($M_{[0]}\!:=\!(0)$) is quasi-simple.
In that case $\ql(M)\!:=\!n$ is called the quasi-length of $M$, and we refer to $M_{[1]}$ and $M/M_{[n-1]}$ as the quasi-socle and quasi-top of $M$, respectively.
The filtration

\[ M_{[1]} \subsetneq \cdots \subsetneq M_{[n-1]} \subsetneq M\]

is called the quasi-composition series of $M$ with quasi-composition factors $\{\tau^{-i}_{K_r}(M_{[1]}) \ ; \ 0 \!\le\! i\!\le \ql(M)\!-\!1\}$.

For each regular component $\cC$ the set $\repp(K_r,d) \cap \cC$ forms a non-empty cone in $\cC$, consisting of the successors $(M_{\cC,d}\!\rightarrow)$ of a uniquely determined
quasi-simple representation $M_{\cC,d}$ in $\cC$.

This follows from Theorem <ref> and <cit.>.

[very thick, scale=1]
[every node/.style=fill, circle, inner sep = 1pt]

[color=black]  at (**2-*2*3,--0.3) $M_{\mathcal{C},1}$;
[color=black]  at (**2-*2*2,--0.3) $M_{\mathcal{C},r-1}$;

in 0,...,
i̱n 0,...,

= <̱
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;

=̱ <
[color=black] (,) at (*2*+,*̱2*+) $\circ$;
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;

=̱ =
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;
[color=black] (,) at (*2*+,*̱2*+) $\circ$;
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;

in 0,...,
in 0,...,

= <
[->] (*2*+,*2*+) to (*2*+-,*2*+-);
[->] (*2*++,*2*+-) to (*2*+2*-,*2*+);

= <

= =

[->] (*2*+,*2*+) to (*2*+-,*2*+-);
[->] (*2*++,*2*++) to (*2*+2*-,*2*+2*-);
[->] (*2*+,*2*+2*-) to (*2*+-,*2*++);
[->] (*2*++,*2*+-) to (*2*+2*-,*2*+);

[color=black] (Dots1) at (0,*+2*) $\cdots$;
[color=black] (Dots2) at (*2*,*+2*) $\cdots$;

[color=black] (Dots3) at (0.5**2*,2**+2*) $\vdots$;
[color=black] (Dots3) at (0.5**2*,2**+) $\vdots$;

[color=black] (Dots1) at (0,*+) $\cdots$;
[color=black] (Dots2) at (*2*,*+) $\cdots$;

[color=black] (Dots3) at (0.5**2*,2**+2*) $\vdots$;
[color=black] (Dots3) at (0.5**2*,2**+) $\vdots$;

= 1
in 0,...,
in 0,...,
= 0
= [->,dotted,thin] (*2*-,*2*) to (*2*-2*+,*2*);
[->,dotted,thin] (*2*-,*2*) to (*2*-2*+,*2*);
[->,dotted,thin] (*2*-+,*2*+) to (*2*-2*++,*2*+);

[on background layer]

= 1
[fill = !20] (**2+,-) node[anchor=north]
– (**2-**2-0.7*,-) node[anchor=north]
– (**2+,**2+0.7*) node[anchor=south];

= 1
[fill= !40](**2+,-) node[anchor=north]
– (**2-**2-0.7*,-) node[anchor=north]
– (**2+,**2+0.7*) node[anchor=south];

Cones of successors of $M_{\mathcal{C},1}$ and $M_{\mathcal{C},r-1}$ in a regular component $\mathcal{C} \subseteq \Gamma(K_r)$ with $M_{\mathcal{C},i} \neq M_{\mathcal{C},i+1}$ for some
$1\!\leq\!i\!<\! r\! -\! 1$ (cf. Theorem <ref>). The Auslander-Reiten translation is indicated by dotted arrows.

Let $M \in \repp(K_r,d)$ be an indecomposable representation. Then $\TilTheta_d(M)$ is an indecomposable object in $\Coh(\Gr_d(A_r))$.

By Theorem <ref>, the functor $\TilTheta_d$ is full and faithful, so that the ring $\End_{\Gr_d(A_r)}(\TilTheta_d(M))\cong \End_{K_r}(M)$ is local. Consequently, $\TilTheta_d(M)$ is

We denote by $\msInd\StVect(\Gr_d(A_r))$ the full subcategory of indecomposable Steiner bundles. Theorem <ref> and Lemma <ref> provide a decomposition

\[ \msInd\StVect(\Gr_d(A_r)) = \TilTheta_d(\cP)\sqcup \bigsqcup_{\cC} \TilTheta_d((M_{\cC,d}\!\rightarrow)),\]

with $\cC$ running through the set of regular AR-components of $K_r$.[Abusing notation, we write $\msInd\StVect(\Gr_d(A_r))$ for the set of isoclasses of the objects of $\msInd\StVect(\Gr_d(A_r))$.]

Recall that $\Phi_r^{-1}\!=\! \left(\begin{smallmatrix} -1 & r \\ -r & r^2\!-\!1 \end{smallmatrix}\right)$, while $\udim \tau^{-1}_{K_r}(M)\!=\!\Phi^{-1}_{K_r}.\udim M$ for every $M \in \rep(K_r)$ without non-zero injective direct
summands, cf. <cit.>.

If $M \not \cong P_0(r)$ is indecomposable, then

\[ \udim \sigma_{K_r}(M) = (r \dim_\KK M_1\!-\!\dim_\KK M_2, \dim_\KK M_1),\]

while $\sigma_{K_r}(P_0(r))\!=\!(0)$ and $\sigma_{K_r}(P_{i+1}(r)) \cong  P_{i}(r)$ for all $i\! \geq\! 0$. Similarly,

\[\udim \sigma_{K_r}^{-1}(M)  = (\dim_\KK M_2, r\dim_\KK M_2\!-\!\dim_\KK M_1)\]

for $M \not\cong S_1$. Thus, viewing $M \in \EKP(K_r)\!\smallsetminus\!\{0\}$ as a $\KK K_r$-module, we obtain

\[ (\dagger) \ \ \ \ \ \ \dim_\KK\sigma^{-1}_{K_r}(M) = \dim_\KK M\!+\!2\Delta_M(1)\!\!+\!(r\!-\!2)\dim_\KK M_2 > \dim_\KK M.\]

In particular, all irreducible morphisms between preprojective representations in $\rep(K_r)$ are injective, and $P_i(r)$ is isomorphic to a subrepresentation of $P_j(r)$ if and only if $i\! \leq\! j$.

Let $M \in \repp(K_r,d)$ be indecomposable. Then the following statements hold:

* If $M\not\cong P_0(r)$, then $\Delta_{\sigma_{K_r}^{-1}(M)}(d)\!>\!\Delta_M(d)$.

* $\Delta_{\tau^{-1}_{K_r}(M)}(d)\!>\!\Delta_M(d)$.

(1) As $M$ is not isomorphic to $I_0(r)\!=\!S_1$, we have

\[\udim \sigma^{-1}_{K_r}(M) = (\dim_\KK M_2, r\dim_\KK M_2\!-\!\dim_\KK M_1).\]

We thus obtain, observing $\Delta_M(d)\!\ge\!0$ and $M \not \cong P_0(r)$,

\begin{eqnarray*}
\Delta_{\sigma^{-1}_{K_r}(M)}(d)\!-\!\Delta_M(d) & = & r\dim_\KK M_2\!-\!\dim_\KK M_1\!-\!d\dim_\KK M_2\!-\!\dim_\KK M_2\!+\!d\dim_\KK M_1 \\
& = & (r\!-\!d\!-\!1)\dim_\KK M_2\!+\!(d\!-\!1)\dim_\KK M_1 \\
&\ge & ((r\!-\!d)d\!-\!1)\dim_\KK M_1 > 0.
\end{eqnarray*}

(2) This follows from (1), as $\tau^{-1}_{K_r}(M)\cong \sigma_{K_r}^{-2}(M)$ and $\sigma^{-1}_{K_r}(M)\not \cong P_0(r)$.

Let $r\!\ge\!3$, $d \in \{1,\ldots,r\!-\!1\}$. The following statements hold:

* Let $\cC \subseteq \Gamma(K_r)$ be a regular AR-component.

(a) We have $\rk(\TilTheta_d(M_{\cC,d}))\!<\!\rk(\TilTheta_d(M))$ for all $M \in (M_{\cC,d}\!\rightarrow)\!\smallsetminus\!\{M_{\cC,d}\}$.

(b) If $M_{\cC,d}$ is a brick, then every $\cF \in \TilTheta_d((M_{\cC,d}\!\rightarrow))$ has a filtration by simple Steiner bundles, whose filtration factors are pairwise non-isomorphic.

* If $M \in \repp(K_r,d)$ is indecomposable of minimal type, then either $M\cong P_1(r)$ and $d\!=\!1$, or $M \cong P_2(r)$ and $d\!=\!r\!-\!1$, or there is a regular AR-component $\cC$ such that $M \cong

* If $\cF \in \StVect(\Gr_d(A_r))$ is simple, then either $\cF \in \TilTheta_d(\cP)$, or there is a regular AR-component $\cC$ and a natural number $n \in \NN_0$ such that $\cF \cong \TilTheta_d(\tau^{-n}_{K_r}

Let $\cC$ be a regular component, $f : (M_\cC\!\rightarrow) \lra \NN_0$ be a function that is additive on almost split sequences and split exact sequences. For $M \in (M_{\cC,d}\!\rightarrow)$, we let
$\cW(M) \subseteq \cC$ be the wing of $M$. (This is the full subgraph of $\cC$ bounded by the triangle with vertices $M$, the quasi-socle of $M$ and the quasi-top of $M$.

[very thick, scale=1]
[every node/.style=fill, circle, inner sep = 1pt]

[color=black]  at (**2-1.5***2,-+*+) $M$;

in 0,...,
i̱n 0,...,

= <̱
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;

=̱ <
[color=black] (,) at (*2*+,*̱2*+) $\circ$;
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;

=̱ =
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;
[color=black] (,) at (*2*+,*̱2*+) $\circ$;
[color=black] (,,̱5)at (*2*,*̱2*) $\circ$;

in 0,...,
in 0,...,

= <
[->] (*2*+,*2*+) to (*2*+-,*2*+-);
[->] (*2*++,*2*+-) to (*2*+2*-,*2*+);

= <

= =

[->] (*2*+,*2*+) to (*2*+-,*2*+-);
[->] (*2*++,*2*++) to (*2*+2*-,*2*+2*-);
[->] (*2*+,*2*+2*-) to (*2*+-,*2*++);
[->] (*2*++,*2*+-) to (*2*+2*-,*2*+);

[color=black] (Dots1) at (0,*+2*) $\cdots$;
[color=black] (Dots2) at (*2*,*+2*) $\cdots$;

[color=black] (Dots3) at (0.5**2*,2**+2*) $\vdots$;
[color=black] (Dots3) at (0.5**2*,2**+) $\vdots$;

[color=black] (Dots1) at (0,*+) $\cdots$;
[color=black] (Dots2) at (*2*,*+) $\cdots$;

[color=black] (Dots3) at (0.5**2*,2**+2*) $\vdots$;
[color=black] (Dots3) at (0.5**2*,2**+) $\vdots$;

= 1
in 0,...,
in 0,...,
= 0
= [->,dotted,thin] (*2*-,*2*) to (*2*-2*+,*2*);
[->,dotted,thin] (*2*-,*2*) to (*2*-2*+,*2*);
[->,dotted,thin] (*2*-+,*2*+) to (*2*-2*++,*2*+);

[on background layer]

= 1
[fill = !20] (**2-**2+0.7*,-) node[anchor=north]
– (**2-2***2-0.7*,-) node[anchor=north]
– (**2-1.5***2,-+*+0.7*) node[anchor=south]
– (**2-**2+0.7*,-) node[anchor=north];

Wing $\mathcal{W}(M)$ of a regular representation $M$ of quasi-length $4$ in a regular component.

Then $\cW(M) \subseteq
(M_{\cC.d}\!\rightarrow)$, and we put

\[ \cW(M)_1 := \{ X \in \cW(M) \ ; \ \ql(X)\!=\!1\}.\]

(This is the set of quasi-composition factors of $M$.) Using induction on $\ql(M)$ one verifies that

\[ (\ast) \ \ \ \ \ \ \ \ f(M) = \sum_{X \in \cW(M)_1} f(X).\]

(1a) Let $M \in (M_{\cC,d}\!\rightarrow)\!\smallsetminus\!\{M_{\cC,d}\}$. According to Theorem <ref>(3), the function

\[ f : (M_{\cC,d}\!\rightarrow) \lra \NN_0 \ \ ; \ \ M \mapsto \rk(\TilTheta_d(M))\]

is additive on almost split sequences and split exact sequences. Consequently, identity ($\ast$) in conjunction with Lemma <ref>(2) and Theorem <ref>(3) yields

\[ \rk(\TilTheta_d(M)) = \sum_{X \in \cW(M)_1} \rk(\TilTheta_d(X)) > \rk(\TilTheta_d(M_{\cC,d})).\]

(1b) Since $M_{\cC,d}$ is a brick, it follows that $\tau_{K_r}^{-n}(M_{\cC,d})$ is a brick for all $n \in \NN_0$.  Let $M \in (M_{\cC,d}\!\rightarrow)$. The quasi-composition series of $M$ has $\cW(M)_1
\subseteq \{\tau_{K_r}^{-n}(M_{\cC,d}) \ ; \ n \in \NN_0\}$ as set of filtration factors, with each factor occurring with multiplicity $1$. By Corollary <ref> and Theorem <ref>, the functor $\TilTheta_d : \repp(K_r,d)
\lra \StVect(\Gr_d(A_r))$ is an exact equivalence, so that the assertion follows.

(2) Suppose that $M$ is regular and let $\cC$ be the regular AR-component containing $M$. A consecutive application of Corollary <ref>(2b) and <cit.> ensures that $M$ is quasi-simple.
Hence there is $n\!\ge\!0$ such that $M \cong \tau_{K_r}^{-n}(M_{\cC,d})$. Since $M$ has minimal type, Lemma <ref>(2) and Corollary <ref>(1a) yield $n\!=\!0$, whence $M\cong M_{\cC,d}$.

If $M$ is preprojective, then $M\cong P_i(r)$ for some $i\!\ge\!0$. As $r\!\ge\!3$, we have $\Delta_{P_0(r)}(d)\!=\!1\!<\!d(r\!-\!d)$, while $\Delta_{P_1(r)}(d)\!=\!r\!-\!d$ and

\[ \Delta_{P_2(r)}(d)\!-\! d(r\!-\!d) = \Delta_{\sigma^{-1}_{K_r}(P_1(r))}(d)\!-\!d(r\!-\!d) = (r\!-\!d)^2\!-\!1.\]

Lemma <ref>(1) thus implies $M\!\cong P_1(r)$ and $d\!=\!1$, or $M\cong P_2(r)$ and $d\!=\!r\!-\!1$.

(3) Theorem <ref> provides $M \in \repp(K_r,d)$ such that $\cF \cong \TilTheta_d(M)$. Suppose that $M \not\in \cP$. As $\cF$ is simple, $M$ is a brick, and <cit.> shows that $M$ is quasi-simple.
Hence there is a regular AR-component $\cC$ and a natural number $n \in \NN_0$ such that $M\cong \tau^{-n}_{K_r}(M_{\cC,d})$.

For future reference we record structural features of regular AR-components that are related to the functor $\sigma_{K_r}$. Recall that $\sigma_{K_r}$ and $\sigma^{-1}_{K_r}$ induce quasi-inverse auto-equivalences
on the full subcategory category $\cR(K_r) \subseteq \rep(K_r)$ of regular representations. Moreover, we have a natural equivalence $\sigma_{K_r} \circ \sigma_{K_r} \cong \tau_{K_r}$.

Let $1\! \leq\! d\! <\! r$ and $M \in \rep(K_r)$ be regular indecomposable. Then there exists a unique representation $M_{\sigma,d}$ in the $\sigma_{K_r}$-orbit $\langle\sigma_{K_r}\rangle.M$
of $M$ such that

\[ \repp(K_r,d) \cap \langle\sigma_{K_r}\rangle.M = \{ \sigma^{-n}_{K_r}(M_{\sigma,d}) \ ; \ n \in \NN_0\}.\]

Moreover, we have $M_{\sigma,d} \in \{M_{\sigma,1},\sigma_{K_r}^{-1}(M_{\sigma,1})\}$.

In view of Proposition <ref>, we have $\repp(K_r,d) \cap \langle\sigma_{K_r}\rangle.M\!\ne\!\emptyset$. Let $M_{\sigma,d} \in \repp(K_r,d) \cap \langle\sigma_{K_r}\rangle.M$ be a vertex of minimal
dimension. Identity ($\dagger$) right before Lemma <ref> in conjunction with Theorem <ref>(3) yields $\repp(K_r,d) \cap \langle\sigma_{K_r}\rangle.M\!=\!\{ \sigma^{-n}_{K_r}(M_{\sigma,d}) \ ;
\ n \in \NN_0\}$.

Thanks to Proposition <ref>(3), we have $M_{\sigma,d} \in \repp(K_r,1)\cap\langle \sigma_{K_r} \rangle.M$ and there exists $n\!\ge\!0$ such that $M_{\sigma,d}\!=\!\sigma_{K_r}^{-n}(M_{\sigma,1})$. Moreover,
Theorem <ref>(3) yields $\sigma_{K_r}^{-1}(M_{\sigma,1}) \in \repp(K_r,d)$, whence $n\!\le\!1$.

Let $\cC \subseteq \Gamma(K_r)$ be a regular component and $M\!:=\!M_{\cC,1}$.

* We have $M_{\sigma,1} \in \{M,\sigma_{K_r}(M)\}$.

* If $M_{\sigma,1}\!=\!\sigma_{K_r}{(M)}$, then $(M\!\rightarrow)\!=\!\cC\cap \repp(K_r,r\!-\!1)$.

* If ${M}_{\sigma,1}\!=\!M$, then $(\tau^{-1}_{K_r}(M)\!\rightarrow)\!=\!\cC\cap\repp(K_r,r\!-\!1)$.

* We have

\[ ({M_{[i]}})|_{\fv} \cong M|_{\fv}\!\oplus\! \Delta_{\tau^{-1}_{K_r} (M_{[i-1]})}(d) P_0(d)\!\oplus\!\dim_\KK \tau^{-1}_{K_r}(M_{[i-1]})_1 P_1(d)\]

for all $i \in \NN_{\geq 2}$, $1\!\leq\!d\!\leq\! r\!-\!1$ and $\fv \in \Gr_d(A_r)$.

(1) Since $\sigma^2_{K_r}(M) \not\in \EKP(K_r)\!=\!\repp(K_r,1)$, we have ${M}_{\sigma,1} \in \{M,\sigma_{K_r}(M)\}$.

(2) If $M_{\sigma,1}\!=\!\sigma_{K_r}(M)$, then Theorem <ref> implies $M \in \repp(K_r,r\!-\!1)$. By Proposition <ref>, the same is true for every element of $(M\! \rightarrow)\!=\!
\cC \cap \EKP(K_r)$. Consequently, $(M\!\rightarrow) \subseteq \cC\cap \repp(K_r,r\!-\!1) \subseteq \cC\cap \EKP(K_r)\!=\!(M\!\rightarrow)$.

(3) If $M\! =\! M_{\sigma,1}$, then $\sigma_{K_r}(M)\not \in \EKP(K_r)$ and Theorem <ref>(2) shows that $M \not \in \repp(K_r,r\!-\!1)$. On the other hand, Theorem
<ref>(3) implies  $\tau^{-1}_{K_r}(M) \cong \sigma_{K_r}^{-2}(M) \in \repp(K_r,r\!-\!1)$, so that $\tau^{-1}_{K_r}(M)\!=\!M_{\cC,r-1}$. Consequently, $(\tau^{-1}_{K_r}(M)\!\rightarrow)\!=\!
\cC\cap\repp(K_r,r\!-\!1)$.

(4) Let $i \in \NN_{\geq 2}$, $1\!\leq\!d\!<\! r$, and $\fv \in \Gr_d(A_r)$. General theory provides a short exact sequence

\[(0) \lra  M \lra M_{[i]} \lra \tau^{-1}_{K_r} (M_{[i-1]}) \lra (0).\]

By virtue of (2), (3) and Proposition <ref>(3), we have  $\tau^{-1}_{K_r}(M_{[i-1]}) \in \repp(K_r,d)$, so that

\[\tau^{-1}_{K_r}(M_{[i-1]})|_{\fv} \cong \Delta_{\tau^{-1}_{K_r} (M_{[i-1]})}(d)P_0(d)\!\oplus\! \dim_\KK \tau^{-1}_{K_r}(M_{[i-1]})_1 P_1(d)\]

is projective. In particular, the above short exact sequence splits upon restriction to $\KK.\fv$.

§ HOMOGENEITY AND UNIFORMITY FOR REPRESENTATIONS AND STEINER BUNDLES

Throughout this section, we assume that $r\!\ge\!3$. We will employ AR-theory to obtain results on indecomposable modules and Steiner bundles, whose restrictions to $K_2$ and $\PP^1$ all have the
same isomorphism type, respectively. We shall see that all preprojective representations as well as ”almost all“ regular indecomposable $\EKP$-representations are uniform in an even stronger sense. As a result,
uniform representations belonging to $\EKP(K_r)$ and Steiner bundles are usually not homogeneous.

§.§ Homogeneous modules and bundles

Given $d \in \{1,\ldots, r\}$, a linear map $\alpha \in \Inj_\KK(A_d,A_r)$ gives rise to an injective morphism $\hat{\alpha} : \PP(A_d) \lra \PP(A_r)$. There results the inverse image functor

\[ \hat{\alpha}^\ast : \Coh(\PP(A_r)) \lra \Coh(\PP(A_d)),\]

which is known to be right exact, cf. <cit.>. The additive functor $\hat{\alpha}^\ast$ commutes with direct sums and the map that sends a fraction $\frac{f}{g}$ of homogeneous polynomials to
$\frac{f\circ\alpha}{g\circ\alpha}$ induces isomorphisms $\hat{\alpha}^\ast(\cO_{\PP^{r-1}}(j)) \cong \cO_{\PP^{d-1}}(j)$ for $j \in \ZZ$.[Here we interpret $f$ and $g$ as elements of the algebra $\KK[A_r]\cong
S(A_r^\ast)$ of polynomial functions on $A_r$.] Moreover, for $\gamma^\ast \in A_r^\ast$, the morphism

\[ \gamma^\ast\cdot : \cO_{\PP^{r-1}}(j) \lra \cO_{\PP^{r-1}}(j\!+\!1) \ \ ; \ \ f \mapsto \gamma^\ast f\]

is being sent to

\[ \hat{\alpha}^\ast(\gamma^\ast\cdot) : \cO_{\PP^{d-1}}(j) \lra \cO_{\PP^{d-1}}(j\!+\!1) \ \ ; \ \ g \mapsto (\gamma^\ast\!\circ\!\alpha)g.\]

For ease of notation, we will write $\TilTheta\!:=\!\TilTheta_1$ and $\TilTheta_M\!:=\!\TilTheta_{M,d}$.

Let $M \in \rep(K_r)$. For $\alpha \in \Inj_\KK(A_d,A_r)$ we have

\[ \TilTheta(\alpha^\ast(M)) \cong \hat{\alpha}^\ast(\TilTheta(M)).\]

Let $\{\gamma_1,\ldots, \gamma_d\}$ be a basis for $A_d$ and put $\delta_i\!:=\!\alpha(\gamma_i)$. We choose $\delta_{d+1}, \ldots, \delta_r \in A_r$ such that $\{\delta_1,\ldots, \delta_r\}$
is a basis for $A_r$. For the corresponding dual bases we obtain

\[ \delta_i^\ast\circ \alpha = \left\{ \begin{array}{cc} \gamma_i^\ast & i \in \{1,\ldots, d\} \\ 0 & \text{else.} \end{array}\right.\]

We compute the morphism $\hat{\alpha}^\ast(\TilTheta_M)$. Let $U \subseteq \PP^{d-1}$ be open, $g \in \cO_{\PP^{d-1}}(-1)(U)$ and $m \in M$. In view of our foregoing observations we obtain

\begin{eqnarray*}
\hat{\alpha}^\ast(\TilTheta_M)(m\otimes g) & = & \sum_{i=1}^r M(\delta_i)(m)\otimes \hat{\alpha}^\ast(\delta_i^\ast\cdot)(g) = \sum_{i=1}^d M(\delta_i)(m)\otimes (\delta_i^\ast\!\circ\alpha)g \\
& = & \sum_{i=1}^d M(\alpha(\gamma_i))(m)\otimes\gamma_i^\ast g  = \TilTheta_{\alpha^\ast(M)}(m\otimes g),
\end{eqnarray*}

so that

\[ \hat{\alpha}^\ast(\TilTheta_M) = \TilTheta_{\alpha^\ast(M)}.\]

In view of the right exactness of $\hat{\alpha}^\ast$, this implies

\[ \hat{\alpha}^\ast(\TilTheta(M)) = \hat{\alpha}^\ast(\coker\TilTheta_M) \cong \coker \hat{\alpha}^\ast(\TilTheta_M) = \coker \TilTheta_{\alpha^\ast(M)} = \TilTheta(\alpha^\ast(M)),\]

as desired.

Recall that the general linear group $\GL(A_r)$ naturally acts on $A_r$ and $\KK K_r$, on the latter via automorphisms of associative $\KK$-algebras. For every $M \in \modd \KK K_r$ and $g \in \GL(A_r)$, the twisted
module $M^{(g)}$ has underlying $\KK$-space $M$ and action given by

\[ a\dact m := (g^{-1}(a)).m \ \ \ \ \forall \ g \in \GL_r(A_r), a \in \KK K_r, m \in M.\]

In this fashion, $\GL(A_r)$ acts on $\modd \KK K_r$ and $\rep(K_r)$ via auto-equivalences. For $g \in \GL_r(\KK)\cong\GL(A_r)$ with $g^{-1}\!=\!(a_{ij})_{1\le i,j\le r}$ we obtain

\[ M^{(g)}(\gamma_j) = \sum_{i=1}^ra_{ij}M(\gamma_i) \ \ \ \ \ \ \ \ 1\!\le\!j\!\le\!r.\]

Given $M \in \rep(K_r)$, the stabilizer

\[ \GL(A_r)_M := \{ g \in \GL_r(A_r) \ ;  \ M^{(g)} \cong M\}\]

is a closed subgroup <cit.>, and we say that $M$ is homogeneous, provided $\GL(A_r)_M\!=\!\GL(A_r)$.

The group $\GL_r(A_r)\!=\!\Inj_\KK(A_r,A_r)$ acts on $\PP^{r-1}\!=\!\PP(A_r)$ via automorphisms and hence on $\Coh(\PP^{r-1})$: Given $g \in \GL(A_r)$ and $\cF \in \Coh(\PP^{r-1})$, we define

\[ (g^\ast.\cF)(U) := \cF(g^{-1}.U)\]

for every open subset $U \subseteq \PP^{r-1}$. A coherent sheaf $\cF \in \Coh(\PP^{r-1})$ is called homogeneous, provided $g^\ast.\cF \cong \cF$ for every $g \in \GL(A_r)$.

Let $M \in \EKP(K_r)$. Then $M$ is homogeneous if and only if $\TilTheta(M)$ is homogeneous.[This result can readily be seen to hold for $M \in \repp(K_r,d)$ and $\TilTheta_d(M) \in \Vect(\Gr_d(A_r))$ as well.]

In view of $g^\ast.\cF\!=\!\widehat{g^{-1}}^\ast(\cF)$ and $M^{(g)}\!=\!(g^{-1})^\ast(M)$, our assertion follows directly from Lemma <ref>.

Homogeneous vector bundles of small rank are scarce: Suppose that $\Char(\KK)\!=\!0$ and let $\cF \in \StVect(\PP^{r-1})$ be an indecomposable homogeneous Steiner bundle of rank at most $2r\!-\!3$.
In view of <cit.>, we have:

(a) $r\!=\!3$ and $\cF \in \{\cO_{\PP^2}, \cT_{\PP^{2}}(-1)\}, S^2 \cT_{\PP^2}(-2)$}, or

(b) $r\!\geq\!4$ and $\cF \in \{ \cO_{\PP^{r-1}},\cT_{\PP^{r-1}}(-1)\}$.

Let $\{\gamma^\ast_1,\ldots, \gamma^\ast_r\} \subseteq A_r^\ast$ be the dual basis of $\{\gamma_1,\ldots, \gamma_r\}$. For $M \in \rep(K_r)$ and $i \in \{1,\ldots, r\}$, we consider the linear map

\[ \widehat{\gamma}_i : A_r\!\otimes_\KK\!M_1 \lra M_1 \ \ ; \ \ a\otimes m \mapsto \gamma^\ast_i(a) m.\]

Then we have

\[ \sigma_{K_r}(M)(\gamma_i) = \widehat{\gamma}_i|_{\ker\psi_M}.\]

Let $M \in \rep(K_r)$.

* We have $\sigma_{K_r}(M^{(g)})\cong \sigma_{K_r}(M)^{((g^{-1})^{\tr})}$ for all $g \in \GL(A_r)\cong \GL_r(\KK)$.

* $M$ is homogeneous if and only if $\sigma_{K_r}(M)$ is homogeneous.

(1) For $g \in \GL(A_r)$ we have

\[ g(\gamma_j) = \sum_{i=1}^r g_{ij}\gamma_i \ \ \ \ \ \ \ 1\!\le\!j\!\le\!r.\]

Direct computation shows that $\ker\psi_{M^{(g)}}\!=\!(g\otimes\id_{M_1})(\ker\psi_M)$.

Setting $\omega_{K_r}(M)\!:=\!(A_r\!\otimes_\KK\!M_1,M_1)$ and

\[ \omega_{K_r}(M)(\gamma_i)\!=\!\widehat{\gamma_i},\]

we obtain

\begin{eqnarray*}
\omega_{K_r}(M^{(g)})(\gamma_\ell)((g\otimes\id_{M_1})(\gamma_j\otimes m)) & = & \widehat{\gamma_\ell}(\sum_{i=1}^rg_{ij}\gamma_i\otimes m) =  g_{\ell j}m \\
& = & (\sum_{i=1}^r g_{\ell i} \omega_{K_r}(M)(\gamma_i))(\gamma_j\otimes m) \\
& = & \omega_{K_r}(M)^{((g^{-1})^{\tr})}(\gamma_\ell)(\gamma_j\otimes m)
\end{eqnarray*}

for $m \in M_1$.  By the above, the subrepresentation $\sigma_{K_r}(M)\subseteq \omega_{K_r}(M)$ is $(g\otimes\id_{M_1})$-stable, and hence $(g\!\otimes\!\id_{M_1},\id_{M_1})$ induces an isomorphism
$\sigma_{K_r}(M)^{((g^{-1})^{\tr})} \cong \sigma_{K_r}(M^{(g)})$.

(2) This is a direct consequence of (1).

§.§ Generalized uniform modules

Recall from Section <ref> that every $M \in \EKP(K_r)$ affords a generic decomposition $M_{\gen}\!=\!\bigoplus_{i\ge 0} n_i(M)P_i(2)$, with

\[ O_M := \{ \fv \in \Gr_2(A_r) \ ; \ n_i(M,\fv)\!=\!n_i(M)\}\]

being a dense open subset of $\Gr_2(A_r)$. We say that $M \in \EKP(K_r)$ is uniform, provided $O_M\!=\!\Gr_2(A_r)$. In <cit.>, the authors provide a notion
that encompasses those of homogeneity and uniformity. In our context the corresponding property reads as follows:

Let $d \in \{1,\ldots, r\}$. We say that a representation $M \in \rep(K_r)$ is $d$-homogeneous, provided $\alpha^\ast(M) \cong \beta^\ast(M)$ for all $\alpha,\beta \in \Inj_\KK(A_d,A_r)$.

Note that every $d$-homogeneous representation belongs to $\CR(K_r,d)$. (Here we set $\CR(K_r,r)\!:=\!\rep(K_r)$.) For $d\!=\!1$, we have in fact equality. Moreover, $M$ is $r$-homogeneous if and only if it is
homogeneous, while the uniform $\EKP$-representations correspond to those that are $2$-homogeneous.[If $M \in \rep(K_r)$ is $2$-homogeneous, then $M \in \CR(K_r)$ has constant rank. The observations at
the end of Section <ref> show that $M$ has a generic decomposition $M_{\gen}\!=\!\bigoplus_{i\ge 0} n_i(M)P_i(2)\!\oplus\!\bigoplus_{j\ge\!0}m_j(M)I_j(2)$. Hence $M$ is $2$-homogeneous if and only if $
\alpha^\ast(M)\cong M_{\gen}$ for all $\alpha \in \Inj_\KK(A_2,A_r)$.]

(1) If $M$ is $d$-homogeneous, then we either have $\cV(K_r,d)_M\!=\!\emptyset$ or $\cV(K_r,d)_M\!=\!\Gr_d(A_r)$. (Here $\cV(K_r,r)_M\!:=\!\emptyset$ in case $M$ is projective and $\cV(K_r,r)_M\!:=\!
\{A_r\}$ otherwise.)

(2) If $\fv \not \in \cV(K_r,d)_M$, then

\[ M|_\fv \cong \Delta_M(d) P_0(d)\!\oplus\!(\dim_\KK M_1) P_1(d).\]

Consequently, every $M \in \repp(K_r,d)$ is $d$-homogeneous for $1\!\le\!d\!\le\!r\!-\!1$.

(3) If $M$ is homogeneous, then $M$ is $d$-homogeneous for all $d \in \{1,\ldots, r\}$.

Let $d \in \{1,\ldots, r\!-\!1\}$. Then there is a brick $M \in \rep(K_r,d)$ such that $M$ not $(d\!+\!1)$-homogeneous.

We first assume that $d\!\le\!r\!-\!2$ and pick a pair $(V_1,V_2)$ of $\KK$-vector spaces such that $\udim (V_1,V_2)\!=\!(2d,d(r\!+\!d))$. Then we have

(a) $\Delta_{(V_1,V_2)}(d)\!=\!d(r\!-\!d)$, and

(b) $\Delta_{(V_1,V_2)}(d\!+\!1)\!\ge\!0$.
We consider $\rep(K_r;V_1,V_2)\!=\!\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)$. Given $\fv \in \Gr_{d+1}(A_r)$, the restriction map

\[ \res_\fv : \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2) \lra \Hom_\KK(\fv\!\otimes_\KK\!V_1,V_2) \  \  ;  \ \ \psi \mapsto \psi|_{\fv\otimes_\KK V_1}\]

is linear and hence a morphism. Consequently,

\[ \cO_{\fv} := \res_\fv^{-1}(\Inj_\KK(\fv\!\otimes_\KK\!V_1,V_2))\]

is an open subset of $\rep(K_r;V_1,V_2)$, which, by virtue of (b), is not empty. In view of (a), Proposition <ref> implies $\repp(K_r,d)\cap\cO_\fv\ne \emptyset$. Theorem <ref> thus provides $M \in \repp(K_r,d)
\cap\rep(K_r;V_1,V_2)$ such that $\fv \not \in \cV(K_r,d\!+\!1)_M$. As $M \in \repp(K_r,d)$ has minimal type, Corollary <ref>(2b,2d) and Corollary <ref> ensure that $M \not \in \rep(K_r,d\!+\!1)$ is
a brick. In particular, $\cV(K_r,d\!+\!1)_M\ne\emptyset$.

Since $M$ belongs to $\repp(K_r,d)$, it is clearly $d$-homogeneous. In view of $\emptyset \ne \cV(K_r,d\!+\!1)_M\!\ne\!\Gr_{d+1}(A_r)$, there are  $\alpha, \beta\in \Inj_\KK(A_{d+1},A_r)$ such that
$\alpha^\ast(M)$ is projective, while $\beta^\ast(M)$ is not. Consequently, $M$ is not $(d\!+\!1)$-homogeneous.

If $d\!=\!r\!-\!1$, we pick the Schwarzenberger module $M_\cS[r\!+\!2] \in \EKP(K_r)$. This module has minimal type and Corollary <ref>(2b,2d) implies that $M$ is a regular brick. In view of Corollary <ref>
and its succeeding remark $M_\cS[r\!+\!2]$ is not homogeneous. Thanks to Theorem <ref>, Lemma <ref> and $\sigma_{K_r}$ being an auto-equivalence on regular modules, the representation
$N\!:=\!\sigma_{K_r}^{-1}(M) \in \repp(K_r,r\!-\!1)$ is an $(r\!-\!1)$-homogeneous brick, which is not $r$-homogeneous.

Every $\alpha \in \Inj_\KK(A_d,A_r)$ gives rise to a linear embedding $\hat{\alpha} : \PP^{d-1} \hookrightarrow \PP^{r-1}$.

Let $d \in \{0,\ldots, r\!-\!1\}$. A vector bundle $\cF \in \Vect(\PP^{r-1})$ is said to be $d$-homogeneous, provided

\[ \hat{\alpha}^\ast(\cF) \cong \hat{\beta}^\ast(\cF)\]

for all $\alpha, \beta \in \Inj_\KK(A_{d+1},A_r)$.

It directly follows from Lemma <ref> that $M \in \EKP(K_r)$ is $d$-homogeneous if and only if $\TilTheta(M)$ is $(d\!-\!1)$-homogeneous.

A closed subset $L \subseteq \PP^{r-1}$ is called a line, provided $L\!=\!\im\hat{\alpha}$ for some $\alpha \in \Inj_\KK(A_2,A_r)$. Let $\iota : L \hookrightarrow \PP^{r-1}$ be the inclusion. Given
$\cF \in \Vect(\PP^{r-1})$, we put

\[\cF|_L:= \iota^\ast(\cF).\]

Since vector bundles on $\PP^1$ are homogeneous, we have

\[ \cF|_L \cong \hat{\alpha}^\ast(\cF) \ \ \ \ \ \ \text{whenever} \ \ L= \hat{\alpha}(\PP^1).\]

A vector bundle $\cF \in \Vect(\PP^{r-1})$ is said to be uniform, if there are $a_i(\cF) \in \NN_0$ such that

\[ \cF|_L \cong \bigoplus_{i\ge 0} a_i(\cF)\cO_L(i)\]

for every line $L \subseteq \PP^{r-1}$.

It follows that $\cF$ is $(r\!-\!1)$-homogeneous if and only if it is homogeneous. Moreover, $\cF$ is uniform if and only if $\cF$ is $1$-homogeneous.

Since the projective representation $P_1(3)$ is homogeneous, so is  $M\!:=\!P_2(3) \cong \sigma_{K_r}^{-1}(P_1(3))$, cf. Lemma <ref>. Then $M$ has dimension vector $(3,8)$ and Proposition
<ref>(2) ensures that $M \in \repp(K_3,2)$. Consequently, $M$ is uniform with $M_{\gen}\!=\!\Delta_M(2)P_0(2)\!\oplus\!(\dim_\KK M_1)P_1(2)$. Note that $\cF\!:=\!\TilTheta_1(M)$ is an indecomposable Steiner
bundle such that $c_1(\cF)\!=\!3$ and $\rk(\cF)\!=\!5$.

Let $N \in \repp(K_3,2)$ be such that $\udim N\!=\!(3,8)$. Since $\Delta_N(2)\!=\!2$, (2b) and (2d) of Corollary <ref> ensure that $N$ is a brick and not projective. Consequently, $\udim \sigma_{K_3}(N)\!=\!(1,3)$,
so that $\sigma_{K_3}(N)\cong P_1(3)$ and $N\cong P_2(3)\!=\!M$. As a result, $\cF$ is isomorphic to the dual of the bundle constructed in <cit.>.

Corollary <ref> and Lemma <ref> imply that $\cF$ is homogeneous (and hence uniform) such that

\[\cF|_L \cong 2\cO_L\!\oplus 3\cO_L(1)\]

for every line $L \subseteq \PP^2$.

According to <cit.>, there exists, for every $d\!\in \{1,\ldots,\!r\!-\!2\}$, a holomorphic vector bundle $\cF_d$ on $\PP^{r-1}(\CC)$ such that $\cF_d$ is $(d\!-\!1)$-homogeneous but not $d$-homogeneous.
The following result, which slightly strengthens that of loc. cit., implies that such examples may also be found within the category of Steiner bundles on $\PP^{r-1}$.

Given $d \in \{1,\ldots, r\!-\!1\}$, there exists a simple Steiner bundle $\cF_d$ that is $(d\!-\!1)$-homogeneous, but not $d$-homogeneous.

Proposition <ref> provides a brick $M \in \rep(K_r,d)$ such that $M$ is not $(d\!+\!1)$-homogeneous. Consequently, the bundle $\cF_d\!:=\!\TilTheta(M)$ has the requisite properties.

In [51] the authors introduced the notion of uniform vector bundles of $k$-type and studied the case $k\!=\!1$ in detail.[The vector bundles studied in [51] are dual to those considered here.]
Given $d \in \{1,\ldots, r\!-\!1\}$ and $k \in \NN_0$, we say that a $d$-homogeneous Steiner bundle $\cF$ is of  $k$-type, provided there are $n_0, \ldots, n_k \in \NN_0$, $n_k\!\ne\!0$ such that

\[ \alpha^\ast(\cF) \cong \bigoplus_{i=0}^k \cE_i[d\!+\!1]^{n_i}\]

for all $\alpha \in \Inj_\KK(A_{d+1},A_r)$. Here the $\cE_i[d\!+\!1] \in \StVect(\PP(A_{d+1}))$ are the exceptional Steiner bundles discussed in Section <ref>.

Our next result implies that, for $k\!=\!1$, we obtain the essential image of $\repp(K_r,d\!+\!1)$.

Let $d \in \{2,\ldots, r\!-\!1\}$, $\cF \in \StVect(\PP^{r-1})$ be a Steiner bundle. Then the following statements are equivalent:

* $\cF \in \TilTheta(\repp(K_r,d))$.

* $\hat{\alpha}^\ast(\cF)\cong \cO^{c_1(\cF)}_{\PP(A_d)}\!\oplus\!\cT_{\PP(A_d)}^{\rk(\cF)-(d-1)c_1(\cF)}(-1)$ for every $\alpha \in \Inj_\KK(A_d,A_r)$.

(1) $\Rightarrow$ (2). By assumption, there exists $M \in \repp(K_r,d)$ such that $\cF \cong \TilTheta(M)$. Given $\alpha \in \Inj_\KK(A_d,A_r)$, it follows that $\alpha^\ast(M)$ is projective. Observing Lemma
<ref>, we thus obtain

\begin{eqnarray*}
\hat{\alpha}^\ast(\TilTheta(M)) & \cong & \TilTheta(\alpha^\ast(M)) \cong \TilTheta(P_0(d))^{\Delta_M(d)}\!\oplus\!\TilTheta(P_1(d))^{\dim_\KK M_1}  \\
& \cong & \cO_{\PP(A_d)}^{\rk(\cF)-(d-1)c_1(\cF)}\!\oplus\!\cT_{\PP(A_d)}^{c_1(\cF)}(-1).
\end{eqnarray*}

(2) $\Rightarrow$ (1). There is $M \in \EKP(K_r)$ such that $\cF \cong \TilTheta(M)$. The foregoing calculation in conjunction with Lemma <ref> and Theorem <ref> now shows that

\[ \alpha^\ast(M) \cong \Delta_M(d) P_0(d)\!\oplus\!(\dim_\KK M_1)P_1(d)\]

for all $\alpha\in \Inj_\KK(A_d,A_r)$, so that each $\alpha^\ast(M)$ is projective. It follows that $M \in \repp(K_r,d)$.

Let $\cC \subseteq \Gamma(K_r)$ be a regular AR-component.

* Every $M \in (\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow)$ is $(r\!-\!1)$-homogeneous.

* If $M \in (\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow)$, then $\TilTheta(M)$ is $(r\!-\!2)$-homogeneous of $1$-type.

* Let $M_{\cC,1}$ be $d$-homogeneous for some $d\!\le\!r$.

* Every $M \in (M_{\cC,1}\!\rightarrow)$ is $d$-homogeneous.

* Let $d\!\ge\!2$. If $\TilTheta(M_{\cC,1})$ is $(d\!-\!1)$-homogeneous of $k$-type, then $\TilTheta(M)$ is $(d\!-\!1)$-homogeneous of $k$-type for every $M \in (M_{\cC,1}\rightarrow) \!\smallsetminus\!(\tau_{K_r}^{-1}

(1) In view of Theorem <ref>(3), we have $(\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow) \subseteq \repp(K_r,r\!-\!1)$, so that every $M \in (\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow)$ is in particular

(2) This is a direct consequence of (1), Lemma <ref> and Lemma <ref>.

(3a) If $d\!=\!r$, then $M$ is homogeneous, and our assertion follows from <cit.> in conjunction with $\GL(A_r)$ being connected. Alternatively, Theorem <ref>(3) implies that $(M_{\cC,1})_{[i]}$ is
$d$-homogeneous for every $i\!\ge\!2$. The result thus follows from (1).

(3b) Since $\TilTheta(M_{\cC,1})$ $(d\!-\!1)$-homogeneous of $k$-type, Theorem <ref>(2) in conjunction with Lemma
<ref> and Theorem <ref>(2) implies that there are $n_0,\ldots, n_k \in \NN_0$, $n_k\!\ne\!0$ such that

\[ \alpha^\ast(M_{\cC,1}) \cong \bigoplus_{i=0}^k n_i P_i(d)  \ \ \ \ \  \ \forall \ \alpha \in \Inj_\KK(A_d,A_r).\]

Parts (3) of Theorem <ref> now shows that for every $M \in (M_{\cC,1}\rightarrow) \!\smallsetminus\!(\tau_{K_r}^{-1}(M_{\cC,1})\rightarrow) $ its restriction $\alpha^\ast(M)$ can be written in such a way.
This implies (b).

If $d\!\le\!r\!-\!2$, then Proposition <ref> provides an indecomposable representation $M \in \rep(K_r,d)$ such that $M$ is not $(d\!+\!1)$-homogeneous. Let $\cC_M$ be the AR-component containing $M$. As “most” vertices of the cone $\cC_M \cap\repp(K_r,d)$ are $(d\!+\!1)$-homogenous, we see that the natural analogue of <cit.> does not hold in this context.

In 1960 Schwarzenberger <cit.> raised the question, whether all uniform vector bundles on $\PP^{r-1}$ are homogeneous. The first counterexample was given by Elencwajg [27] in 1979. The following
result implies that every indecomposable inhomogeneous Steiner bundle gives rise to families of inhomogeneous, but $(r\!-\!2)$-homogeneous indecomposable vector bundles of $1$-type.

Let $\cC \subseteq \Gamma(K_r)$ be a component that contains a representation $M \in \cC$ that is not homogeneous. Then every $X \in (\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow)$ is
$(r\!-\!1)$-homogeneous, but not homogeneous.

Since preprojective and preinjective representations are homogeneous, it follows that $\cC$ is regular.  By Corollary <ref>, the cone $(\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow) \subseteq \repp(K_r,r\!-\!1)$
consists of $(r\!-\!1)$-homogeneous representations. As $M \in \cC$ is not homogeneous and $\GL(A_r)$ is connected, <cit.> ensures that all representations in $\cC$ are inhomogeneous.

In the situation above, every $X \in (\tau_{K_r}^{-1}(M_{\cC,1})\!\rightarrow)$ gives rise to a family of Steiner bundles that are not homogeneous but $(r\!-\!2)$-homogeneous: In view of [29],

\[ \GL(A_r)_X := \{ g \in \GL(A_r) \ ; \ X^{(g)} \cong X\}\]

is a closed, proper subgroup of $\GL(A_r)$, so that the variety $\GL(A_r)/\GL(A_r)_X$ has dimension $\ge\!1$. Hence there are at least $|\KK|$ pairwise non-isomorphic inhomogeneous but $(r\!-\!1)$-homogeneous
representations of dimension vector $\udim X$. Upon application of $\TilTheta$, we obtain a family of Steiner bundles, whose members are inhomogeneous but $(r\!-\!2)$-homogeneous.

§.§ Constructions
In this section we turn to the existence of modules and bundles with certain prescribed properties. This contributes to understanding the conceptual
sources of several computations and ad hoc arguments in [51].

Let $d\!\ge\!2$. For every $r\!\ge\!d\!+\!1$ and $n\!\ge\!d\!+\!2$, there exists $N \in \repp(K_r,d)$ such that

(a) $\dim_\KK N_1\!=\!n$, and

(b) $N$ is not homogeneous and of minimal type.

We proceed by induction on $r$, starting with $r\!=\!d\!+\!1$. Let $n\!\ge\!d\!+\!2$, so that $\ell\!:=\!n\!-\!r\!+\!1\!\ge\!2$. We consider the Schwarzenberger module $M\!:=\!M_\cS[\ell+r] \in \EKP(K_r)\!=\!
\repp(K_r,1)$ of dimension vector $\udim M\!=\!(\ell,\ell\!+\!r\!-\!1)$. Thanks to Theorem <ref> the representation $N\!:=\!\sigma_{K_r}^{-1}(M)$ belongs to $\repp(K_r,d)$. It has dimension vector $\udim N\!=\!(n,(r\!-\!1)n\!+\!d)$ and
hence is of minimal type. Since $\ell\!\ge\!2$, Corollary <ref> and its succeeding remark imply that $M$ is not homogeneous. By Lemma <ref>, the representation $N$ is also not homogeneous.

Suppose the statements hold for some $r\!\ge\!d\!+\!1$. Given $n\!\ge\!d\!+\!2$, there is $M \in \repp(K_r,d)$ of minimal type such that (a) and (b) hold. Proposition <ref> provides $N \in \repp(K_{r+1},d)$ of
minimal type such that $N|_{K_r} \cong M\!\!\oplus\!d P_0(r)$. In particular, (a) holds. If $N$ is homogeneous, then its restriction $N|_{K_r}$ inherits this property. As $\GL(A_r)$ is connected, this then holds for every
direct summand of $N|_{K_r}$, a contradiction.[One can show that the category of uniform representations is closed under taking direct summands. Hence one can also find $N$ of minimal type that is not

Upon application of $\TilTheta_d$, Corollary <ref> yields a generalization of <cit.>: One obtains inhomogeneous vector bundles $\cF_d \in \StVect(\Gr_d(A_r))$ with $c_1(\cF_d)\!=\!
n\sigma_1$ and $\rk(\cF_d)\!=\!\dim \Gr_d(A_r)$.

We provide a conceptual proof for the following generalization of  <cit.>:

Let $d \in \{2,\ldots,r\!-\!1\}$. The following statements hold:

* Let $\cF \in \StVect(\PP^{r-1})$ be a Steiner bundle.

* If $c_1(\cF)\!\ge\!d$ and $\cF$ is $(d\!-\!1)$-homogeneous of $1$-type, then $\rk(\cF)\!\ge\!d(r\!-\!d)\!+\!(d\!-\!1)c_1(\cF)$.

* If $\cO_{\PP^{r-1}}$ is not a direct summand of $\cF$, then $\rk(\cF)\!\le\!c_1(\cF)(r\!-\!1)$.

* For each pair $(c,\ell) \in \NN^2$ such that $\ell\!\ge\!d(r\!-\!d)\!+\!c(d\!-\!1)$ there is $\cF \in \StVect(\PP^{r-1})$ which is $(d\!-\!1)$-homogeneous of $1$-type and such that $(c_1(\cF),\rk(\cF))\!=\!(c,\ell)$.

(1) Let $M \in \EKP(K_r)$ be such that $\TilTheta(M)\cong\cF$.

(a) Lemma <ref> implies $M\in\repp(K_r,d)$ with $\dim_\KK M_1\!=\!c_1(\cF)\!\ge\!d$ and $\Delta_M\!=\!\rk(\cF)$. Theorem <ref> now yields $\rk(\cF)\!=\!\Delta_M(d)\!+\!(d\!-\!1)\dim_\KK M_1\!\ge
\!d(r\!-\!d)\!+\!(d\!-\!1)\dim_\KK M_1\!=\!d(r\!-\!d)\!+\!(d\!-\!1)c_1(\cF)$.

(b) Since $\cO_{\PP^{r-1}}$ is not a direct summand of $\cF$, the map $\psi_M$ is surjective, so that there is a projective resolution

\[ (0) \lra -\Delta_M(r)P_0(r) \lra (\dim_\KK M_1)P_1(r) \lra M \lra (0).\]

In particular, $\Delta_M(r)\!\le\!0$, whence $\rk(\cF)\!=\!\Delta_M\!=\!\Delta_M(r)\!+\!(r\!-\!1)\dim_\KK M_1\!\le\! c_1(\cF)(r\!-\!1)$.

(2) Let $(V_1,V_2)$ be a pair of $\KK$-vector spaces such that $\dim_\KK V_1\!=\!c$ and $\dim_\KK V_2\!=\!c\!+\!\ell$. By assumption, we have $\Delta_{(V_1,V_2)}(d)\!=\!\ell\!-\!(d\!-\!1)c\!\ge\!d(r\!-\!d)$ and
Proposition <ref> provides $M \in \rep(K_r;V_1,V_2)\cap\repp(K_r,d)$. Hence $\cF\!:=\!\TilTheta(M)$ has the requisite properties.

We continue by providing a general method of constructing uniform $1$-type Steiner bundles which are not homogeneous relative to certain subgroups of $\GL(A_r)$.

Let $H \subseteq \GL(A_r)$ be a subgroup. Then $M \in \rep(K_r)$ is called $H$-homogeneous, provided

\[ M^{(h)} \cong M \ \ \ \ \ \forall \ h \in H.\]

There is an analogous notion of $H$-homogeneous vector bundles on $\PP^{r-1}$.

For $s\!<\!r$, we put $A_s\!:=\!\bigoplus_{j=1}^s\KK\gamma_j$ and $A_{r-s}\!:=\!\bigoplus_{j=s+1}^r\KK\gamma_j$, so that $A_r\!=\!A_s\!\oplus\!A_{r-s}$. Given $M \in \rep(K_s)$, we denote by $\Inf(M) \in \rep(K_r)$ the
representation with underlying pair of spaces $(M_1,M_2)$ and structure map

\[ \psi_{\Inf(M)}((a\!+\!b)\otimes m) := \psi_M(a\otimes m) \ \ \ \ \ \ a \in A_s, b \in A_{r-s}, m \in M_1.\]

\[\Inf(M)(\gamma_i) = \left\{\begin{array}{cc} M(\gamma_i) & 1\!\le\!i\! \le\! s \\ 0 & s\!+\!1\!\le\!i\!\le\! r.\end{array} \right.\]

Note that

\[ \iota : \GL(A_s) \lra \GL(A_r) \ \ ; \ \ h \mapsto h\!\oplus\!\id_{A_{r-s}}\]

is an injective homomorphism of algebraic groups such that

\[ (\ast) \ \ \ \ \ \Inf(M)^{\iota(h)} = \Inf(M^{(h)}) \ \ \ \ \ \forall \ h \in \GL(A_s).\]

In the sequel, we shall identify $\GL(A_s)$ with its image under $\iota$ in $\GL(A_r)$.

If $N \in \rep(K_s)$ is homogeneous, then ($\ast$) shows that $\Inf(N)$ is $\GL(A_s)$-homogeneous. However, $\Inf(N)$ is usually not $s$-homogeneous.

Let $d \in \{1,\ldots, r\!-\!1\}$. The following statements hold:

* For $\fv \in \Gr_d(A_r)$, we have $\udim E(\fv)\!=\!(d,rd\!-\!1)$, $\udim \sigma_{K_r}(E(\fv))\!=\!(1,d)$ and $\udim \tau_{K_r}(E(\fv))\!=\!(r\!-\!d,1)$.

* Let $M \in \rep(K_r)$ be regular such that $\cV(K_r,d)_M\!\ne\!\Gr_d(A_r)$.

* We have $\tau^{-1}_{K_r}(M) \in \repp(K_r,d)$.

* If $d\!>\!1$, then $\sigma_{K_r}^{-1}(M) \in \repp(K_r,d\!-\!1)$.

* If $d\!>\!1$, then $\tau_{K_r}^{-1}(M)\in \repp(K_r,r\!-\!1)$.

(1) Let $\alpha \in \Inj_\KK(A_d,A_r)$ be such that $\im \alpha\!=\!\fv$. Directly from the definition we obtain

\[ \udim D_{K_r}(E(\fv)) = \udim\tau_{K_r}(\coker\bar{\alpha}) = \Phi_r\dact(1,r\!-\!d) = (rd\!-\!1,d),\]

whence $\udim E(\fv)\!=\!(d,rd\!-\!1)$. Writing $\sigma_r(a,b)\!:=\!(ra\!-\!b,a)$, we have $\udim \sigma_{K_r}(N)\!=\!\sigma_r(\udim N)$ for every $N \in \rep(K_r)$. The remaining assertions now follow by applying $\sigma_r$
to $(d,rd\!-\!1)$ and $(1,d)$, respectively.

(2) In view of Theorem <ref>, our current assumption provides $\fv \in \Gr_d(A_r)$ with $\Hom_{K_r}(E(\fv),M)$ $=\!(0)$.

(a) Let $\fu \in \Gr_d(A_r)$ be such that $\Hom_{K_r}(E(\fu),\tau_{K_r}^{-1}(M))\!\ne\!(0)$. Then $\Hom_{K_r}(\tau_{K_r}(E(\fu)),M)\!\ne\!(0)$ and since
$d\dim_\KK \tau_{K_r}(E(\fu))_1\!=\!d(r\!-\!d)\!>\!1\!=\!\dim_\KK\tau_{K_r}(E(\fu))_2$, an application of Proposition <ref> ensures that

\[ 0 \ne \dim_\KK\ker\psi_{\tau_{K_r}(E(\fu)),\fv} = \dim_\KK\Hom_{K_r}(E(\fv),\tau_{K_r}(E(\fu)).\]

Since $\tau_{K_r}(E(\fu))$ is elementary, while $E(\fv)$ and $M$ are regular, it follows from <cit.> that $\Hom_{K_r}(E(\fv),M)\!\ne\!(0)$, a contradiction.

(b) This is analogous to (a).

(c) Thanks to (b), we have $\sigma_{K_r}(\tau^{-1}_{K_r}(M))\cong \sigma^{-1}_{K_r}(M) \in \repp(K_r,d\!-\!1) \subseteq \EKP(K_r)$. The assertion now follows from Theorem <ref>.

Our next result produces $(s\!-\!1)$-homogeneous $1$-type vector bundles $\cF$ that fail to be $\GL(A_s)$-homogeneous.

Let $r\!>\!s\!>\!d\geq\!1$. Suppose that $M \in \rep(K_s)$ is an indecomposable, non-simple representation such that $\cV(K_s,d)_M\!\ne\!\Gr_d(A_s)$.
Then the following statements hold:

* $\Inf(M)$ and $\sigma^{-3}_{K_r}(\Inf(M))$ are regular, quasi-simple representations.

* $\sigma^{-2}_{K_r}(\Inf(M)) \in \repp(K_r,d)$.

* We have $(\sigma^{-3}_{K_r}(\Inf(M))\!\rightarrow) \subseteq \repp(K_r,r\!-\!1)$. For each $N \in (\sigma^{-3}_{K_r}(\Inf(M))\!\rightarrow)$, the Steiner bundle $\TilTheta(N) \in \StVect(\PP^{r-1})$ is $(r\!-\!2)$-homogenous
of $1$-type, but not homogeneous.

* Suppose that $M$ is not homogeneous. For each $N \in (\sigma^{-3}_{K_r}(\Inf(M))\!\rightarrow)$, the Steiner bundle $\TilTheta(N) \in \StVect(\PP^{r-1})$ is $(s\!-\!1)$-homogeneous of $1$-type, but not

(1) By <cit.>, the representation $\Inf(M) \in \rep(K_r)$ is indecomposable, regular and quasi-simple. Therefore, the same holds for $\sigma^{-3}_{K_r}(\Inf(M))$.

(2) By assumption, there is $\alpha \in \Inj_\KK(A_d,A_s)$ such that $\alpha^\ast(M)$ is projective. Let $\iota : A_s \lra A_r$ be the inclusion. Then $\iota\circ\alpha \in \Inj_\KK(A_d,A_r)$ and $(\iota\circ \alpha)^\ast(\Inf(M)) \cong \alpha^\ast(\iota^\ast(\Inf(M))) \cong \alpha^\ast(M)$, so that $\im (\iota\circ \alpha) \not \in \cV(K_r,d)_{\Inf(M)}$. The assertion now follows from (1) and Lemma <ref>(2a).

(3) By (1),(2) and Theorem <ref>(3) we know that $\sigma_{K_r}(\sigma^{-3}_{K_r}(\Inf(M))) \in \EKP(K_r)$ is quasi-simple. Now Theorem <ref>(2) yields $\sigma^{-3}_{K_r}(\Inf(M))
\in \repp(K_r,r\!-\!1)$. We conclude with Theorem <ref> that $(\sigma^{-3}_{K_r}(\Inf(M))\!\rightarrow) \subseteq \repp(K_r,r\!-\!1)$. As $\Inf(M)(\gamma_r)\!=\!0$, the
representation $\Inf(M)$ is not homogeneous. Hence Lemma <ref> shows that $\sigma^{-3}_{K_r}(\Inf(M))$ enjoys the same property. Consequently, every representation belonging to the (regular) component of
$\sigma^{-3}_{K_r}(\Inf(M))$ is not homogeneous. In conclusion, each bundle $\TilTheta (N) \in \StVect(\PP^{r-1})$ with $N \in (\sigma^{-3}_{K_r}(\Inf(M))\!\rightarrow)$ is $(r\!-\!2)$-homogeneous of $1$-type and not

(4) The arguments of Corollary <ref> in conjunction with ($\ast$) show that the representations belonging to $(\sigma^{-3}_{K_r}(\Inf(M))\!\rightarrow) \subseteq \repp(K_r,r\!-\!1)$ are $(r\!-\!1)$-homogeneous and therefore belong to $\repp(K_r,s).$ Hence the corresponding Steiner bundles are $(s\!-\!1)$-homogeneous of $1$-type.  Since $\GL(A_s)$ is connected, none of these representations is  $\GL(A_s)$-homogeneous.

Let $s\!=\!2$ and consider $M\!=\!(\KK,\KK,(\id_\KK,\id_\KK)) \in \rep(K_2)$. Then $\cV(K_2,1)_M\!\ne\!\PP^1$ and $M$ is indecomposable, non-simple and not homogeneous. Proposition <ref>
thus provides Steiner bundles $\cF \in \StVect(\PP^{r-1})$ that are uniform but not $\GL(A_2)$-homogeneous.

§ STABILITY

Throughout this section, we assume that $r\!\ge\!3$. Let $\cF \in \Coh(\PP^{r-1})$ be a torsion-free sheaf. By definition, the $\cO_{\PP^{r-1},x}$-module $\cF_x$ is torsion-free for every $x \in \PP^{r-1}$. If $\cF\!\ne\!(0)$,
then $\rk(\cF)\!\ne\!0$ and we denote by

\[ \mu(\cF) := \frac{c_1(\cF)}{\rk(\cF)}\]

the slope of $\cF$. Recall that $\cF$ is called semistable (stable), provided

\[ \mu(\cG) \le \mu(\cF) \ \ \ \ (\mu(\cG) < \mu(\cF))\]

for every proper subsheaf $(0) \subsetneq \cG \subsetneq \cF$ (such that $\rk(\cG)\!<\!\rk(\cF)$).

In this section, we relate the stability of Steiner bundles on $\PP^{r-1}$ to that of the corresponding representations of $\EKP(K_r)\!=\!\repp(K_r,1)$. Let $M \in \EKP(K_r)$. As before, write $\TilTheta(M)\!:=\!\TilTheta_1(M)
$ and recall the defining exact sequence

\[ (0) \lra \widetilde{M_1} \stackrel{\TilTheta_M}{\lra} \widetilde{M_2} \lra \TilTheta(M) \lra (0),\]

where $\widetilde{M_i}\!=\!M_i\!\otimes_\KK\!\cO_{\PP^{r-1}}(i\!-\!2)$ for $i \in \{1,2\}$. We continue to write $\Delta_M\!:=\!\Delta_M(1)$, so that Corollary <ref> yields $\Delta_M\!\ge\!r\!-\!1$ in case $M_1\!\ne\!(0)$.

We begin by recalling the See-Saw Lemma:

Let $(0) \lra \cE \lra \cF \lra \cG \lra (0)$ be a short exact sequence of non-zero torsion-free coherent sheaves. Then we have $\mu(\cE)\! \leq\! \mu(\cF)\!\leq\!\mu(\cG)$ or $\mu(\cG)\!
\leq\!\mu(\cF)\!\leq\!\mu(\cE)$. Moreover, $\mu(\cH)\!=\!\mu(\cF)$ for $\cH \in \{\cE,\cG\}$ implies $\mu(\cE)\!=\!\mu(\cF)\!=\!\mu(\cG)$.

The Whitney sum formula implies

\[\mu(\cF)\!=\!\frac{\rk(\cE) \mu(\cE)+\rk(\cG) \mu(\cG)}{\rk(\cF)}.\]

If $\mu(\cF)\!\le \!\mu(\cE)$ and $\mu(\cF)\!< \!\mu(\cG)$, the foregoing formula yields

\[ \mu(\cF) = \frac{\rk(\cE)\!+\!\rk(\cG)}{\rk(\cF)} \mu(\cF) < \frac{\rk(\cE) \mu(\cE)\!+\!\rk(\cG) \mu(\cG)}{\rk(\cF)} = \mu(\cF),\]

a contradiction. Thus, $\mu(\cG)\!\le\!\mu(\cF)$, and the other assertions follow analogously.

§.§ Stability for $\EKP$-representations and Steiner bundles
The main result of this section recasts <cit.>, which was formulated for $\KK\!=\!\CC$, into our set-up.

For $(0)\! \neq \!M \in \EKP(K_r)$, we define

\[\mu(M) := \frac{\dim_\KK M_1}{\Delta_M}.\]

We say that $M$ is stable, provided $\mu(N)\! <\!\mu(M)$ for every subrepresentation $(0) \subsetneq N \subsetneq M$. We say that $M$ is semistable, provided $\mu(N)\! \leq\! \mu(M)$ for every
subrepresentation $(0) \subsetneq N \subsetneq M$.

We identify the Chow ring $A(\PP^{r-1})$ with the truncated polynomial ring $\ZZ[X]/(X^r)$. Theorem <ref> and Corollary <ref> then show that

\[ \mu(M) = \mu(\TilTheta(M))\]

for every non-zero $M \in \EKP(K_r)$. In particular, $\mu(\TilTheta(M))\!\ge\!0$ for every such $M$.

Let $M \in \EKP(K_r)\!\smallsetminus\!\{(0)\}$. The following statements hold:

* If $\TilTheta(M)$ is semistable, then $M$ is semistable.

* If $\TilTheta(M)$ is stable, then $M$ is stable.

Let $(0) \subsetneq N \subsetneq M$ be a proper submodule with canonical inclusion $\iota : N \hookrightarrow M$. There results a commutative diagram

\[ \begin{tikzcd}   (0) \arrow[r] & \widetilde{N_1} \arrow[r, "\TilTheta_N"] \arrow[d,"\tilde{\iota}_1"] & \widetilde{N_2} \arrow[d,"\tilde{\iota}_2"] \arrow[r] & \TilTheta(N) \arrow[r] \arrow[d,"\TilTheta(\iota)"] & (0) \\
(0) \arrow[r] & \widetilde{M_1} \arrow[r, "\TilTheta_M"]   & \widetilde{M_2}  \arrow[r] & \TilTheta(M) \arrow[r] & (0)
\end{tikzcd} \]

with exact rows. In view of $\msCoker \tilde{\iota}_1 \cong \widetilde{(M/N)_1}$ the Snake Lemma provides a monomorphism $\msKer \TilTheta(\iota) \hookrightarrow \widetilde{(M/N)_1}$. By definition,
$\widetilde{(M/N)_1} \cong (M_1/N_1)\!\otimes_\KK\cO_{\PP^{r-1}}(-1)$ is semistable (see <cit.>) and with slope $\mu( \widetilde{(M/N)_1})\!=\!-1$ whenever $ \widetilde{(M/N)_1}\!\ne\!(0).$

Since $\Coh(\PP^{r-1})$ is an abelian category, there is an exact sequence

\[ (\ast) \ \ \ \ \ \ (0) \lra \msKer \TilTheta(\iota) \lra \TilTheta(N) \lra \msIm\TilTheta(\iota) \lra (0)\]

and $\TilTheta$ being faithful entails $\msIm\TilTheta(\iota)\!\ne\!(0)$.

(1) Suppose that $\TilTheta(M)$ is semistable. If $\msKer \TilTheta(\iota)\!=\!(0)$, we have $\mu(N)\!=\!\mu(\TilTheta(N))\!=\!\mu(\msIm\TilTheta(\iota))\!\le\!\mu(\TilTheta(M))\!=\mu(M)$. Alternatively,
$\mu(\msKer\TilTheta(\iota))\!\le\!-1\!<\!\mu(\TilTheta(N))$ and Lemma <ref> in conjunction with ($\ast$) implies $\mu(N)\!=\!\mu(\TilTheta(N))\!<\!\mu(\msIm \TilTheta(\iota))\!\le\!\mu(\TilTheta(M))\!=\mu(M)
$. Hence $M$ is semistable.

(2) Suppose that $\TilTheta(M)$ is stable. Then $M$ is indecomposable and the assumption $N_1\!=\!M_1$ either implies $M\!=\!P_0(r)\!=\!N$, or $N_2 \subseteq M_2\!=\im\psi_M\!=\!\im\psi_N\subseteq N_2$, a
contradiction. If $\rk(\TilTheta(N))\!\geq\!\rk(\TilTheta(M))$, we obtain $\Delta_N\!\geq\!\Delta_M$, whence $\mu(N)\!<\!\mu(M)$. We may therefore assume that $\rk(\TilTheta(N))\!<\!\rk(\TilTheta(M))$.

If $\msKer\TilTheta(\iota)\!=\!(0)$, we have $\rk(\msIm\TilTheta(\iota))\!=\!\rk(\TilTheta(N))\!<\!\rk(\TilTheta(M))$ as well as $\mu(N)\!=\!\mu(\TilTheta(N))\!=\!\mu(\msIm\TilTheta(\iota))\!<\!\mu(\TilTheta(M))\!=\!\mu(M)$.
Alternatively, we may proceed as in (1).

There is a slope $\mu$, defined for non-zero, torsion-free coherent sheaves of $\Gr_d(A_r)$, such that

\[ \mu(\TilTheta_d(M)) = \mu(M)\]

for all $M \in \repp(K_r,d)\!\smallsetminus\!\{(0)\}$. In view of [11], the bundle $\cU_{(r,d)}$ is semistable relative to this slope, so that the foregoing result may be generalized to this context.

Let $M \in \repp(K_r,d)$ be non-projective and of minimal type. Then $M$ is stable.

Let $N \subsetneq M$ be a proper submodule. We shall show that $\mu(N)\!<\!\mu(M)$. Decomposing

\[ N = \bigoplus_{i=1}^n N_i\]

into its indecomposable constituents, Lemma <ref> implies $\mu(N) \le \max_{1\le i \le n} \mu(N_i)$. Hence we may assume $N$ to be indecomposable.

If $N$ is projective, then $\mu(N)\!\le\!\frac{1}{r-1}$. Assuming $\mu(N)\!\ge\!\mu(M)$, we have $d(r\!-\!d)\!+\!(d\!-\!1)\dim_\KK M_1=\!\Delta_M\!\ge\!(r\!-\!1)\dim_\KK M_1$, so that
$(r\!-\!d)\dim_\KK M_1\!\le\!d(r\!-\!d)$. Hence $\dim_\KK M_1\!\le\!d$, and the remark at the beginning of Section <ref> shows that $M$ is projective, a contradiction.

We may thus assume that $N$ is not projective, so that Corollary <ref>(1) implies $\Delta_N(d)\!\ge\!\Delta_M(d)$. In view of Corollary <ref>(2), the non-projective representation $M$ is indecomposable.
The assumption $N_1\!=\!M_1$ thus entails $N_2\!=\!\psi_M(A_r\!\otimes_\KK\! N_1)\!=\!\psi_M(A_r\otimes_\KK\! M_1)\!=\!M_2$, a contradiction. Consequently,

\begin{eqnarray*}
\dim_\KK N_1 \Delta_M & = & (\dim_\KK N_1)(\Delta_M(d)\!+\!(d\!-\!1)\dim_\KK M_1) = \dim_\KK N_1\Delta_M(d)\!+\!(d\!-\!1)\dim_\KK M_1\dim_\KK N_1\\
& < & \dim_\KK M_1 \Delta_N(d)\!+\!(d\!-\!1)\dim_\KK M_1\dim_\KK N_1 =  (\dim_\KK M_1)(\Delta_N(d)\!+\!(d\!-\!1)\dim_\KK N_1)\\
& = & \dim_\KK M_1\Delta_N,
\end{eqnarray*}

whence $\mu(N)\!<\!\mu(M)$. As a result, $M$ is stable.

For $d\!=\!1$ and $\Char(\KK)\!=\!0$, the foregoing result is a consequence of Proposition <ref>(2) and <cit.>.

§.§ The distribution of slopes
For $N \in \rep(K_r)$ with $N_1\!\ne\!(0)$ we put

\[ \nu(N) := \frac{\dim_\KK N_2}{\dim_\KK N_1}\]

and note that

\[ \mu(M) = \frac{1}{\nu(M)\!-\!1}\]

for every $M \in \EKP(K_r)$ such that $\dim_\KK M_1\!\ne\!0$. Given $r\! \geq\! 3$, recall that $L_r\! =\! \frac{r+\sqrt{r^2\!-\!4}}{2}$. We have $\frac{1}{L_r}\!=\! \frac{r\!-\!\sqrt{r^2\!-\!4}}{2}$ and $r\!-\!L_r\!=\!\frac{1}{L_r}$.

Let $M,N \in \EKP(K_r)$ be indecomposable with $M \not \cong P_0(r)$.

* We have $\lim_{n \to \infty} \nu(\sigma^{-n}_{K_r}(M))\!=\!L_r$.

* The sequence $(\mu(\sigma^{-n}_{K_r}(M)))_{n \in \NN_0}$ of slopes converges to $\frac{1}{L_r - 1}$.

* If $M$ is regular, then the sequence $(\mu(\sigma^{-n}_{K_r}(M)))_{n \in \NN_0}$ is strictly decreasing.

* If $M$ is regular and $N$ is preprojective, then $\mu(N)\! <\! \frac{1}{L_r -1}\!<\!\mu(M)$.

We write $\udim\sigma^{-n}_{K_r}(M)\!=:\!(x_n,y_n)$ for all $n \in \NN_0$ and recall that

\[\udim \sigma^{-1}_{K_r}(X) = (\dim_\KK X_2,r\dim_\KK X_2\!-\!\dim_\KK X_1)\]

for each representation $X$ without $S_1$ as a direct summand. Theorem <ref> yields $\sigma^{-n}_{K_r}(M) \in \EKP(K_r)$ and $\sigma^{-n}_{K_r}(M)$ is regular or preprojective
with $x_n\!\neq\!0$. For $n \in \NN_0$ we set

\[d_n(M) := \nu(\sigma^{-n}_{K_r}(M))\]

and note that $\sigma^{-n}_{K_r}(M) \in \EKP(K_r)$ implies $d_n(M)\!>\!1$.

(1) We first consider the case where $M$ is preprojective.  Kac's Theorem implies $0\!<\!q_r(\udim \sigma^{-n}_{K_r}(M))$ $=\!q_r(x_n,y_n)\!=\!x_n^2\!+\!y_n^2\!-\!rx_n y_n$, whence

\[ (\ast) \ \ \ \ \ \ d_n(M) = \frac{y_n}{x_n} > r\!-\!\frac{x_n}{y_n} = \frac{r y_n\!-\!x_n}{y_n} = d_{n+1}(M) > 1.\]

Now let $M$ be regular and note that $f\!:=\!X^2\!-\!rX\!+\!1 \in \RR[X]$ has roots $L_r,\frac{1}{L_r}$. As $r\!\geq\!3$, Kac's Theorem yields $0\!>\!x_n^2\!+\!y_n^2\!-\!rx_n y_n$, so that $f(d_n(M))\!<\!0$ and $d_n(M)\!<\!L_r$.
Moreover, the computation of ($\ast$) implies $d_n(M)\!<\!d_{n+1}(M)$.

We have shown that the sequence $(d_n(M))_{n \in \NN_0}$ converges in both cases and the identity

\[ d_{n+1}(M) = \frac{ry_n-x_n}{y_n} = r\!-\!\frac{1}{d_n(M)}\]

implies that $d\!:=\!\lim_{n \to \infty} d_n(M)$ is a root of $f$. Since $\frac{1}{L_r}\!<\!1\!<\!d_n(M)$ for all $n \in \NN_0$, it follows that $d\!=\!L_r$.

(2) We have $\mu(\sigma^{-n}_{K_r}(M))\!=\!\frac{1}{d_n(M)-1}$. Since $d_n(M)\!>\!1$ for all $n \in \NN_0$, we conclude with (1) that $(\mu(\sigma^{-1}_{K_r}(M)))_{n \in \NN_0}$ converges to $\frac{1}{L_r-1}$.

(3) Let $M$ be regular. As noted in the proof of (1), the sequence $(d_n(M))_{n \in \NN_0}$ is strictly increasing, so that the identity $\mu(\sigma^{-n}_{K_r}(M))\!=\!\frac{1}{d_n(M)-1}$ implies the assertion.

(4) Let $M$ be regular. Then $1\!<\!d_0(M)\!<\!L_r$ and therefore $\mu(M)\!=\!\frac{1}{d_0(M)-1}\!>\!\frac{1}{L_r-1}$. If $N\!=\!P_0(r)$, then $\mu(N)\!=\!0$. Alternatively, $d_0(N)$ is well-defined and $q_r(\udim N)\!>\!0$
yields $d_0(N)\!>\!L_r$. This shows $\mu(N)\!=\!\frac{1}{d_0(N)-1}\!<\!\frac{1}{L_r-1}$.

We continue by recording three important consequences.

Let $M \in \EKP(K_r)$ be indecomposable.

* The representation $M$ is (semi)stable if and only if $\sigma_{K_r}^{-1}(M) \in \EKP(K_r)$ is (semi)stable.

* The following statements are equivalent:

* All representations in $\{ \sigma_{K_r}^z(M) ; z \in \ZZ\} \cap \EKP(K_r)$ are (semi)stable[For $M = P_{\ell}(r)$ we only consider $z \in \ZZ_{\geq -\ell}$.].

* The representation $M$ is (semi)stable.

* If $M$ is regular and semistable, then $M$ is quasi-simple.

Let $Z \in \EKP(K_r)$. Recall that we have $\sigma_{K_r}^{-1}(Z) \in \EKP(K_r)$ and

\[\udim \sigma_{K_r}^{-1}(Z) = ( \dim_{\KK} Z_2,r \dim_{\KK} Z_2\! -\! \dim_{\KK} Z_1).\]

Hence we obtain for $U,V \in \EKP(K_r)$:

\[ (\ast) \quad \mu(\sigma_{K_r}^{-1}(U)) \leq(<)  \mu(\sigma_{K_r}^{-1}(V)) \Leftrightarrow \dim_{\KK} U_1 \dim_{\KK} V_2 \leq(<) \dim_{\KK} V_1 \dim_{\KK} U_2 \Leftrightarrow \mu(U) \leq(<) \mu(V).\]

(1) Since $\mu(P_0(r)) = 0 < \mu(P_1(r))$ and $\sigma^{-1}_{K_r}(P_{\ell}(r)) = P_{\ell+1}(r))$ for all $\ell \in \NN_0$, we obtain $\mu(P_{\ell}(r)) < \mu(P_{\ell+1}(r))$ for all $\ell \in \NN_0$. Clearly, $P_0(r)$ is stable. Now we fix $\ell \in \NN$ and let $(0) \subsetneq X \subsetneq P_{\ell}(r)$ be a subrepresentation. We fix a direct summand $Y \subseteq X$ of maximal slope. Then Lemma <ref> yields $\mu(Y) \geq \mu(X)$. Since the only proper indecomposable subrepresentations of $P_{\ell}(r)$ are (up to isomorphism) $P_0(r),\ldots,P_{\ell-1}(r)$, we conclude $\mu(P_{\ell}(r))\! >\! \mu(P_{\ell-1}(r))\! \geq\! \mu(Y)\! \geq\! \mu(X)$. Hence every preprojective indecomposable representation is stable and we only have to consider the case that $M$ is regular.

At first we assume that $M$ is stable. Let $(0) \subsetneq X \subsetneq N\!:=\!\sigma_{K_r}^{-1}(M)$, and assume that $X$ is indecomposable. If $X$ is preprojective, then Proposition <ref>(4) implies $\mu(X)\!<\!\mu(N)$.
Alternatively, $X$ is regular. Since $\sigma_{K_r}$ is left exact and an equivalence on regular modules, there is a proper inclusion $(0)\!\ne\!\sigma_{K_r}(X) \hookrightarrow M$. As $M$ is stable, we obtain $\mu(\sigma_{K_r}(X))\!<\!
\mu(M)$ and $(\ast)$ implies $\mu(X)\!\le\!\mu(Y)$. If $X$ is not indecomposable, we consider an indecomposable summand $Y \subseteq X$ of maximal slope and argue as in the preprojective case. This shows that $\sigma_{K_r}^{-1}(M)$ is stable.

Now we assume that $\sigma_{K_r}^{-1}(M)$ is stable and let $(0) \subsetneq X \subsetneq M$ be a subrepresentation of maximal slope. We show that $\mu(X)\! <\! \mu(M)$. If $M/X \in \add(I_0(r))$, then we find $a \in \NN$ such that $\udim X = (\dim_{\KK} M_1-a,\dim_{\KK}M_2)$ and conclude with $\dim_{\KK} M_2 \neq 0$ that

\[ \mu(X) = \frac{\dim_{\KK} M_1-a}{\Delta_M + a} < \frac{\dim_{\KK} M_1}{\Delta_M} = \mu(M).\]

Hence we may assume that $M/X\!=\!Q\! \oplus \! aI_0(r)$ with $a \in \NN_0$ and $Q \in \rep_1(K_r)$ non-zero. We find a subrepresentation $0 \subsetneq Y \subsetneq M$ with $\udim Y\!=\!\udim X\!+\!(a,0)$ and maximality of $\mu(X)$ implies

\[  \frac{\dim_{\KK} X_1+a}{\Delta_X - a} = \mu(Y) \leq \mu(X) =\frac{\dim_{\KK} X_1}{\Delta_X}.\]

This can only happen if $a\!=\!0$. Hence we have a short exact sequence

\[ (0) \lra X \lra M \lra Q \lra (0)\]

in $\rep_1(K_r)$. Since $\sigma^{-1}_{K_r}: \rep_1(K_r)\lra \rep_2(K_r)$ is exact (cf. Section <ref>), we obtain  a proper  inclusion $(0) \neq \sigma^{-1}_{K_r}(X) \hookrightarrow \sigma_{K_r}^{-1}(M)$ and $\sigma_{K_r}^{-1}(M)$ being stable in conjunction with $(\ast)$ implies that $\mu(X) < \mu(M)$. Hence $M$ is stable.
The arguments concerning semistability are analogous.

(2) This is a direct consequence of (1).

(3)  We write $M\!=\!N_{[\ell]}$ with quasi-socle $N$ and assume that $\ell\! \geq\! 2$. General AR-theory provides a short exact sequence

\[ (0) \lra N \lra M \lra \tau_{K_r}^{-1}(N_{[\ell\!-\!1]}) \lra (0)\]

in $\EKP(K_r)$ and the See-Saw Lemma for representations give us $\mu(N)\!\leq\! \mu(M)\! \leq \! \mu(\tau_{K_r}^{-1}(N_{[\ell\!-\!1])})$.
Moreover, $N_{[\ell\!-\!1]}$ being a subrepresentation of $M\!=\!N_{[\ell]}$ and Proposition <ref>(3) imply

\[ \mu(M) \leq \mu(\tau_{K_r}^{-1}(N_{[\ell\!-\!1]})) < \mu(N_{[\ell\!-\!1]}) \leq \mu(M),\]

a contradiction. Hence $\ell =  1$ and $M$ is quasi-simple.

(1) Owing to Corollary <ref>, a cone of a regular component either contains no (semi)stable representations or all of its quasi-simple vertices are (semi)stable.

(2) Let $r \geq 3$ and $B_1,B_2$ be non-isomorphic indecomposable representations with dimension vector $(1,1)$. Clearly, $B_1,B_2$ are elementary representations and by <cit.> $B_1$ and $B_2$ are $\Hom$-orthogonal. We conclude $\dim_{\KK} \Ext^1_{K_r}(B_2,B_1)\!=\!  \dim_{\KK} \Hom_{K_r}(B_2,B_1)\! -\!q_r(1,1)\!  =\! r\!-\!2\! >\! 0$. In particular, we find a non-split short exact sequence

\[(\ast) \quad (0) \lra B_1 \lra N \lra B_2 \lra (0)\]

and conclude with <cit.> $\dim_{\KK} \End_{K_r}(N)\! <\! \dim_{\KK} \End(B_1\! \oplus\! B_2)\! =\! 2$. Hence $N$ is a brick with dimension vector $(2,2)$ and regular by Kac's Theorem. Since regular bricks for $K_r$ are quasi-simple (see <cit.>), we conclude that every quasi-simple vertex in the regular component $\cC$ of $N$ is a brick.

There exists $n \in \NN$ such that $\tau^{-n}_{K_r}(B_1),\tau^{-n}_{K_r}(B_2), \tau^{-n}_{K_r}(N) \in \EKP(K_r)$. Recall that $\tau^{-n}_{K_r}$ is an exact endo-functor of the category $\reg(K_r)$ of regular representations.
In view of ($\ast$), there results an exact sequence

\[ (0) \lra \tau_{K_r}^{-n}(B_1) \lra \tau^{-n}_{K_r}(N) \lra \tau^{-n}_{K_r}(B_2)\lra (0),\]

whose (nonzero) extreme terms have equal dimension vectors and hence equal slopes. The See-Saw Lemma now shows that $\tau^{-n}_{K_r}(N)$ is not stable. By Corollary <ref>(2), $\cC\cap\EKP(K_r)$ does not contain any stable vertices.

Let $E \in \EKP(K_r)$ be elementary. Then $E$ is stable.

Let $(0) \subsetneq X \subsetneq E$ be a proper indecomposable subrepresentation. In view of Proposition <ref> we may assume that $X$ is regular. Thanks to <cit.>, the factor module $E/X$ is
preinjective. Consequently, $\Delta_{E/X}\!<\!0$, so that $\Delta_E\!=\!\Delta_X\!+\!\Delta_{E/X}\!<\!\Delta_X$. As $\dim_\KK X_1\ne\!0$, we obtain

\[ \mu(X) < \frac{\dim_\KK X_1}{\Delta_E} \le \mu(E).\]

As in the proof of Corollary <ref>, this implies the assertion.

§.§ Filtrations
As before, we assume that  $r\!\ge\!3$ and let

\[\sigma_r\!:=\!\begin{pmatrix}
r & -1 \\
1 & 0
\end{pmatrix} \in \GL_2(\ZZ)\]

be the invertible matrix that describes the $\sigma_{K_r}$-orbits on the level of the Grothendieck group $K_0(\rep(K_r))$: For $M \in \rep_2(K_r)$ we have $\udim \sigma_{K_r}(M)\!=\!\sigma_r.\udim M$, while
$\udim \sigma^{-1}_{K_r}(N)\!=\!\sigma_r^{-1}.\udim N$ for $N \in \rep_1(K_r)$, cf. Section <ref>.

We denote by $\lfloor \ \rfloor : \RR \lra \ZZ$ the floor function.

Let $(a,b) \in \NN^2\!\smallsetminus\!\{0\}$ be such that $q_r(a,b)\!<\!0$ and $b\!=\!\lfloor a L_r \rfloor$. Let $M \in \rep(K_r)$ be an indecomposable representation with $\udim M\!=\! \sigma_{r}^{-
\ell}.(a,b)$ for some $\ell \in \NN$. Then $M$ is regular and there is a filtration

\[(0) = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_{n-1} \subsetneq M_n = M\]

such that for each $i \in \{1,\ldots,n\}$ the filtration factor $M_i/M_{i-1}$ is elementary and belongs to $\EKP(K_r)$.

As $q_r(\udim M)\!=\!q_r(a,b)\!<\!0$, Kac's Theorem ensures that the representation $M$ is regular. We define $N\!:=\!\sigma_{K_r}^{\ell}(M)$ and note that $\udim N\!=\!(a,b)$. Thanks to <cit.>
there is a filtration

\[(0) = N_0 \subsetneq N_1 \subsetneq \cdots \subsetneq N_{n-1} \subsetneq N_n = N\]

such that for each $j \in \{1,\ldots,n\}$ the filtration factor $E^j := N_j/N_{j-1}$ is elementary. By the same token, there is at most one $i \in \{1,\ldots,n\}$ such that $E^i$ is not in $\EKP(K_r)$ and in this case there is $\fv \in
\Gr_1(A_r)$ such that $E^i \cong E(\fv)$. We have $\udim E(\fv) = (1,r\!-\!1)$ and therefore $\udim \sigma^{-1}_{K_r}(E^i) = (r\!-\!1,r^2\!-\!r\!-\!1)$. The representation $X\!:=\!\sigma^{-1}_{K_r}(E^i)$ satisfies

\[ q_r(\udim X)\!+\!\dim_\KK X_2\!-\!\dim_\KK X_1 = (r\!-\!2)(r\!-\!1)\geq 1\]

and <cit.> ensures that $X \in \EKP(K_r)$. Since $\sigma^{-1}_{K_r}$ is exact on the category of regular representations and $\EKP(K_r)\!=\!\repp(K_r,1)$ is closed under $\sigma_{K_r}^{-1}$ (see Theorem
<ref>), we conclude that

\[(0) = \sigma_{K_r}^{-\ell}(N_0) \subsetneq \sigma_{K_r}^{-\ell}(N_1) \subsetneq \cdots \subsetneq \sigma_{K_r}^{-\ell}(N_{n-1}) \subsetneq \sigma_{K_r}^{-\ell}(N_n) = M\]

is the desired filtration.

For $(a,b) \in \NN^2_0$ and $d \in \{1,\ldots, r\!-\!1\}$ we put

\[ \Delta_{(a,b)}(d):=b\!-\!da.\]

Let $(a,b) \in \NN^2\!\smallsetminus\!\{0\}$ be such that $q_r(a,b)\!<\!0$ and $q_r(a,b)\!+\!\Delta_{(a,b)}(r\!-\!1)\!\geq\!1$. For $\ell \in \NN_0$, we let $(a_\ell,b_\ell)\!:=\!\sigma_r^{-\ell}.(a,b)$. Then the
following statements hold:

* We have $q_r(a_\ell,b_\ell)\!+\Delta_{(a_{\ell},b_{\ell})}(r\!-\!1)\geq\! 1$.

* Let $M$ be an indecomposable representation with dimension vector $(a_\ell,b_\ell)$. Then $M$ has a filtration such that each filtration factor is an elementary representation belonging to $\EKP(K_r)$. In particular,
$M \in \EKP(K_r)$.

Let $(x,y) \in \NN^2$. Direct computation shows that $\Delta_{\sigma_r^{-1}.(x,y)} (r\!-\!1)= \Delta_{(x,y)}$.

(1) We prove the statement by induction on $\ell\!\geq\!0$, the case $\ell\!=\!0$ being trivial. For $\ell\!>\!0$ we obtain

\begin{align*}
q_r(a_\ell,b_\ell)\!+\!\Delta_{(a_{\ell},b_{\ell})}(r\!-\!1) &= q_r(a_\ell,b_\ell)\!+\!\Delta_{\sigma_r^{-1}.(a_{\ell-1},b_{\ell-1})}(r\!-\!1) \\
&=  q_r(a_{\ell-1},b_{\ell-1})\!+\! \Delta_{(a_{\ell-1},b_{\ell-1})}\\
&> q_r(a_{\ell-1},b_{\ell-1})\!+\!\Delta_{(a_{\ell-1},b_{\ell-1})}(r\!-\!1) \geq 1.
\end{align*}

(2) In view of Lemma <ref>, it suffices to show that $(x,y)\!:=\! \sigma_r.(a_\ell,b_\ell)$ satisfies $q_r(x,y)\!<\!0$ and $y\!=\!\lfloor x L_r \rfloor$. Clearly, we have $q_r(x,y)\!=\!q_r(a_\ell,b_\ell)\!=\!q_r(a,b)\!<\!0$.
By definition, we obtain

\[ q_r(x,y)\! +\! y\! -\! x = q_r(a_\ell,b_\ell)\!+\! \Delta_{\sigma_r.(a_{\ell},b_{\ell})} =
q_r(a_\ell,b_\ell)\!+\! \Delta_{(a_{\ell},b_{\ell})}(r\!-\!1),\]

so that (1) implies $q_r(x,y)\!+\!y\!-\!x\!\geq\!1$. As $q_r(x,y)\!<\!0$, we have $x\!<\!y$ and <cit.> yields $y\!=\! \lfloor x L_r \rfloor$.

Let $\cC \subseteq \Gamma(K_r)$ be a regular component and assume that $X \in \cC$ is quasi-simple such that $q_r(\udim X)\!+\!\Delta_X(r\!-\!1)\!\geq\!1$. Then every representation in
$(X\! \rightarrow)$ possesses a filtration whose filtration factors are elementary and belong to $\EKP(K_r)$.

Let $M \in (X\!\rightarrow)$, then each of the filtration factors $M_{[i]}/M_{[i-1]}$ of the quasi-composition series of $M$ is isomorphic to $\tau^{-\ell_i}_{K_r}(X) \cong \sigma_{K_r}^{-2\ell_i}(X)$ for some
$\ell_i \in \NN_0$. By Lemma <ref>, each of these representations possesses a filtration such that each filtration factor is elementary and in $\EKP(K_r)$.

Let $M \in \rep(K_r)$ be regular indecomposable. In view of Lemma <ref> and $q_r(\udim\sigma^{-1}_{K_r}(M))\!=\!q_r(\udim M)$, it follows that every regular
AR-component $\cC \subseteq \Gamma(K_r)$ contains a quasi-simple representation $X$ such that $q_r(\udim X)\!+\!\Delta_X(r\!-\!1)\!\ge\!1$.

§.§ The Harder-Narasimhan filtration

Let $\cF \in \Coh(\PP^{r-1})$ be a torsion-free coherent sheaf. There exists a uniquely determined filtration, the Harder-Narasimhan filtration (cf. <cit.>),

\[ (0) = \cF_0 \subsetneq \cF_1 \subsetneq \cF_2 \subsetneq \cdots \subsetneq \cF_{t-1} \subsetneq \cF_t = \cF\]

that satisfies the following two properties:

* Each quotient $\cF^i\!:=\!\cF_i/\cF_{i-1}$ is semistable ($i \in \{1,\ldots,t\}$).

* We have $\mu(\cF^1)\!>\!\mu(\cF^2)\!> \cdots >\!\mu(\cF^{t-1})\!>\!\mu(\cF^t)$.

The sheaf $\cF$ is semistable if and only if $t\!=\!1$.

Let $\cF \in \Coh(\PP^{r-1})$ be  a non-zero, torsion-free sheaf that is not semistable. There exists a non-zero subsheaf $\cU \subsetneq \cF$ with the following properties:

(a) $\cF/\cU$ is semistable,

(b) $\mu(\cU)\!>\!\mu(\cF)$,

(c) $\Hom_{\PP^{r-1}}(\cS,\cF/\cU)\!=\!(0)$ for each semistable sheaf $\cS \in \Coh(\PP^{r-1})$ with $\mu(\cF)\!\leq\!\mu(\cS)$.

We consider the Harder-Narasimhan filtration

\[ (0) = \cF_0 \subsetneq \cF_1 \subsetneq \cF_2 \subsetneq  \cdots \subsetneq \cF_{t-1} \subsetneq \cF_t = \cF\]

with $t\!\geq\!2$ and set $\cU\!:=\!\cF_{t-1}$. Then (a) holds by definition. Using Lemma <ref> and the defining property (2), induction on $s$ implies that $\mu(\cF_{s-1})\!>\!\mu(\cF_s)\!>\!\mu(\cF^s)$ for $s \in
\{2,\ldots,t\}$. Consequently, (b) holds as well. Lemma <ref> also yields

\[ \mu(\cF/\cU) = \mu(\cF^t) < \mu(\cF) \leq \mu(\cS)\]

for each semistable sheaf $\cS \in \Coh(\PP^{r-1})$ with $\mu(\cF)\!\leq\!\mu(\cS)$.
Let $f : \cS \lra \cF/\cU$ be a morphism for such a sheaf $\cS$. As $\cF/\cU$ and $\cS$ are semistable, the assumption $f\!\neq\!0$ in conjunction with Lemma <ref> implies

\[  \mu(\cF/\cU) \geq \mu(\im f) \geq \mu(\cS), \]

which contradicts $\mu(\cF/\cU)\!<\!\mu(\cS)$.

§.§ The Beilinson Spectral Sequence
For $\cF \in \Coh(\PP^{r-1})$ and $n \in \ZZ$, we write $\cF(n)$ for the $n$-th Serre twist of $\cF$.

We denote by $\Omega$ the cotangent bundle of $\PP^{r-1}$ (see <cit.>), and, for $m \in \NN_0$, we let $\Omega^m\!:=\!\bigwedge^m_{\PP^{r-1}} \Omega$ be
the $m$-th exterior power of $\Omega$, where $\Omega^0\!:=\!\cO_{\PP^{r-1}}$.

Let $-r\!<\!b\!\leq\!0$ and $i \in \NN_0$. For $m \in \NN_0$ we have

\[ \HH^i(\PP^{r-1},\Omega^{m}(m\!+\!b)) \cong  \left\{ \begin{array}{cc}  \KK & \text{if} \ i\!=\!m\!=\!-b \\ (0) & \text{otherwise.} \end{array} \right. \]

For $i\!\geq\!r$ there is nothing to show (see <cit.>). Hence we assume that $0\!\leq\!i\!\leq\!r\!-\!1$. By Bott's formula (see <cit.>) the only cases to consider are

* $i\!=\!0$, $0\!\leq\!m\!<\!r$ and $m\!+\!b\!>\!m$: This can not happen since $b\!\leq\!0$.

* $m\!+\!b\!=\!0$ and $m\!=\!i$: Then $\dim_\KK \HH^i(\PP^{r-1},\Omega^{m}(m\!+\!b))\!=\!1$.

* $i\!=\!r\!-\!1$, $0\!\leq\!m\!<\!r$, $m\!+\!b\!\leq\!m\!-\!r$: This can not happen since $b\!> \!-r$.

(cf. <cit.>)
Let $\cF \in \Coh(\PP^{r-1})$ be a coherent sheaf, $-(r\!-\!1)\!<\!d\!\leq\!0$ and

\[ (0) \lra \widetilde{V_1}(d) \lra \widetilde{V_2}(d) \lra \cF \lra(0)\]

be a short exact sequence. If $\HH^i(\PP^{r-1},\cF\!\otimes_{\PP^{r-1}}\!\Omega^m(m))\!\neq\!(0)$, then $i\!=\!-d$ and $m \in \{-d,-(d\!-\!1)\}$,

Upon tensoring the exact sequence over $\cO_{\PP^{r-1}}$ with the vector bundle $\Omega^m(m)$, we obtain the exact sequence

\[ (0) \lra V_1\!\otimes_\KK\!\Omega^m(m\!+\!d\!-\!1)  \lra V_2\!\otimes_\KK\!\Omega^m(m\!+\!d) \lra  \cF\!\otimes_{\PP^{r-1}}\!\Omega^m(m) \lra (0).\]

The long exact sequence in cohomology provides an exact sequence

\[V_2\! \otimes_\KK\! \HH^{i}(\PP^{r-1},\Omega^m(m\!+\!d)) \lra \HH^{i}(\PP^{r-1},\cF\!\otimes_{\PP^{r-1}}\!\Omega^m(m)) \lra V_1\!\otimes_\KK\!\HH^{i+1}(\PP^{r-1},\Omega^m(m\!+\!d\!-\!1)).\]

Since $-r\!<\!d\!-\!1,d\! \leq\! 0$,  Lemma <ref> shows, that the assumption $\HH^i(\PP^{r-1},\cF\!\otimes_{\PP^{r-1}}\!\Omega^m(m))\!\neq\!(0)$ entails either $i\!=\!m\!=\!-d$ or
$i\!+\!1\!=\!m\!=-(d\!-\!1)$. Hence $(i,m)\!=\!(-d,-d)$ or $(i,m)\!=\!(-d,-(d\!-\!1))$.

(see <cit.>) Let $\cF \in \Coh(\PP^{r-1})$ be a coherent sheaf. There is a second quadrant spectral sequence $E\!=\!E_{\cF}$ with first page

\[ E^{pq}_1 = \HH^q(\PP^{r-1},\cF\!\otimes_{\PP^{r-1}}\!\Omega^{-p}(-p))\!\otimes_\KK\!\cO_{\PP^{r-1}}(p)\]

for $(-p,q) \in \{0,\ldots,r\!-\!1\}^2$ and zero otherwise, which converges to

\[E^{p+q} \Rightarrow \begin{cases}
\cF, \ \text{for} \ p\!+\!q \!=\!0 \\
(0), \ \text{for} \ p\!+\!q\!\neq\!0.
\end{cases}
\]

Moreover, this sequence is natural in $\cF$.

Given $-(r\!-\!1)\!<\!d\!\leq\!0$ and $\cF \in \Coh(\PP^{r-1})$, we denote the differential $d_1^{d-1,-d}$ on the first page of the spectral sequence $E_\cF$ by

\[ \alpha_{\cF,d}  : E_1^{d-1,-d} \lra E_1^{d,-d}.\]

Let $\cF$ be a Steiner bundle, $-(r\!-\!1)\!<\!d\!\leq\!0$. There exists a short exact sequence

\[ (0) \lra E_1^{d-1,-d} \stackrel{\alpha_{\cF(d),d}}{\lra} E_1^{d,-d} \lra \cF(d) \lra (0).\]

By Corollary <ref> we know that the first page of the spectral sequence $E_{\cF(d)}$ has at most two non-zero entries $E_1^{d-1,-d}\!=
\!\HH^{-d}(\PP^{r-1},\cF(d)\!\otimes_{\PP^{r-1}}\! \Omega^{-(d-1)}(-(d\!-\!1)))\!\otimes_\KK\!\cO_{\PP^{r-1}}(d\!-\!1)$ and $E_1^{d,-d}\!=\!\HH^{-d}(\PP^{r-1},\cF(d)\!\otimes_{\PP^{r-1}}\!\Omega^{-d}(-d))\!
\otimes_\KK\!\cO_{\PP^{r-1}}(d)$. In particular, $E_{\cF(d)}$ collapses at $E_2$, and we have $E_\infty\!=\!E_2$. Since $E_1^{d+1,-d}\!=\!(0)\!=\!E_1^{d-2,-d}$, we obtain

\[ E_2^{d,-d} = \ker d_1^{d,-d}/\im \alpha_{\cF(d),d} = E_1^{d,-d} / \im \alpha_{\cF(d),d} = \coker \alpha_{\cF(d),d} \ \ \text{and} \ \ E_2^{d-1,-d} = \ker \alpha_{\cF(d),d} \]

while all other terms vanish. Now Theorem <ref> implies $\ker \alpha_{\cF(d),d}\!=\!(0)$ and $\cF(d) \cong \coker \alpha_{\cF(d),d}$.
There results the short exact sequence

\[ (0) \lra E_1^{d-1,-d} \stackrel{\alpha_{\cF(d),d}}{\lra} E_1^{d,-d}  \lra \cF(d) \lra (0). \qedhere\]

§.§ Stability in $\EKP(K_3)$ and $\StVect(\PP^2)$

In terms of its methods, this section is based on those set forth in <cit.> (see also <cit.>).

Let $\cF \in \Coh(\PP^{r-1})$ be torsion-free, $b \in \ZZ$. Then $\cF$ is (semi)stable if and only if the same is true for $\cF(b)$.

Note that $\cO_{\PP^{r-1}}(b)\!\otimes_{\PP^{r-1}}\!-$ induces an exact auto-equivalence on the category of coherent sheaves. Given a torsion-free coherent sheaf $\cG\!\neq\!(0)$, the sheaf $\cG(b)$ is locally
isomorphic to $\cG$ and hence also torsion-free and of the same rank. We conclude with <cit.>

\[ \mu(\cO_{\PP^{r-1}}(b)\!\otimes_{\PP^{r-1}}\!\cG) = \mu(\cG(b)) = \frac{c_1(\cG(b))}{\rk(\cG)} = \frac{c_1(\cG)\!+\!b\rk(\cG)}{\rk(\cG)} = \mu(\cG)\!+\!b.\]

Hence the equivalence respects the order of slopes.

Let $M \in \EKP(K_3)$. Then the following statements hold:

* If $\Delta_M(2)\!\ge\!0$ and $M$ is semistable, then $\TilTheta(M)$ is semistable.

* If $\Delta_M(2)\!>\!0$ and $M$ is stable, then $\TilTheta(M)$ is stable.

We begin the proof by listing several properties that both cases have in common.

Given $\cU \in \Coh(\PP^2)$, we consider the differentials

\[\alpha_{\cU,-1} \colon \HH^{1}(\PP^{2},\cU\!\otimes_{\PP^2}\!\Omega^{2}(2))\!\otimes_\KK\!\cO_{\PP^2}(-2)  \lra  \HH^{1}(\PP^{2},\cU\!\otimes_{\PP^2}\!\Omega^{1}(1))\!\otimes_\KK\!\cO_{\PP^2}(-1)\]

\[\delta_{\cU,-1} \colon \HH^{2}(\PP^{2},\cU\!\otimes_{\PP^2}\!\Omega^{2}(2))\!\otimes_\KK\! \cO_{\PP^2}(-2)  \lra  \HH^{2}(\PP^{2},\cU\!\otimes_{\PP^2}\!\Omega^{1}(1))\!\otimes_\KK\!\cO_{\PP^2}(-1)\]

from the first page of the Beilinson spectral sequence $E_\cU$, see Theorem <ref>.

Let $\cF\!:=\!\TilTheta(M)(-1)$ and $\cU \subsetneq \cF$ be a subsheaf. The inclusion $\cU \hookrightarrow \cF$
induces $\KK$-linear maps

\[ \tau_i : \HH^1(\PP^{2},\cU\!\otimes_{\PP^2}\!\Omega^i(i)) \lra  \HH^1(\PP^{2},\cF\!\otimes_{\PP^2}\!\Omega^i(i))\]

for $i \in \{1,2\}$.

Throughout, we assume that $\cU \subsetneq \cF$ is a non-zero subsheaf such that

\[ (\star) \ \ \ \ \ \ \ \Hom_{\PP^2}(\cO_{\PP^2}(\ell),\cF/\cU)\!=\!(0) \ \ \text{for all} \ \ \ell\!\geq\!0.\]

(a) $\tau_1$ is injective.

Recall that $\HH^0(\PP^2,\cG)$ is naturally isomorphic to the global section of $\cG$ for every $\cO_{\PP^2}$-module (see <cit.>). Hence the isomorphism $\Omega^1(1)^\vee \cong \cT(-1)$ and <cit.> imply

\[
\HH^0(\PP^2,(\cF/\cU)\!\otimes_{\PP^2}\!\Omega^1(1)) \cong \HH^0(\PP^2,(\Omega^1(1)^{\vee})^{\vee}\!\otimes_{\PP^2}\!(\cF/\cU)) \cong\Hom_{\PP^2}(\cT(-1),\cF/\cU).
\]

Application of the left exact functor $\Hom_{\PP^2}(-,\cF/\cU)$ to the Euler sequence $(0) \lra \cO_{\PP^2}(-1) \lra \cO_{\PP^2}^3 \lra \cT(-1) \lra (0)$ yields $\dim_\KK \Hom_{\PP^2}(\cT(-1),\cF/\cU)\!\leq\!
\dim_\KK\Hom_{\PP^2}(\cO_{\PP^2}^3,\cF/\cU) \stackrel{(\star)}{=}0$. Hence we have $\HH^0(\cO_{\PP^2},(\cF/\cU)\!\otimes_{\PP^2}\!\Omega^1(1))\!=\!(0)$. The long exact sequence in cohomology yields the exactness of

\[ \HH^0(\PP^2,(\cF/\cU)\!\otimes_{\PP^2}\!\Omega^{1}(1)) \lra \HH^1(\PP^{2},\cU\!\otimes_{\PP^2}\!\Omega^{1}(1)) \stackrel{\tau_1}{\lra} \HH^{1}(\PP^{2},\cF\!\otimes_{\PP^2}\! \Omega^{1}(1)).\]

Consequently, $\tau_1$ is injective. $\diamond$

(b) $\tau_2$ is injective.

Upon tensoring the sequence $(0) \lra \cU \lra \cF \lra \cF/\cU \lra (0)$ with the vector bundle $\Omega^2(2)$ the long exact sequence in cohomology yields the exactness of

\[ \HH^{0}(\PP^2,(\cF/\cU)\!\otimes_{\PP^2}\!\Omega^{2}(2)) \lra \HH^1(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^2(2)) \stackrel{\tau_2}{\lra} \HH^{1}(\PP^2,\cF\!\otimes_{\PP^2}\!\Omega^{2}(2)).\]

By <cit.>, we have $\Omega^2(2) \cong \cO_{\PP^2}(-1)$. The choice of $\cU$ implies

\[ \HH^0(\PP^2,(\cF/\cU)\!\otimes_{\PP^2}\!\Omega^2(2)) \cong \Hom_{\PP^2}(\cO_{\PP^2}(1),\cF/\cU) \stackrel{(\star)}{=}(0),\]

so that $\tau_2$ is injective.  $\diamond$

(c) $\alpha_{\cU,-1}$ is injective.

Corollary <ref> and the naturality of the Beilinson spectral sequence yield the following commutative diagram

\[ (\dagger) \ \ \ \ \ \
\xymatrix{
& \HH^1(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^2(2))\!\otimes_\KK\!\cO_{\PP^2}(-2) \ar[d]^{\tau_2 \otimes \id_{\cO_{\PP^2}(-2)}} \ar^{\alpha_{\cU,-1}}[r] & \HH^1(\PP^{2},\cU\!\otimes_{\PP^2}\!
\Omega^1(1))\!\otimes_\KK\! \cO_{\PP^2}(-1)  \ar[d]^{\tau_1 \otimes \id_{\cO_{\PP^2}(-1)}} \\
(0) \ar[r] &\HH^1(\PP^2,\cF\!\otimes_{\PP^2}\! \Omega^2(2))\!\otimes_\KK\!\cO_{\PP^2}(-2) \ar^{\alpha_{\cF,-1}}[r] & \HH^{1}(\PP^2,\cF\!\otimes_{\PP^2}\! \Omega^1(1))\!\otimes_\KK\!\cO_{\PP^2}(-1),}
\]

whose lower row is exact. In view of (b), the morphism $\tau_2\otimes \id_{\cO_{\PP^2}(-2)}$ is a monomorphism, so that $\ker\alpha_{\cU,-1}\!=\!(0)$.  $\diamond$

(d) There is an exact sequence

\[ (0) \lra \coker \alpha_{\cU,-1} \lra \cU \lra \ker \delta_{\cU,-1} \lra (0)\]

of torsion-free coherent sheaves.

Owing to Corollary <ref>, we have

\[ \dim_\KK \HH^0(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^m(m)) \leq \dim_\KK \HH^0(\PP^2,\cF\!\otimes_{\PP^2}\!\Omega^m(m)) = 0\]

for $m \in \{0,1,2\}$ as well as $\HH^1(\PP^2,\cF)\!=\!(0)$. Since $(0)\!\stackrel{(\star)}{=}\!\Hom_{\PP^2}(\cO_{\PP^2},\cF/\cU)\!=\!\HH^0(\PP^2,\cF/\cU)$, we conclude with the exactness of

\[ \HH^0(\PP^2,\cF/\cU) \lra  \HH^1(\PP^2,\cU) \lra \HH^1(\PP^2,\cF)\]

that $\HH^1(\PP^2,\cU)\!=\!(0)$. Consequently, the first page of the Beilinson spectral sequence $E_\cU$ reads

$\HH^2(\PP^2,\cU\otimes_{\PP^2}\! \Omega^2(2))\!\otimes_\KK\!\cO_{\PP^2}(-2)$     $\HH^2(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^1(1))\!\otimes_\KK \cO_{\PP^2}(-1)$     $\HH^2(\PP^2,\cU)\! \otimes_\KK\! \cO_{\PP^2}$

$\HH^1(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^2(2))\!\otimes_\KK\! \cO_{\PP^2}(-2)$      $\HH^1(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^1(1))\!\otimes_\KK\! \cO_{\PP^2}(-1)$      $(0)$

$(0)$     $(0)$     $(0)$

with all other terms being zero. In view of (c) the second page thus is of the following form:

$\ker \delta_{\cU,-1}$     $\ast$     $\ast$

$(0)$     $\coker \alpha_{\cU,-1}$     $(0)$

$(0)$     $(0)$     $(0)$

Theorem <ref> in conjunction with <cit.>[One sets $r\!=\!2\!=\!q'$, $q\!=\!1$ and $n\!=\!0$.] now provides an exact sequence

\[ (0) \lra \coker \alpha_{\cU,-1} \lra \cU \lra \ker \delta_{\cU,-1} \lra (0)\]

of torsion-free coherent sheaves. $\diamond$

(e) If $\mu(\cU)\!\ge\!(>)\mu(\cF)$, then $\mu(\coker \alpha_{\cU,-1})\!\ge\!(>)\mu(\cF)$.

Suppose that $\ker \delta_{\cU,-1}\!\neq\!(0)$. Since $\cO_{\PP^2}(-2)$ is semistable, we conclude with $\ker \delta_{\cU,-1}  \subseteq  \HH^2(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^2(2))\!\otimes_\KK\! \cO_{\PP^2}(-2)$, <cit.> and $\frac{\dim_{\KK} M_1}{\Delta_M} \geq 0$ that

\[\mu(\ker \delta_{\cU,-1}) \leq \mu(\cO_{\PP^2}(-2)) = - 2 < -1\!+\!\frac{\dim_\KK M_1}{\Delta_M} =  \mu(\cF) \le \mu(\cU).\]

In particular, $\ker \delta_{\cU,-1} \not \cong \cU$ and therefore (d) forces $\coker \alpha_{\cU,-1}\!\neq\!(0)$.
Now (d) in conjunction with Lemma <ref> implies $\mu(\coker \alpha_{\cU,-1})\!>\!\mu(\cU)\!\ge\!\mu(\cF)$.

Alternatively, $\ker \delta_{\cU,-1}\!=\!(0)$, so that $\coker \alpha_{\cU,-1} \cong \cU$ and $\mu(\coker \alpha_{\cU,-1})\!=\!\mu(\cU)\!\ge\!(>)\mu(\cF)$. $\diamond$

(f) There exists a subrepresentation $(0) \subsetneq N \subsetneq M$ of $M$ such that $\mu(N)\!=\!\mu(\coker \alpha_{\cU,-1}(1))$.

Setting $V_i\!:=\!\HH^1(\PP^2,\cU\!\otimes_{\PP^2}\!\Omega^i(i))$ and $W_i\!:=\!\HH^1(\PP^2,\cF\!\otimes_{\PP^2}\!\Omega^i(i))$ for $i \in \{1,2\}$, application of $-\!\otimes_{\PP^2}\cO_{\PP^2}(1)$ to the
diagram ($\dagger$) in conjunction with (c) and Corollary <ref> yields

\[
\xymatrix{
(0) \ar[r] & \widetilde{V_1} \ar[d]^{\tilde{\tau}_2} \ar^{\alpha_{\cU,-1}(1)}[r] & \widetilde{V_2}  \ar[d]^{\tilde{\tau}_1} \ar[r] & \coker\alpha_{\cU,-1}(1) \ar[d] \ar[r] & (0)\\
(0) \ar[r] &\widetilde{W_1} \ar^{\alpha_{\cF,-1}(1)}[r] & \widetilde{W_2} \ar[r] & \TilTheta(M) \ar[r] & (0),
\]

where we have used the notation introduced in Section <ref>. Let $x \in \PP^2$. By (b) and diagram <ref> (see Section <ref>), the map $\tilde{\tau}_2(x)$ is injective and since $\TilTheta(M)$ is a vector bundle, <cit.> shows that this also holds for $\alpha_{\cF,-1}(1)(x)$. Passing to fibers in the diagram above, we see that $\alpha_{\cU,-1}(1)(x)$ is injective, and another application of  <cit.>
ensures that $\coker\alpha_{\cU,-1}(1)$ is a vector bundle.

The proof of Theorem <ref>(1) provides $\KK$-linear maps

\[ V(\gamma_i) : V_1 \lra V_2 \ \ \text{and} \ \ W(\gamma_i) : W_1 \lra W_2 \ \ \ \ \ \ \  (i \in \{1,\ldots, r\})\]

such that

\[ \TilTheta_V = \alpha_{\cU,-1}(1) \ \ \text{and} \ \ \TilTheta_W = \alpha_{\cF,-1}(1),\]

where $X\!:=\!(X_1,X_2,(X(\gamma_i))_{1\le i\le r})$ for $X \in \{V,W\}$. It follows from Theorem <ref>(1) that $V$ and $W$ are $\EKP(K_3)$-representations such that $\TilTheta(W) \cong \TilTheta(M)$ and $\TilTheta(V)
\cong \coker \alpha_{\cU,-1}(1)$. Theorem <ref>(2) implies $M\cong W$ and the diagram above yields

\[ \tilde{\tau}_1\circ \TilTheta_V = \TilTheta_W\circ \tilde{\tau}_2,\]

so that evaluation at $v\otimes \frac{1}{\gamma_j^\ast} \in \widetilde{V_1}(D(\gamma_j^\ast))$ gives[Observe that the identification $\cU_{(r,1)}\cong \cO_{\PP^{r-1}}(-1)$ yields $\TilTheta_X(U)(x\otimes f)\!=\!
\sum_{i=1}^r X(\gamma_i)(x)\otimes \gamma_i^\ast f$ for $x \in X$ and $f \in \cO_{\PP^{r-1}}(-1)(U)$.]

\[ \sum_{i=1}^r (\tau_1\circ V(\gamma_i))(v)\otimes \frac{\gamma_i^\ast}{\gamma_j^\ast} = \sum_{i=1}^r (W(\gamma_i)\circ \tau_2)(v)\otimes \frac{\gamma_i^\ast}{\gamma_j^\ast}.\]

Evaluation at $[\gamma_j] \in \PP(A_r)\!=\!\PP^{r-1}$ shows that $(\tau_2,\tau_1): V \lra W$ is a morphism, which, by virtue of (a) and (b), is injective. Thus, $V$ is isomorphic to a non-zero subrepresentation $N$ of $M$ such that
$\mu(N)\!=\!\mu(\coker \alpha_{\cU,-1}(1))$. $\diamond$

(1) Suppose that $\TilTheta(M)$ is not semistable. By Lemma <ref>, the vector bundle $\cF:=\!\widetilde{\Theta}(M)(-1)$ is not semistable. For a representation $(0)\!\neq\!N \in \EKP(K_3)$ with $N_1\!=\!(0)$, the
bundle $\TilTheta(N)(-1) \cong N_2\!\otimes_{\KK}\!\cO_{\PP^2}(-1)$ is semistable (see <cit.>). In view of $\Delta_M(2)\!\ge\!0$, we thus have $0\!<\!2\dim_\KK M_1\!\leq\!\dim_\KK M_2$ as well as $\mu(\cF)\!=\!\mu(\TilTheta(M))\!-\!1\!
=\!\frac{\dim_\KK M_1}{\Delta_M}\!-\!1\!\le\!0$. Lemma <ref> provides $\cU \subsetneq \cF$ non-zero such that $\cF/\cU$ is semistable with

(i) $\mu(\cU)\!>\!\mu(\cF)$, and

(ii) $\Hom_{\PP^2}(\cO_{\PP^2}(\ell),\cF/\cU)\!=\!(0)$ for all $\ell\!\geq\!0$.

In particular,  $(\star)$ is satisfied. Thanks to (e) and (f) there exists a subrepresentation $(0) \subsetneq N \subsetneq M$ such that

\[ \mu(N) = \mu(\coker \alpha_{\cU,-1}(1)) > \mu(\TilTheta(M)) = \mu(M),\]

which contradicts $M$ being semistable.

(2) As before, we set $\cF\!:=\!\widetilde{\Theta}(M)(-1)$. Since the assumptions of (1) are satisfied, $\cF$ is semistable. Suppose that $\cF$ is not stable. Proceeding as in (1), we first observe that $M$ is indecomposable (even a brick), so that we may assume that $M_1\!\neq\!(0)$. We apply <cit.>[This result holds for algebraically closed fields of arbitrary characteristic.] in conjunction with Lemma <ref> to obtain a short
exact sequence

\[ (0) \lra \cU \lra \cF \lra \cF/\cU \lra (0)\]

such that $0\! <\! \rk(\cU), \rk(\cF/\cU)\!<\!\rk(\cF)$ with $\cF/\cU$ being torsion-free and $\mu(\cU)\!=\!\mu(\cF/\cU)\!=\!\mu(\cF)$. We choose $\cU$ such that $\cU$ is of maximal rank subject to these properties.

Let $\cQ$ be a torsion-free quotient of $\cF/\cU$ with $0\!<\!\rk(\cQ)\!<\!\rk(\cF/\cU)$. There results a short exact sequence $(0) \lra \cV \lra \cF \lra \cQ \lra (0)$ with $0\!<\!\rk(\cV),\rk(\cQ)\!<\!\rk(\cF)$, and, observing
$\rk(\cQ)\!<\!\rk(\cF/\cU)$, we conclude that $\rk(\cV)\!>\!\rk(\cU)$. Since $\cF$ is semistable, the choice of $\cU$ in conjunction with Lemma <ref> implies $\mu(\cV)\!<\!\mu(\cF)$ and Lemma <ref> yields $\mu(\cQ)\!>\!\mu(\cF)\!=\!\mu(\cF/\cU)$. Consequently, <cit.> shows that $\cF/\cU$ is stable.

Assume that $f \colon \cO_{\PP^2}(\ell) \lra \cF/\cU$ is a non-zero morphism. Since $\cO_{\PP^2}(\ell)$ is stable and $\im f \subseteq \cF/\cU$ is a non-zero torsion-free sheaf, we conclude

\[ \ell = \mu(\cO_{\PP^2}(\ell)) \leq \mu(\im f) \leq \mu(\cF/\cU) = \mu(\cF) = \frac{\dim_\KK M_1}{\Delta_M}\!-\!1 < 0. \]

We have proven:

(i) $\mu(\cU)\!=\!\mu(\cF)\!=\!\mu(\cF/\cU)$, and

(ii) $\Hom_{\PP^2}(\cO_{\PP^2}(\ell),\cF/\cU)\!=\!(0)$ for all $\ell\!\geq\!0$.

Thanks to (e) and (f) the module $M$ is not stable, a contradiction. It follows that $\TilTheta(M)\!=\!\cF(1)$ is stable.

We summarize our findings in the following

Let $M \in \EKP(K_3)\!\smallsetminus\!\{0\}$.

* If $\Delta_M(2)\!\ge\!0$, then $\TilTheta(M)$ is semistable if and only if $M$ is semistable.

* If $\Delta_M(2)\!>\!0$, then $\TilTheta(M)$ is stable if and only if $M$ is stable.

This is just a combination of Proposition <ref> and Proposition <ref>.

(a) Exceptional Steiner bundles. According to <cit.>, exceptional vector bundles on $\PP^2(\CC)$ are stable. Thanks to <cit.>, this also holds for exceptional Steiner bundles on $\PP^{r-1}(\CC)$. Returning to our context, we let $\cF$ be an exceptional Steiner bundle on $\PP^2$. According to Proposition <ref>, there is a preprojective indecomposable representation $M \in \EKP(K_3)$ such that
$\cF\cong \TilTheta(M)$. A consecutive application of Proposition <ref> and Theorem <ref> ensures that $\Delta_M(2)\!>\!0$. Now Theorem <ref> in conjunction with the remark
following Corollary <ref> shows that $\cF$ is stable.

(b) "Regular" indecomposable Steiner bundles. Let $(V_1,V_2)$ be a pair of $\KK$-vector spaces such that $\Delta_{(V_1,V_2)}(2)\!>\!0$ and $q_3(\udim(V_1,V_2))\!<\!0$. The stable representations of $\rep(K_3;V_1,V_2)\cap\EKP(K_3)$ may
be interpreted as the stable points of the variety $\PP(\rep(K_3;V_1,V_2))$ relative to the canonical action of
$\SL(V_2)\!\times\!\SL(V_1)$. This can be used to show that the subset $\cS(V_1,V_2)$ of stable representations is non-empty open (and hence dense) whenever $\Delta_{(V_1,V_2)}\!\ge\!2$; see also Proposition <ref> below for the case, where $\KK\!=\!\CC$.

§.§ Applications: Filtrations for Steiner bundles

Suppose that $M \in \EKP(K_3)$ is semistable. If $P_0(r)$ is a direct summand of $M$, then a two-fold application of the See-Saw Lemma implies $M_1\!=\!(0)$, whence $M\cong nP_0(r)$.  Alternatively $\psi_M$ is surjective, whence $\dim_\KK M_2\!\le\! 3\dim_\KK M_1$. Hence the conditions of Theorem <ref> may seem to limit its utility. The results of this section illustrate that a combination of Theorem <ref> with techniques involving AR-theory provides many examples of stable Steiner bundles on $\PP^2$.

Let $E \in \EKP(K_3)$ be an elementary representation. Then $\TilTheta(E)$ is a stable.

Thanks to Proposition <ref>, the representation $E$ is stable and $E \in \EKP(K_3)\!\smallsetminus\!\{0\}$ implies $\dim_\KK E_1\!<\!\dim_\KK E_2$. In view of Theorem <ref>, it therefore
suffices to consider the case, where $\dim_\KK E_1\!< \!\dim_\KK E_2\!\leq\!2\dim_\KK E_1$. Thanks to <cit.>, $\udim E$ belongs to one of the $\sigma_3$-orbits generated by the dimension vectors
$(1,1),(2,2)$. Since $\Delta_{\sigma_{K_3}(M)}\!=\!\Delta_M(2)\!<\!\Delta_M$ for every regular indecomposable $M \in \rep(K_3)$, there is a unique dimension vector $(x,y) \in \langle\sigma_3\rangle.\udim E$ such that
$\Delta_{(x,y)}\!=\!y\!-\!x \in \NN$ and $\Delta_{(x,y)}(2)\!=\!y\!-\!2x\!\le\!0$. Hence the only dimension vectors in these orbits that satisfy $x\!<\!y\!\leq\!2x$ are $(1,2)$ and $(2,4)$. By virtue of Theorem <ref>,
representations with dimension vector $(1,2)$ do not belong to $\EKP(K_3)$, whence $\udim E\!=\!(2,4)$. As elementary representations are bricks, $\TilTheta(E)$ is a simple vector bundle of rank $2$. We conclude with
<cit.>, which holds in arbitrary characteristic (cf. also <cit.>), that $\TilTheta(E)$ is stable.

Let $\cC \subseteq \Gamma(K_r)$ be a regular Auslander-Reiten component. We say that $\cC$ is (semi)stable, provided $\cC\cap\EKP(K_r)$ contains a (semi)stable vertex[Recall from <ref>(3) that semistable vertices are quasi-simple.]. The following result
shows that for such components the Harder-Narasimhan filtrations of most of the bundles $\TilTheta(M)$ correspond to the quasi-composition series of $M$.

Let $\cC \subseteq \Gamma(K_3)$ be a semistable AR-component. If $M \in (\tau^{-1}_{K_3}(M_{\cC,1})\!\rightarrow)$ has quasi-length $\ell$, then the Harder-Narasimhan filtration of $\TilTheta(M)$ is given by

\[  (0) \subsetneq \TilTheta(M_{[1]}) \subsetneq \TilTheta(M_{[2]})\subsetneq \cdots \subsetneq \TilTheta(M_{[\ell\!-\!1]}) \subsetneq \TilTheta(M_{[\ell]}) = \TilTheta(M) \]

with filtration factors $\TilTheta(M_{[j]})/\TilTheta(M_{[j\!-\!1]}) \cong \TilTheta(\tau^{-j+1}_{K_3}(M_{[1]}]))$ for all $j \in \{2,\ldots,\ell\}$.

We consider the quasi-composition series

\[ (0) \subsetneq M_{[1]} \subsetneq \cdots \subsetneq M_{[\ell\!-\!1]} \subsetneq M\]

of $M$. For $1\!< \!j\!\leq\!\ell$ general AR-theory provides an exact sequence

\[ (0) \lra M_{[j-1]} \lra M_{[j]} \lra \tau^{-j+1}_{K_3}(M_{[1]}) \lra (0)\]

in $\EKP(K_3)$. According to Corollary <ref>, application of $\TilTheta$ yields an exact sequence

\[ (0) \lra \TilTheta(M_{[j-1]}) \lra \TilTheta(M_{[j]}) \lra \TilTheta(\tau^{-j+1}_{K_3}(M_{[1]})) \lra (0)\]

so that $\TilTheta(M_{[j]})/\TilTheta(M_{[j-1]}) \cong \TilTheta(\tau^{-j+1}_{K_3}(M_{[1]}))$. Our assumption on
$M$ implies that the quasi socle $M_{[1]}$ belongs to the cone $(\tau^{-1}_{K_3}(M_{\cC,1})\!\rightarrow)$. It
follows from Theorem <ref>(3) that $\tau^{-j+1}_{K_3}(M_{[1]}) \in \rep(K_3,2)$, whence $\Delta_{\tau^{-

As $\cC$ is semistable, Corollary <ref> ensures that the same is true for $\tau^{-n}_{K_3}(M_{[1]})$ for all $n \in \NN_0$. We apply Theorem <ref> and conclude that
$\TilTheta(M_{[j]})/\TilTheta(M_{[j-1]})$ is semistable. Proposition <ref>(3) yields

\[\mu(\TilTheta(\tau^{-j+1}_{K_3}(M_{[1]}))) = \mu(\tau^{-j+1}_{K_3}(M_{[1]})) > \mu(\tau^{-j}_{K_3}(M_{[1]})) = \mu(\TilTheta(\tau^{-j}_{K_3}(M_{[1]})))\]

for all $1\!\le\!j\!\leq\!\ell$. The statement now follows from the uniqueness of the Harder-Narasimhan filtration.

Let  $\cC \subseteq \Gamma(K_3)$ be a regular component. If $M \in \cC\cap\repp(K_3,d)$ has minimal type, then $\TilTheta(N)$ is stable for every quasi-simple $N \in (M_{\cC,d} \!\rightarrow)$.

Corollary <ref> and Proposition <ref> show that $M\!=\!M_{\cC,d}$ is a brick. If $d\!=\!2$, then Lemma <ref> and Theorem <ref> imply that $\TilTheta(M)$ is stable. Alternatively,
$\TilTheta(M) \in \Vect(\PP^2)$ is simple and of rank $2$, so that <cit.> implies the stability of $\TilTheta(M)$. As $N \in \{M_{\cC,1}\}\cup (M_{\cC,2}\!\rightarrow)$, it follows from Corollary
<ref> that $\TilTheta(N)$
is stable.

Let $\cC \subseteq \Gamma(K_3)$ be a regular AR-component, $\cF$ be an indecomposable Steiner bundle.

* If $\cC$ contains an elementary representation and $\cF \in \TilTheta(M_{\cC,1}\!\rightarrow)$, then $\cF$ possesses a Harder-Narasimhan filtration by Steiner bundles, whose filtration factors are stable.

* If $\cF$ has a resolution

\[ (0) \to \cO_{\PP^2}(-1)^a \to \cO^b_{\PP^2} \to \cF \to (0)\]

such that $(a,b)$ is an imaginary root and $q_3(a,b)\!+\!\Delta_{(a,b)}(2)\! \geq\! 1$, then $\cF$ possesses a filtration

\[ (0) = \cF_0 \subseteq \cF_1 \subseteq \cdots \subseteq \cF_n = \cF\]

by Steiner bundles such that each quotient $\cF_i/\cF_{i-1}$ is a stable Steiner bundle.

* Suppose $X \in \cC$ is quasi-simple such that $ q_3(\udim X)\!+\!\Delta_X(2)\!\ge\!1$. If $\cF \in \TilTheta(X\!\rightarrow)$, then $\cF$ possesses a

\[ (0) = \cF_0 \subseteq \cF_1 \subseteq \cdots \subseteq \cF_n = \cF\]

by Steiner bundles such that each quotient $\cF_i/\cF_{i-1}$ is a stable Steiner bundle.

(1) Let $E \in \cC$ be elementary. By definition, $E$ is quasi-simple, so that, for every $n \in \NN_0$, the module $\tau_{K_3}^{-n}(M_{\cC,1})$ is elementary and contained $\EKP(K_3)$. Given $\cF\!:=\!\TilTheta(M) \in \TilTheta(M_{\cC,1}\!\rightarrow)$, we consider the quasi-composition series

\[ (0) = M_{[0]}\subsetneq M_{[1]} \subsetneq \cdots \subsetneq M_{[\ql(M)]} = M,\]

whose filtration factors belong to $\{\tau_{K_3}^{-\ell}(M_{\cC,1}) \ ; \ \ell \in \NN_0\}$. Application of $\TilTheta$ provides a filtration

\[ (0) = \TilTheta(M_{[0]})\subsetneq \TilTheta(M_{[1]}) \subsetneq \cdots \subsetneq \TilTheta(M_{[\ql(M)]}) = \cF,\]

with filtration factors $\TilTheta(M_{[j]})/\TilTheta(M_{[j-1]}) \cong \TilTheta(\tau^{-j+1}_{K_3}(M_{[1]}))$. Thanks to Theorem <ref>, the factors are stable and with strictly decreasing slopes (cf. Proposition <ref>), so
that we have found the Harder-Narasimhan filtration of $\cF$.

(2) By Theorem <ref> we can find an indecomposable representation $M \in \EKP(K_3)$ such that $\TilTheta(M) \cong \cF$. We have $b\!-\! a\! = \!\rk( \cF)\! =\! \rk(\TilTheta(M))\!=\! \Delta_M$ and $a\!=\!c_1(\cF)\!=\!
c_1(\TilTheta(M))\! =\!\dim_\KK M_1$. Hence

\[  q_3(\udim M)\!+\!\Delta_M(2) = q_3(a,b)\!+\!\Delta_{(a,b)}(2) \geq 1.\]

We conclude with Lemma <ref>  that there is a filtration

\[ (0) = M_0 \subsetneq M_1 \subsetneq \cdots \subsetneq M_n = M\]

such that each quotient $M_i/M_{i-1}$ is an elementary representation in $\EKP(K_3)$. Exactness of $\TilTheta$ on $\EKP(K_3)$ yields a filtration

\[ (0) = \TilTheta(M_0) \subsetneq \TilTheta(M_1) \subsetneq \cdots \subsetneq \TilTheta(M_n) = \TilTheta(M) \cong \cF\]

of Steiner bundles. By Theorem <ref>, each quotient $\TilTheta(M_i)/\TilTheta(M_{i-1}) \cong \TilTheta(M_i/M_{i-1})$ is a stable Steiner bundle.

(3) This follows as in (2), using Corollary <ref>.

§.§ Applications: Restrictions to hyperplanes

We assume throughout that $\KK\!= \!\CC$. Let $\cF$ be a stable (respectively, semistable) vector bundle of rank $n$ on $\PP^{r-1}$ with $r\!\geq\!4$. In <cit.> it was asked whether for almost all linear
hyperplanes $H \subseteq \PP^{r-1}$ the restriction of $\cF$ to $H$ is stable (respectively, semistable).

The question was answered affirmatively by Barth [6] for $n\!=\!2$ and for $n\!<\! r\!-\!1$ by Maruyama [49] (in both cases with one exception). In the context of Steiner bundles, these rank conditions
imply that only (direct sums of) the structure sheaf of $\PP^{r-1}$ can occur.

Consider the tangent bundle $\cT(-1)$ of rank $r\!-\!1$. Then $\cT(-1) \cong \TilTheta(P_1(r))$ is stable. On the other hand, $P_1(r\!-\!1) \subseteq \alpha^\ast(P_1(r))$ for all $\alpha \in \Inj_\KK(A_{r-1},A_r)$, while $
\mu(\alpha^\ast(P_1(r)))\!=\!\frac{1}{r-1}\! <\! \frac{1}{r-2} \! = \! \mu(P_1(r\!-\!1))$. Thanks to Lemma <ref> and Proposition <ref>, $\cT(-1)|_H$ is not semistable for any hyperplane
$H \subseteq \PP^{r-1}$. This example, which does not depend on the ground field, is well-known (cf. [30]). Building on recent work of Coskun, Huizenga and Smith [21], we will show below that generic Steiner bundles on $\PP^{r-1}$ behave similarly.

Let $r\!\geq\!3$, $(V_1,V_2)$ be a pair of vector spaces such that

(a) $\Delta_{(V_1,V_2)}\!\geq\!r\!-\!1$, and

(b) $q_r(\udim (V_1,V_2))\!<\!1$.
There exists a non-empty open subset $O \subseteq \rep(K_r;V_1,V_2) \cap \EKP(K_r)$ such that for every $M \in O$ the Steiner bundle $\TilTheta(M)$ is stable.

Following [21], we define $\psi_{r-1} := \frac{r-2+\sqrt{(r-1)^2+2(r-1)-3}}{2}$, so that

\[\psi_{r-1}\!+\!1 = \frac{r+\sqrt{(r\!-\!1)^2\!+\!2(r\!-\!1)\!-\!3}}{2} = \frac{r\!+\!\sqrt{r^2\!-\!4}}{2} = L_r.\]

As $V_1\!\ne\!(0)$, our current assumptions imply

\[\frac{r\!-\!1}{\dim_\KK V_1} \leq \frac{\Delta_{(V_1,V_2)}}{\dim_\KK V_1} \ \ \ \text{and} \ \ \ \frac{\Delta_{(V_1,V_2)}}{\dim_\KK V_1} = \frac{\dim_\KK V_2}{\dim_\KK V_1}\!-\!1 < L_r\!-\!1 = \psi_{r-1},\]

respectively. Thanks to <cit.>, there is a dense open subset

\[ O \subseteq \Hom_{\PP^{r-1}}(V_1\! \otimes_\KK\! \cO_{\PP^{r-1}}(-1),V_2\!\otimes_\KK\! \cO_{\PP^{r-1}}) \cong \Hom_\KK(V_1,V_2)^r = \rep(K_r;V_1,V_2)\]

such that $O \subseteq \EKP(K_r)$, and for every $M \in O$ the Steiner bundle $\TilTheta(M)$ is stable.

The statement is also true if $q_r(\udim(V_1,V_2))\!=\!1$ and $\dim_\KK V_1\!\leq\!\dim_\KK V_2$. In this case there exists a non-empty open subset $O \subseteq \rep(K_r;V_1,V_2)$ such that every
representation in $O$ is isomorphic to some $P_i(r)$. By Proposition <ref> and <cit.>, the bundles $\TilTheta(P_i(r))$ are stable.[In [14] Gieseker stability was considered. It can be
shown, however, that the relevant result of loc. cit. also holds in our context.]

Assume that  $q_r(\udim(V_1,V_2))\!<\!1$ and $\Delta_{(V_1,V_2)}(r\!-\!1)\!\geq\!r\!-\!1$. A general Steiner bundle $\cF$ of rank $\Delta_{(V_1,V_2)}$ and with first Chern class $\dim_\KK
V_1$ is stable and $\cF|_H$ is not semistable for each hyperplane $H \subseteq \PP^{r-1}$.

Let $O \subseteq \rep(K_r;V_1,V_2)$ be the non-empty open set defined in Proposition <ref>. By Proposition <ref> the set $O\cap\repp(K_r,r\!-\!1)$ is non-empty and open. Let $M \in O \cap
\repp(K_r,r\!-\!1)$, then $\cF\!:=\!\TilTheta(M)$ is stable. Since $M \in \repp(K_r,r\!-\!1)$, we have $\alpha^\ast(M) \cong \Delta_M(r\!-\!1)P_0(r\!-\!1)\!\oplus\!(\dim_\KK M_1)P_1(r\!-\!1)$ for all $\alpha \in
\Inj_\KK(A_{r-1},A_r)$. Thus, $\mu(\alpha^\ast(M))\!<\!\mu(P_1(r\!-\!1))$, and the above arguments show that for every hyperplane $H \subseteq \PP^{r-1}$, the bundle $\cF|_H$  is not semistable.

Corollary <ref> provides negative answers to <cit.> for a fixed rank and Chern class on an open subset of $ \rep(K_r;V_1,V_2)$.

Our next result shows that we can find an infinite family of Steiner bundles of fixed rank and Chern class that are counter examples and even belong to different $\GL(A_r)$-orbits. Moreover, these bundles are
isomorphic when we restrict them to any hyperplane.

Let $(V_1,V_2)$ be a pair of vector spaces such that

(a) $q_r(\udim(V_1,V_1))\!<\!-r^2$, and

(b) $\Delta_{(V_1,V_2)}(r\!-\!1)\!\geq\!r\!-\!1$.
There exists an infinite family $(\cF_i)_{i \in I}$ of Steiner bundles such that for all $i,j \in I$ the following statements hold:

* $\cF_i$ is stable.

* $c(\cF_i)\!=\!\dim_\KK V_1$, $\rk(\cF_i)\!=\!\Delta_{(V_1,V_2)}$.

* $\cF_i|_H \cong \cF_j|_H$ for each hyperplane $H \subseteq \PP^{r-1}$.

* $\cF_i|_H$ is not semistable for each hyperplane $H \subseteq \PP^{r-1}$.

* All $\cF_i$ lie in different orbits under the natural action of $\GL(A_r)$ on $\Coh(\PP^{r-1})$.

Let $O \subseteq \rep(K_r;V_1,V_2)$ be the non-empty open set defined in Proposition <ref>. The algebraic group $G\!:= \GL(A_r)\!\times\!\GL(V_2)\!\times\!\GL(V_1)$ acts canonically on
$\rep(K_r;V_1,V_2) \cong \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)$. We consider the $G$-stable and open set $U\!:=\!\bigcup_{g \in G} g.O \subseteq \rep(K_r;V_1,V_2)$ and set $\cU\!:=\!U \cap \rep(K_r;V_1,V_2)
\cap \repp(K_r,r\!-\!1)$. By Proposition <ref> the $G$-stable set $\cU$ is non-empty and open.

Let $M \in \cU$. We find $(a,h_2,h_1) \in G$ and $S \in O \cap \rep(K_r;V_1,V_2) \cap \repp(K_r,r\!-\!1)$ such that $M\!=\!(a,h_2,h_1).S$. By definition, the representations $M$ and $(a,\id_{V_2},\id_{V_1}).S \!=\!S^{(a)}$ are isomorphic. We obtain

\[ \cF := \TilTheta(M) \cong \TilTheta(S^{(a)}) \cong a^\ast.\TilTheta(S),\]

see Section <ref>. Since $S \in O$, we know that $\TilTheta(S)$ is stable. Therefore, the same is true for $a^\ast.\TilTheta(S) \cong \cF$. This shows that every bundle arising in this way has property (1). Owing to Theorem <ref> and
Corollary <ref>, each bundle of this form satisfies (2). The arguments in the proof of Corollary <ref> now show that (3) and (4) hold. Since $q_r(\udim(V_1,V_2))\!<\!-r^2$ we can apply <cit.> and
conclude that $\cU/G$ is not finite. By construction, elements of $\rep(K_r;V_1,V_2)$ belonging to different $G$-orbits correspond to Steiner bundles, whose isomorphisms classes belong to different orbits under the
action of $\GL(A_r)$. This shows (5).

We fix $r\!=\!4$.

(1) $q_r(5,18)\!=\!-11$ and $\Delta_{(5,18)}(3)\!=\!3$. We apply Corollary <ref> and find a stable Steiner bundle of rank $13$ on $\PP^3$ such that $\cF|_H$ is not semistable for each hyperplane
$H \subseteq \PP^3$.

(2) $q_r(6,21)\!=\!-27\!<\!-4^2$ and $\Delta_{(6,21)}(3)\!=\!3$. We apply Corollary <ref> and  find an infinite family of stable Steiner bundles $(\cF_i)_{i \in I}$ of rank $15$ on $\PP^3$ such that $\cF_i|_H$ is not
semistable for all $i \in I$ and all hyperplanes $H \subseteq \PP^3$. Moreover, we have $\cF_i|_H \cong \cF_j|_H$ for each hyperplane and all $i,j \in I$, while $\cF_i$ are $\cF_j$ are not in the same $\GL(A_r)$-orbit.

[1] E. Arrondo and S. Marchesi, Jumping pairs of Steiner bundles. Forum Math. 27 (2015), 3233–3267.
[2] I. Assem, D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, I. London Mathematical Society Student Texts 65.
Cambridge University Press, 2006.
[3] M. Atiyah, On the Krull-Schmidt Theorem with applications to sheaves. Bull. Soc. Math. France 84 (1956), 307–317.
[4] M. Auslander, I. Reiten and S. Smalø, Representation Theory of Artin Algebras.  Cambridge Studies in Advanced Mathematics 36. Cambridge University Press, 1995.
[5] E. Ballico and Ph. Ellia, Fibrés homogènes sur $\PP^n$. C.R. Acad. Sci. Paris 294 (1982), 403–406.
[6] W. Barth, Some properties of stable rank-2 vector bundles on $\PP_n$. Math. Ann. 226 (1977), 125–150.
[7] D. Benson, Representations of Elementary Abelian $p$-Groups and Vector Bundles. Cambridge Tracts in Mathematics 208. Cambridge University Press, 2017.
[8] D. Bissinger, Representations of constant socle rank for the Kronecker algebra. Forum Math. 32 (2020), 23–43.
[9] , Dimension vectors with the equal kernels property. J. Pure Appl. Algebra 227 (2023), 107424.
[10] , Shift orbits for elementary modules of Kronecker quivers. arXiv: 2403.01824 [math.RT]
[11] I. Biswas,  On the stability of homogeneous vector bundles. J. Math. Sci. Univ. Tokyo 11 (2004), 133–140.
[12] G. Bohnhorst and H. Spindler, The stability of certain vector bundles on $\PP^n$. Lecture Notes in Math. 1507 (1992), 39–50.
[13] M. Brambilla, Simplicity of generic Steiner bundles. Boll. Unione Mat. Ital. (8) 8-B (2005), 723–735.
[14] , Cokernel bundles and Fibonacci bundles. Math. Nachr. 281 (2008), 499–516.
[15] R.-O. Buchweitz, G. Leuschke, and M. Van den Bergh. On the derived category of Grassmannians in arbitrary characteristic. Compos. Math. 151 (2015), 1242–1264.
[16] J. Carlson, E. Friedlander, and J. Pevtsova, Representations of elementary abelian p-groups and bundles of Grassmannians.  Adv. Math. 229 (2012), 2985–3051.
[17] , , , Elementary subalgebras of Lie algebras. J. Algebra 442 (2015), 155–189.
[18] J. Carlson, E. Friedlander and A. Suslin, Modules for $\ZZ/p\!\times\!\ZZ/p$. Commentarii Math. Helvetici 86 (2011), 609–657.
[19] H. Cartan and S. Eilenberg, Homological Algebra. Princeton Mathematical Series 19. Princeton University Press, 1973.
[20] B. Chen, Dimension vectors in regular components over wild Kronecker quivers. Bull. Sci. Math. 137 (2013), 730–745.
[21] I. Coskun, J. Huizenga and G. Smith, Stability and cohomology of kernel bundles on projective space. arXiv:2204.10247 [math.AG]
[22] E. Dade, Endo-permutation modules over $p$-groups II. Ann. of Math. 108 (1978), 317–346.
[23] I. Dolgachev and M. Kapranov, Arrangements of hyperplanes and vector bundles on $\PP_n$. Duke Math. J. 71 (1993), 633–664.
[24]  J.-M. Drezet, Fibrés exceptionels et variétés de modules de faisceaux semi-stables sur $\PP_2(\CC)$. J. reine angew. Math. 49 (1987), 14–58.
[25] , Variétés de modules alternatives. Ann. Inst. Fourier 49 (1999), 57–139.
[26] D. Eisenbud and J. Harris, 3264 and All That. Cambridge University Press, 2016.
[27] G. Elencwajg, Des fibrés uniformes non homogènes. Math. Ann. 239 (1979), 185–192.
[28] R. Farnsteiner, Varieties of tori and Cartan subalgebras of restricted Lie algebras. Trans. Amer. Math. Soc.  356 (2004), 4181–4236.
[29] , Categories of modules given by varieties of $p$-nilpotent operators. arXiv: 1110.2706 [math.RT]
[30] H. Flenner, Restrictions of semistable bundles on projective varieties. Commentarii Math. Helvetici 59 (1984), 635–650.
[31] E. Friedlander and J. Pevtsova, Representation-theoretic support spaces for finite group schemes. Amer. J. Math. 127 (2005), 379–420. Erratum:
Amer. J. Math. 128 (2006), 1067–1068.
[32] , , Constructions for infinitesimal group schemes. Trans. Amer. Math. Soc. 363 (2011), 6007–6061.
[33] W. Geigle and H. Lenzing, A class of weighted projective curves arising in representation theory of finite dimensional algebras. Lect. Notes Math. 1273 (1987), 265–297.
[34] U. Görtz and T. Wedhorn, Algebraic Geometry I. Advanced Lectures in Mathematics. Vieweg+Teubner, 2010.
[35] A. Gorodentsev and A. Rudakov, Exceptional vector bundles on projective spaces. Duke Math. J. 54 (1987), 115–130.
[36] R. Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics 52. Springer-Verlag, 1977.
[37] , Algebraic vector bundles on projective spaces: a problem list. Topology 18 (1979), 117–128.
[38] G. Hochschild, Relative homological algebra. Trans. Amer. Math. Soc. 82 (1956), 246–269.
[39] K. Hulek, On the classification of stable rank-r vector bundles over the projective plane. Progress in Math. 7 (1981), 113-144.
[40] M. Jardim and D. Prata, Vector bundles on projective varieties and representations of quivers. Algebra Discr. Math. 20 (2015), 217–249.
[41] D. Huybrechts, Fourier-Mukai Transforms in Algebraic Geometry. Clarendon Press, 2006.
[42] V. Kac, Infinite root systems, representations of graphs and invariant theory. Invent. math. 56 (1980), 57–92.
[43] , Infinite root systems, representations of graphs and invariant theory, II. J. Algebra 78 (1982), 141–162.
[44] B. Karpov, Semistable sheaves on a two-dimensional quadric, and Kronecker modules. Russian Acad. Izv. Math. 40 (1993), 33–66.
[45] O. Kerner, Representations of wild quivers. CMS Conf. Proc. 19 (1996), 65–107.
[46] O. Kerner and F. Lukas, Elementary modules. Math. Z. 223 (1996), 421–434.
[47] E. Kunz, Einführung in die kommutative Algebra und algebraische Geometrie. Vieweg Studium 46. Vieweg, 1979.
[48] A. Langer, Lectures on torsion-free sheaves and their moduli. In: Algebraic Cycles, Sheaves, Shtukas, and Moduli, 69–103. Springer-Verlag, 2007.
[49] M. Maruyama, Boundedness of semistable sheaves of small ranks. Nagoya Math. J. 78 (1980), 65–94.
[50] S. Marchesi, Jumping spaces in Steiner bundles. Ph.D.-Thesis, Universita degli Studi di Milano, Universidad Complutense de Madrid, 2012.
[51] S. Marchesi and R. Miró-Roig, Uniform Steiner bundles. Ann. Inst. Fourier 71 (2021), 447–472.
[52] D. Mumford, The Red Book of Varieties and Schemes. Lecture Notes in Mathematics 1358. Springer-Verlag, 1999.
[53] C. Okonek, M. Schneider, and H. Spindler, Vector Bundles on Complex Projective Spaces. Birkäuser-Verlag, 1980 (3rd Ed. 2011).
[54] D. Perrin. Algebraic Geometry. An Introduction.  Springer University Text. Springer-Verlag, 2008.
[55] C. Ringel, Representations of K-species and bimodules. J. Algebra 41 (1976), 269–302.
[56] , Finite-dimensional hereditary algebras of wild representation type. Math. Z. 161 (1978), 235–255.
[57] , The elementary $3$-Kronecker modules. arxiv:1612.09141 [math.RT]
[58] A. Rudakov, Exceptional collections, mutations, and helices. In: Helices and Vector Bundles. London Math. Soc. Lecture Notes 148. Cambridge University Press, 1990.
[59] R. Schwarzenberger, Vector bundles on algebraic surfaces. Proc. London Math. Soc. 11 (1961), 601–622.
[60] , Vector bundles on the projective plane. Proc. London Math. Soc. 11 (1961), 623–640.
[61] I. Shafarevich, Basic Algebraic Geometry 1. Springer-Verlag, 1994.
[62] D. Simson and A. Skowroński, Elements of the Representation Theory of Associative Algebras, III. London Mathematical Society Student Texts 72. Cambridge University Press, 2007.
[63] R. Westwick, Spaces of matrices of fixed rank. Linear & Multilinear Algebra 20 (1987), 171–174.
[64] J. Worch, Categories of modules for elementary abelian p-groups and generalized Beilinson algebras. J. London Math. Soc. 88 (2013), 649–668.