Kronecker Representations and Steiner Bundles]Representations of Kronecker Quivers and Steiner Bundles on Grassmannians

D. Bissinger and R. Farnsteiner]Daniel Bissinger and Rolf Farnsteiner

[Daniel Bissinger]Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel, Germany
[Rolf Farnsteiner]Mathematisches Seminar, Christian-Albrechts-Universität zu Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel, Germany

Let $\KK$ be an algebraically closed field. Connections between representations of the generalized Kronecker quivers $K_r$ and vector bundles on $\PP^{r-1}$ have been known for quite some time. This
article is concerned with a particular aspect of this correspondence, involving more generally Steiner bundles on Grassmannians $\Gr_d(\KK^r)$ and certain full subcategories $\repp(K_r,d)$ of relative projective
$K_r$-representations. Building on a categorical equivalence first explicitly established by Jardim and Prata [40], we employ representation-theoretic techniques provided by Auslander-Reiten theory and reflection
functors to organize indecomposable Steiner bundles in a manner that facilitates the study of bundles enjoying certain properties such as uniformity and homogeneity. Conversely, computational results on Steiner bundles
motivate investigations in $\repp(K_r,d)$, which elicit the conceptual sources of some recent work on the subject.

From a purely representation-theoretic vantage point, our paper initiates the investigation of certain full subcategories of the, for $r\!\ge\!3$, wild category of $K_r$-representations. These may be characterized as being
right Hom-orthogonal to certain algebraic families of elementary test modules.

§ INTRODUCTION

Let $k$ be an algebraically closed field of characteristic $p\!>\!0$. In their groundbreaking article [32], Friedlander and Pevtsova associated vector bundles to representations of infinitesimal group schemes by
means of the so-called universal nilpotent operators. In subsequent work [18], Carlson-Friedlander-Suslin computed some of these bundles for the second Frobenius kernel $\GG_{a(2)}$ of the additive group
$\GG_a$. The bundles considered in [18] are kernels of the nilpotent operators associated to the so-called $W$-modules. These modules are graded with respect to the standard grading of the “group algebra"
$k\GG_{a(2)}$, and the grading induces a decomposition of the associated  vector bundles.

The vector bundles studied in [18] are defined on the projective line $\PP^1$ and hence decompose into a direct sum of Serre shifts of the structure sheaf. For group schemes of type $\GG_{a(r)}$ there are natural
generalizations of $W$-modules, whose bundles have base space $\PP^{r-1}$. When studying their graded pieces as mentioned above, one is naturally led to bundles on $\PP^{r-1}$ that are associated to certain
representations of the $r$-Kronecker quiver $K_r$. The observation that their duals are examples of the so-called Steiner bundles motivated the investigations of this paper.

Since the introduction of the notion of Steiner bundles by Dolgachev-Kapranov [23], the original definition has been generalized to include vector bundles over projective varieties with short resolutions or
coresolutions given by certain exceptional pairs. We will confine our attention to Steiner bundles of Grassmannians, whose defining resolution is given by the universal bundle and trivial bundles, cf. [1]. In the
aforementioned context of group schemes, the study of the interplay between representations of elementary abelian $p$-groups and vector bundles on Grassmannians was initiated in [16]. Being based on
morphisms between Serre shifts  of trivial vector bundles, the approach in loc. cit. differs from the point of view taken here, which can also be adapted to representations of elementary abelian groups.

The by now numerous connections between vector bundles on $\PP^{r-1}$ and representations of the generalized Kronecker quiver $K_r$, were apparently first systematically exploited in Hulek's article
[39].[Hulek considers representations of dimension vectors $(n,n)$, whereas our approach is based on dimension vectors $(m,n)$ with $m\!<\!n$.] While in most cases, the authors are working over the
complex numbers, the examples alluded to above guide us to work in greater generality: Throughout, $\KK$ is assumed to be an algebraically closed field of arbitrary characteristic. All vector spaces are assumed to be
finite-dimensional over $\KK$.

Our paper is organized as follows. After recalling basic features of representations of Kronecker quivers, we introduce in Section <ref> the full subcategories $\repp(K_r,d)$ of the category $\rep(K_r)$ of
representations of the Kronecker quiver that turn out to correspond to Steiner bundles on the Grassmannian of $d$-planes of $\KK^r$. Their objects may be described as those having trivial rank varieties; they may also be
thought of as relative projective modules of certain ring extensions. The category $\repp(K_r,1)$ coincides with the category $\EKP(K_r)$ of equal kernels representations, whose definition was inspired by [18], and
which was studied in our context in a number of papers including [64, 8]. Our first main result, Theorem <ref>, describes the objects $M \in \repp(K_r,d)$ via a family $(E(\fv))_{\fv \in \Gr_d(\KK^r)}$ of
elementary test modules. This observation is the gateway for our applications of the shift functor $\sigma_{K_r} : \rep(K_r) \lra \rep(K_r)$ and Auslander-Reiten theory. We summarize some of our findings as follows:

Suppose that $r\!\ge\!3$ and $d \in \{1,\ldots, r\!-\!1\}$. Then the following statements hold:

* A representation $M \in \rep(K_r)$ belongs to $\repp(K_r,d)$ if and only if $\Hom_{K_r}(E(\fv),M)\!=\!(0)$ for every $\fv \in \Gr_d(\KK^r)$.

* $\repp(K_r,d) \subseteq \EKP(K_r)$ is a torsion-free class containing all preprojective representations.

* We have $\repp(K_r,r\!-\!1) \subseteq \repp(K_r,r\!-\!2) \subseteq \cdots \subseteq \repp(K_r,2) \subseteq \repp(K_r,1)\!=\!\EKP(K_r)$.

* We have $\sigma_{K_r}^{-1}(\repp(K_r,d)) \subseteq \repp(K_r,r\!-\!1)$.

Generalizing a consequence of Westwick's fundamental result [63], we introduce, for a given $M\!=\!(M_1,M_2, (M(\gamma_i))_{1\le i \le r}) \in \rep(K_r)$ and $d \in \{1,\ldots, r\}$, the invariant $\Delta_M(d)\!:=\!
\dim_\KK M_2\!-\!d\dim_\KK M_1 \in \ZZ$ and show that $\Delta_M(d)\!\ge \min\{\dim_\KK M_1,d\}(r\!-\!d)$ whenever $M \in \repp(K_r,d)$. Moreover, if $(V_1,V_2)$ is a pair of vector spaces such that
$\Delta_{(V_1,V_2)}(d)\!\ge\!d(r\!-\!d)$, then there always exists $M \in \repp(K_r,d)$ such that $(M_1,M_2)\!=\!(V_1,V_2)$.

Given $d \in \{1,\ldots, r\!-\!1\},$ we introduce in Section <ref> the functor $\TilTheta_d : \rep(K_r) \lra \Coh(\Gr_d(\KK^r))$ which assigns to a representation of $K_r$ a coherent sheaf on $\Gr_d(\KK^r)$. The essential
image of $\repp(K_r,d)$ under $\TilTheta_d$ coincides with the full subcategory $\StVect(\Gr_d(\KK^r))$ of Steiner bundles. By definition (see [1]), each Steiner bundle $\cF$ affords a short resolution

\[ (0) \lra \cU_{(r,d)}^s \lra \cO_{\Gr_d(\KK^r)}^t \lra \cF \lra (0),\]

where $\cU_{(r,d)}$ denotes the universal subbundle on $\Gr_d(\KK^r)$. Since $\repp(K_r,d) \subseteq \repp(K_r,e)$ for $e\!\le\!d$, one can in fact associate Steiner bundles $\TilTheta_e(M) \in
\StVect(\Gr_e(\KK^r))$ to each $M \in \repp(K_,d)$ ($e\!\le\!d$). The functor $\TilTheta_e : \repp(K_r,d) \lra \StVect(\Gr_e(\KK^r))$ is full and faithful and $\TilTheta_d : \repp(K_r,d) \lra \StVect(\Gr_d(\KK^r))$ is an
equivalence, cf. [40]. We record a few basic properties of $\TilTheta_d$ and, by way of illustration, interpret the results of Section <ref> in the context of Steiner bundles.

The present understanding of the wild category $\rep(K_r)$ mainly rests on Kac's Theorem [43] concerning the distribution of the dimension vectors of indecomposable representations and the Auslander-Reiten
quiver $\Gamma(K_r)$, which describes $\rep(K_r)$ modulo the square of its radical. In Section <ref> we introduce these tools and provide first applications concerning exceptional Steiner bundles, which have also
been referred to as Fibonacci bundles, cf. [14]. Much of our subsequent work on non-exceptional Steiner bundles builds on Ringel's determination [56] of the regular Auslander-Reiten
components of $\Gamma(K_r)$. This enables us to identify the parts of AR-components belonging to $\repp(K_r,d)$ and thus yields a way to efficiently organize the corresponding Steiner bundles of $\Gr_d(\KK^r)$, see
Section <ref>.[In case $r\!=\!2$, it is well-known that $\Coh(\PP^1)$ is a $\Hom$-finite abelian $\KK$-category with Serre duality. It therefore affords almost split sequences, see <cit.>.]

In the next two sections, we apply the aforementioned concepts in the context of homogeneity, uniformity and stability, with the latter two only being discussed in the classical setting of projective spaces. Since the
inhomogeneity of an indecomposable Steiner bundle is inherited by all other members of its component, one can easily construct families of inhomogeneous bundles that are uniform of certain type. There are various
notions of equivariance for modules and vector bundles, which are equivalent to homogeneity in some cases, such as modules with trivial space endomorphisms or over fields of characteristic $0$. We intend to address
these connections along with those involving rational representations of parabolic subgroups of $\SL(A_r)$ in future work, where it turns out to be expedient to interpret vector bundles as locally trivial vector space fibrations.

Given $M \in \EKP(K_r)\!\smallsetminus\!\{(0)\}$ one defines the slope $\mu(M)\!:=\!\frac{\dim_\KK M_1}{\Delta_M(1)}$ and observes that $\mu(\TilTheta_1(M))\!=\!\mu(M)$ for the standard slope on $\Vect(\PP^{r-1})$. This
motivates the investigation of stability for modules and Steiner bundles and our main result in Section <ref> shows that modules corresponding to stable Steiner bundles are stable. Non-projective modules $M \in
\repp(K_r,d)$ of minimal type are stable and our results of Section <ref> concerning the distribution of slopes relative to AR-components imply that elementary $\EKP$-modules also enjoy this property. This is applied in
Section <ref>, where we provide a sufficient condition on a pair $(a,b) \in \NN^2$ involving the Tits form for $\EKP$-modules of dimension vector $(a,b)$ to afford a filtration by elementary $\EKP$-modules. Specializing to $r\!=\!3$, we use methods by Drezet [24] to give a sufficient condition for the stability of a bundle corresponding to a stable Kronecker representation. In particular, we verify:

Let $r\!\ge\!3$.

* Suppose that $M \in \EKP(K_r)\!\smallsetminus\!\{(0)\}$.

* If $\TilTheta_1(M)$ is (semi)stable, so is $M$.

* Let $r\!=\!3$ and $\Delta_M(2)\!\ge\!0$ ($\Delta_M(2)\!>\!0$). If $M$ is semistable (stable), so is $\TilTheta_1(M)$.

* Let $r\!=\!3$. If $M$ is elementary, then $\TilTheta_1(M)$ is stable.

* Let $\cC \subseteq \Gamma(K_3)$ be an Auslander-Reiten component containing an elementary module. Then every Steiner bundle $\cF \in \TilTheta_1(\cC\cap\EKP(K_3))$ possesses a Harder-Narasimhan filtration
by Steiner bundles, whose filtration factors are  stable.

The final section is concerned with an application pertaining to restrictions of Steiner bundles on $\PP^{r-1}(\CC)$ to linear hyperplanes $H \subseteq \PP^{r-1}(\CC)$. We provide conditions on the rank and the first Chern
class ensuring that a generic Steiner bundle $\cF$ with these data is stable with none of its restrictions $\cF|_H$ being semistable.

The authors would like to thank Chiara Brambilla for patiently explaining her work to them, Adrian Langer for bringing reference [11] to their attention, and Enrique Arrondo and Simone Marchesi for correspondence.

§ PRELIMINARIES

§.§ Representations of $K_r$

Let $r\!\ge\!2$ be a natural number. We denote by $K_r$ the (generalized) Kronecker quiver with two vertices $1$ and $2$ and $r$ arrows:

\[ \begin{tikzcd} 1 \arrow[r, draw=none, "\raisebox{+1.5ex}{\vdots}" description] \arrow[r, bend left, "\gamma_1"]  \arrow[r, bend right, swap,  "\gamma_r"] & 2. \end{tikzcd} \]

(For $r\!=\!1$ one obtains the quiver $K_1$ with underlying Dynkin diagram $A_1$.)

It will be convenient to view representations of $K_r$ in the following equivalent ways:

(i) A representation $M$ of the quiver $K_r$ is a triple $(M_1,M_2,(M(\gamma_i))_{1\le i \le r})$, where $M_1, M_2$ are
finite-dimensional vector spaces and each $M(\gamma_i) : M_1 \lra M_2$ is a $\KK$-linear map. A homomorphism $f : M \lra N$ is a pair $(f_1,f_2)$ of $\KK$-linear maps $f_j : M_j \lra N_j \ (j \in \{1,2\})$ such that,
for each $i \in \{1,\ldots,r\}$, the diagram

\[ \begin{tikzcd}   M_1 \arrow[r, "M(\gamma_i)"] \arrow[d,"f_1"] & M_2 \arrow[d,"f_2"] \\
N_1  \arrow[r,"N(\gamma_i)"] & N_2
\end{tikzcd} \]

(ii) Let $A_r\!:=\!\bigoplus_{i=1}^r\KK\gamma_i$ be the arrow space of $K_r$. We consider the triangular matrix algebra

\[ \KK K_r := \begin{pmatrix} \KK & 0 \\ A_r & \KK \end{pmatrix},\]

which is isomorphic to the path algebra of $K_r$. Accordingly, $\rep(K_r)$ is equivalent to the category $\modd \KK K_r$ of finite-dimensional $\KK K_r$-modules. In view of <cit.>, the category $\modd
\KK K_r$ is equivalent to the category $\cK_r$, whose objects are triples $M\!=\!(M_1,M_2, \psi_M)$, where $M_1,M_2$ are finite-dimensional $\KK$-vector spaces and $\psi_M : A_r\!\otimes_\KK\!M_1 \lra M_2$ is $\KK$-
linear. A morphism $M \lra N$ is a pair $(f_1,f_2)$ of $\KK$-linear maps $f_i: M_i \lra N_i$ such that the diagram

\[ \begin{tikzcd}   A_r\!\otimes_\KK\!M_1 \arrow[r, "\psi_M"] \arrow[d,"\id_{A_r}\otimes f_1"] & M_2 \arrow[d,"f_2"] \\
A_r\!\otimes_\KK\!N_1  \arrow[r,"\psi_N"] & N_2
\end{tikzcd} \]

(iii) One can equally well consider triples $(M_1,M_2,\psi_M)$, where $\psi_M : A_r \lra \Hom_\KK(M_1,M_2)$ is $\KK$-linear. The condition for a morphism $(f_1,f_2)$ then reads

\[ \psi_N(a)\circ f_1 = f_2\circ \psi_M(a) \ \ \ \ \forall \ a \in A_r.\]

(iv) For future reference we also record the interpretation as triples $(M_1,M_2,\varphi_M)$, where $\varphi_M : M_1 \lra \Hom_\KK(A_r,M_2)$ is given by $\varphi_M(m)(a)\!:=\!\psi_M(a \otimes m)$ for all
$m \in M_1$ and $a \in A_r$.

We denote by $\rep(K_r)$ the category of all (finite-dimensional) representations of $K_r$ and refer the reader to [4] and [2] for further details.

Every subspace $\fv \subseteq A_r$ gives rise to a subalgebra

\[ \KK.\fv := \begin{pmatrix} \KK & 0 \\ \fv & \KK \end{pmatrix}\]

of $\KK K_r$. If $M$ is a $\KK K_r$-module, the restriction $M|_{\KK.\fv}$ corresponds to $(M_1,M_2,\psi_{M,\fv})$, where $\psi_{M,\fv}\!=\!\psi_M|_{\fv\otimes_\KK M}$. By choosing a basis of $\fv$ one obtains an
equivalence $\modd \KK.\fv \cong \rep(K_{\dim_\KK\!\fv})$, which depends on the choice of the basis.

Given a pair $(V_1,V_2)$ of $\KK$-vector spaces, we let $\udim(V_1,V_2)\!\:=\!(\dim_\KK V_1,\dim_\KK V_2)$ be its dimension vector. For $M \in \rep(K_r)$, we write $\udim M\!:=\!\udim(M_1,M_2)$. The category
$\rep(K_r)$ has two simple objects (representations) $S_1,S_2$ of dimension vectors $(1,0)$ and $(0,1)$, respectively. The object $S_1$ is injective, while $S_2$ is projective. The projective cover $P_1(r)$ of
$S_1$ has dimension vector $\udim P_1(r)\!=\!(1,r)$. We will also write $P_0(r)\!:=\!S_2$.

Note that

\[ \omega_r : \KK K_r \lra \KK K_r \ \ ; \ \ \left(\begin{smallmatrix} \alpha & 0 \\ a & \beta \end{smallmatrix}\right) \mapsto \left(\begin{smallmatrix} \beta & 0 \\ a & \alpha \end{smallmatrix}\right)\]

is an anti-automorphism of order $2$. There results a duality $D_{K_r}: \modd \KK K_r \lra \modd \KK K_r  \   \  ; \  \   M  \mapsto M^\ast$ given by

\[ (x.f)(m) := f(\omega_r(x).m) \ \ \ \ \ \ \forall \ f \in M^\ast, m \in M, x \in \KK K_r.\]

Let $M \in \rep(K_r)$. For $a \in A_r$, we denote by

\[ a_M : M_1 \lra M_2 \ \ ; \ \ m \mapsto a.m\]

the multiplication effected by $a$. Then the dual $D_{K_r}(M)$ of $M \in \modd \KK K_r$ corresponds to $D_{K_r}(M)\!=\!(M_2^\ast, M_1^\ast, \psi_{D_{K_r}(M)})$, where

\[ \psi_{D_{K_r}(M)}(a\otimes f) = f\circ a_M \ \ \ \  \  \forall \ a \in A_r, f \in M_2^\ast.\]

We denote by

\[ \langle \, , \, \rangle_r : \ZZ^2\!\times\!\ZZ^2 \lra \ZZ \ \ ; \ \ (x,y) \mapsto x_1y_1\!+\!x_2y_2\!-\!rx_1y_2\]

the Euler-Ringel bilinear form. The corresponding quadratic form

\[ q_r : \ZZ^2 \lra \ZZ \ \; \ \ x \mapsto \langle x,x\rangle_r\]

is referred to as the Tits quadratic form of $K_r$.

The dimension vectors give rise to an isomorphism

\[ \udim : K_0(\rep(K_r)) \lra \ZZ^2,\]

which identifies the Grothendieck group $K_0(\rep(K_r))$ of $\rep(K_r)$ with $\ZZ^2$. Recall that

\[\langle \udim M,\udim N\rangle_r\!=\!\dim_\KK\Hom_{K_r}(M,N)\!-\!\dim_\KK\Ext^1_{K_r}(M,N)\]

for all $M,N \in \rep(K_r)$. Moreover, as the path algebra $\KK K_r$ is hereditary, we have $\Ext^i_{K_r}(-,-)\!=\!(0)$ for all $i\!\ge\!2$.

Let $M \in \modd \KK K_r$. Then $\Ext^1_{K_r}(M,\KK K_r)$ is a right $\KK K_r$-module, so that

\[ \tau_{K_r}(M)\!:=\!\Ext^1_{K_r}(M,\KK K_r)^\ast\]

is a left $\KK K_r$-module. There results a (covariant) endofunctor

\[ \tau_{K_r} : \modd \KK K_r \lra \modd \KK K_r\]

which, by virtue of $\KK K_r$ being hereditary, is left exact. By the same token, $\tau_{K_r}$ is exact when restricted to the full subcategory of those modules which have no non-zero projective direct summands.

The functor $\tau_{K_r}$ induces an endofunctor on $\rep(K_r)$, which we will also denote by $\tau_{K_r}$. General theory <cit.> implies that

\[ \udim \tau_{K_r}(M) = \Phi_r.\udim M\]

for every non-projective indecomposable $M \in \rep(K_r)$,[When multiplying matrices with vectors, we will always implicitly assume that the vector has the correct shape and thus dispense with transposing vectors.]

\[ \Phi_r\!=\! \begin{pmatrix} r^2\!-\!1 & -r \\ r & -1 \end{pmatrix} \in \GL_2(\ZZ)\]

is the Coxeter matrix of the quiver $K_r$. An indecomposable representation $M \in \rep(K_r)$ is called preprojective, provided $\tau^n_{K_r}(M)\!=\!(0)$ for some $n \in \NN$.

Similarly, we consider the functor

\[ \tau_{K_r}^{-1} : \modd \KK K_r \lra \modd \KK K_r \ \ ; \  \ M \mapsto \Ext^1_{K_r}(M^\ast, \KK K_r).\]

Then we have

\[ \udim \tau^{-1}_{K_r}(M) = \Phi^{-1}_r.\udim M\]

for every non-injective indecomposable representation $M \in \rep(K_r)$. An indecomposable representation $M \in \rep(K_r)$ is called preinjective, provided $\tau^{-n}_{K_r}(M)\!=\!(0)$ for some $n \in \NN$.
Indecomposable representations that are neither preprojective nor preinjective are said to be regular.

The preprojective $\KK K_r$-modules are well understood. For $r\!\ge\!3$, we put $L_r\!:=\!\frac{r+\sqrt{r^2-4}}{2}$ and set

\[ a_i(r) := \left\{ \begin{array}{cc} \frac{(L_r)^i-(\frac{1}{L_r})^i}{\sqrt{r^2-4}} & r\!\ge\!3 \\ i & r\!=\!2 \end{array} \right.  \ \ \ \ \forall\ i \in \NN_0.\]

For each $i \in \NN_0$ there is, up to isomorphism, exactly one preprojective module $P_i(r)$ such that

(a) $P_0(r)\!=\!S_2$; $P_1(r)$ is the projective cover of $S_1$, and

(b) $\udim P_i(r)\!=\!(a_i(r),a_{i+1}(r))$ for all $i \in \NN_0$.
For every $i \in \NN_0$ there is an almost split sequence

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

In particular, $\tau^{-1}_{K_r}(P_i(r))\cong P_{i+2}(r)$.

In case the module category to be considered is clear from the context, we will write $P_i$ instead of $P_i(r)$.

§.§ Module varieties
In this section, we recall a number of general results from the representation theory of quivers, most of which can be found in [43].

Let $A$ be an $n$-dimensional associative $\KK$-algebra, $V$ be a $\KK$-vector space. We denote by $\modd(A;V)$ the affine variety of $A$-module structures on $V$. When
convenient we will identify $M \in \modd(A;V)$ with its representation $\varrho_M : A \lra \End_\KK(V)$. If $\{a_1,\ldots, a_n\}$ is a basis of $A$, then

\[ \iota_V : \modd(A;V) \lra \End_\KK(V)^n \ \ ; \ \ M \mapsto (\varrho_M(a_1), \ldots, \varrho_M(a_n))\]

is a closed embedding.

If $X$ is a variety and $x \in X$, then $\dim_xX$ denotes the local dimension of $X$ at $x$.

Let $V$ and $W$ be $\KK$-vector spaces. The map

\[ d : \modd(A;V)\!\times\!\modd(A;W) \lra \NN_0 \ \ ; \ \ (M,N) \mapsto \dim_\KK\Hom_A(M,N)\]

is upper semicontinuous. $\square$

In analogy with the above, one can consider the variety $\rep(K_r;V_1,V_2)$ of representations of $K_r$ on a pair $(V_1,V_2)$ of vector spaces. There is an isomorphism

\[  \rep(K_r;V_1,V_2) \stackrel{\sim}{\lra} \modd(\KK K_r;V_1\!\oplus\!V_2)^{\rightarrow}\]

between $\rep(K_r;V_1,V_2)$ and the closed subset  $\modd(\KK K_r;V_1\!\oplus\!V_2)^{\rightarrow}$ of the module variety of the path algebra $\KK K_r$, given by conditions $(\begin{smallmatrix} 1 & 0 \\ 0 & 0
\end{smallmatrix})\dact (v_1\!\oplus\!v_2)\!=\!v_1$ and $(\begin{smallmatrix} 0 & 0 \\ 0 & 1\end{smallmatrix})\dact (v_1\!+\!v_2)\!=\!v_2.$

Recall that $M \in \rep(K_r)$ is referred to as a brick, provided $\End_{K_r}(M)\cong \KK$. Preprojective and preinjective representations are known to be bricks.

Suppose that $r\!\ge\!3$ and let $V_1,V_2$ be vector spaces with $V_1\!\oplus\!V_2\!\ne\!(0)$ and such that $q_r(\udim (V_1,V_2))\!\le\!1$. Then

\[ \cB_{(V_1,V_2)} := \{ M \in \rep(K_r;V_1,V_2) \ ; \ M \ \text{is a brick} \}\]

is a dense open subset of $\rep(K_r;V_1,V_2)$.

Since $r\!\ge\!3$, the assumption $q_r(a,b)\!=\!0$ implies $a\!=\!0\!=\!b$.

If $q_r(\udim(V_1,V_2))\!\le\!0$, we may apply <cit.> (see also <cit.> for an elementary proof valid in our context) in conjunction with the observation above to see that the set
$\cB_{(V_1,V_2)}$ is not empty. Alternatively, general theory provides $M \in \rep(K_r;V_1,V_2)$ that is preprojective or preinjective (see Theorem <ref> below). In particular, $M$ is a brick.

Lemma <ref> implies that the map $\rep(K_r;V_1,V_2) \lra \NN_0 \ ; \ M \mapsto \dim_\KK\End_{K_r}(M)$ is upper semicontinuous, so that $\cB_{(V_1,V_2)}$ is open. Since the variety $\rep(K_r;V_1,V_2)
\cong \Hom_\KK(V_1,V_2)^r$ is irreducible, $\cB_{(V_1,V_2)}$ thus lies dense in $\rep(K_r;V_1,V_2)$.

Let $V_1,V_2$ be non-zero $\KK$-vector spaces. The algebraic group $G\!:=\!\GL(V_2)\!\times\!\GL(V_1)$ acts on the irreducible variety $\Hom_\KK(V_1,V_2)^r$ via

\[ (g_2,g_1)\dact (f_i)_{1\le i\le r} := (g_2\circ f_i\circ g_1^{-1})_{1\le i\le r}.\]

Upper semicontinuity of fiber dimension implies that the map

\[ \Hom_\KK(V_1,V_2)^r \lra \NN_0 \ \ ; \ \ x \mapsto \dim G\dact x\]

is lower semicontinuous. Thus, if $s\!:=\!\max\{ \dim G\dact x \ ; \ x \in \Hom_\KK(V_1,V_2)^r\}$, then the open sheet

\[ \cO_{(V_1,V_2)} := \{ x \in \Hom_\KK(V_1,V_2)^r \ ; \ \dim G\dact x\!=\!s\}\]

for the action of $G$ on $\Hom_\KK(V_1,V_2)^r$ is a non-empty open subset of $\Hom_\KK(V_1,V_2)^r\!=\!\rep(K_r;V_1,V_2)$.

The following result, which was first proved by Kac <cit.>, actually shows that for those dimension vectors for which $\rep(K_r;V_1,V_2)$ contains indecomposable modules, the modules belonging to the open
sheet are just the bricks.

Let $r\!\ge\!3$. Suppose that $V_1,V_2$ are vector spaces with $V_1\!\oplus\!V_2\!\ne\!(0)$ and such that $q_r(\udim(V_1,V_2))\!\le\!1$. Then we have $\cB_{(V_1,V_2)}\!=\!\cO_{(V_1,V_2)}$.

Given any $M \in \rep(K_r;V_1,V_2)$, the stabilizer $G_M$ of $M$ coincides with $G\cap\End_{K_r}(M)$, and thus is a dense open subset of $\End_{K_r}(M)$.

By Corollary <ref>, there is $M_0 \in \cB_{(V_1,V_2)}\cap\cO_{(V_1,V_2)}$, so that

\[ s = \dim G\dact M_0 = \dim G\!-\!\dim G_{M_0} = \dim G\!-\!1.\]

Let $M \in \cB_{(V_1,V_2)}$. Then $\dim G_M\!=\!1$, so that $\dim G\dact M\!=\!s$. This implies $\cB_{(V_1,V_2)}\! \subseteq \!\cO_{(V_1,V_2)}$.

On the other hand, if $M \in \cO_{(V_1,V_2)}$, then $\dim_\KK\End_{K_r}(M)\!=\!\dim G_M\!=\!\dim G\!-\!s\!=\!1$, whence $M \in \cB_{(V_1,V_2)}$.

§.§ Restrictions
For $d \in \{1,\ldots, r\}$, we let $\Gr_d(A_r)$ be the Grassmannian of $d$-planes of $A_r$. This is an irreducible projective variety of dimension $\dim \Gr_d(A_r)\!=\!d(r\!-\!d)$.
For $M \in \modd \KK K_r$ and $\fv \in \Gr_d(A_r)$, we denote by $M|_\fv$ the restriction of $M$ to $\KK.\fv \subseteq \KK K_r$.

Given $\KK$-vector spaces $V_1,V_2$, we consider the set

\[ \Iso(K_d;V_1,V_2)\!:=\!\rep(K_d;V_1,V_2)/(\GL(V_2)\!\times\!\GL(V_1))\]

of isomorphism classes together with the projection map $\pi : \rep(K_d;V_1,V_2) \lra \Iso(K_d;V_1,V_2)$. We equip $\Iso(K_d;V_1,V_2)$ with the final topology relative to $\pi$. This renders $\pi$ a continuous map
such that $U \subseteq \Iso(K_d;V_1,V_2)$ is open if and only if $\pi^{-1}(U) \subseteq \rep(K_r;V_1,V_2)$ is open.

\[ \Inj_\KK(A_d,A_r) := \{ \alpha \in \Hom_\KK(A_d,A_r) \ ; \ \alpha \ \text{is injective}\}.\]

This is a conical, open subset of the affine space $\Hom_\KK(A_d,A_r)$. In particular, $\Inj_\KK(A_d,A_r)$ is a quasi-affine variety.

The algebraic groups $\GL(A_d)$ and $\GL(A_r)$ act on the affine variety $\Hom_\KK(A_d,A_r)$ via

\[ g.\alpha := \alpha\circ g^{-1} \ \ \ \forall \ g \in \GL(A_d) \ \ \text{and} \ \ h.\alpha := h\circ \alpha \ \ \ \forall \ h \in \GL(A_r),\]

respectively. The subvariety $\Inj_\KK(A_d,A_r) \subseteq \Hom_k(A_d,A_r)$ is stable with regard to these commuting actions.

Every $\alpha \in \Inj_\KK(A_d,A_r)$ defines an injective homomorphism

\[ \alpha : \KK K_d \lra \KK K_r \   \   ;  \   \   \left(\begin{smallmatrix} x & 0 \\ a & y \end{smallmatrix} \right) \lra \left(\begin{smallmatrix} x & 0 \\ \alpha(a) & y \end{smallmatrix}\right) \]

of $\KK$-algebras. In particular, $\GL(A_r)$ can be considered a group of automorphisms of $\KK K_r$.

Let $M \in \modd \KK K_r$. Given $\alpha \in \Inj_\KK(A_d,A_r)$, we let $\alpha^\ast(M) \in \modd \KK K_d$ be the pull-back of $M$ along $\alpha$. Thus, we have

\[ u \dact m := \alpha(u)m \ \ \ \ \ \forall \ u \in \KK K_d, m \in M.\]

In particular, $\GL(A_r) \subseteq \Aut_\KK(\KK K_r)$ acts on $\modd \KK K_r$ via pull-back: If $M \in \modd \KK K_r$, and $g \in \GL(A_r)$, then

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

We say that $M$ is homogeneous, provided $M^{(g)} \cong M$ for every $g \in \GL(A_r)$.

Let $P_i\!=\!(k[X,Y]_{i-1},k[X,Y]_i, X\cdot, Y\cdot)$ be the preprojective $\KK K_2$-module of dimension vector $(i,i\!+\!1)$. Then $\GL(A_2)\!=\!\GL(\KK X\!\oplus\!\KK Y)$
acts of $\KK [X,Y]$ via algebra homomorphisms of degree zero in such a way that

\[ g(\gamma f) = g(\gamma) g(f) \ \ \ \ \ \ \ \ \ \forall \ g \in \GL(A_2), \gamma \in A_2, f \in \KK[X,Y]_{i-1}.\]

Consequently, $g$ induces a $\KK$-linear isomorphism $P_i^{(g)} \stackrel{\sim}{\lra} P_i$ such that

\[ g(\gamma \dact f) = g(g^{-1}(\gamma)f) = \gamma g(f).\]

As a result, the $\KK K_2$-module $P_i$ is homogeneous. (In fact, the $P_i$ are examples of equivariant $K_r$-representations.)

We let $\GL(A_r)$ act on $\Gr_d(A_r)$ via $h\dact \fv\!:=\!h(\fv)$ for all $h \in \GL(A_r)$ and $\fv \in \Gr_d(A_r)$. This action is transitive.

Following [64], we denote by $\EKP(K_r)$ the full subcategory of $\rep(K_r)$ such that $a_M$ is injective for every $a \in A_r\!\smallsetminus\!\{0\}$.[The notation derives from the
correspondence with the category of equal kernels modules studied in [18].] Note that $\EKP(K_r)$ is closed under taking subobjects.

The classification of indecomposable objects in $\rep(K_2)$ implies that every $N \in \EKP(K_2)$ decomposes into a direct sum

\[ N \cong \bigoplus_{i\ge 0} n_iP_i.\]

Let $M \in \EKP(K_r)$. Then we have

\[ \alpha^\ast(M) \cong \beta^\ast(M)\]

for all $\alpha,\beta \in \Inj_\KK(A_2,A_r)$ such that $\im\alpha\!=\!\im\beta$.

Let $\alpha \in \Inj_\KK(A_2,A_r)$. Since $\alpha$ is injective, it follows that $\alpha^\ast(M) \in \EKP(K_2)$. By virtue of the foregoing remark and the example above, we conclude that  $\alpha^\ast(M)$
is homogeneous.

If $\beta \in \Inj_\KK(A_2,A_r)$ is such that $\im \alpha\!=\!\im \beta$, then $g\!:=\!\beta^{-1}\circ \alpha \in \GL(A_2)$, so that for $m \in (\alpha^\ast(M))^{(g)}$ and $\gamma \in A_2$, we obtain

\[ \gamma\dact m = g^{-1}(\gamma).m = \alpha(g^{-1}(\gamma))m = \beta(\gamma)m.\]

By the example above, this shows that $\beta^\ast(M)\cong (\alpha^\ast(M))^{(g)} \cong \alpha^\ast(M)$.

Let  $M \in \EKP(K_r)$, $\fv \in \Gr_2(A_r)$. In view of the foregoing result, we may define the isomorphism class $[M|_\fv]$ via

\[ [M|_\fv] := [\alpha^\ast(M)] \ \ \ \ \ \alpha \in \Inj_\KK(A_2,A_r)\ ; \  \im \alpha\!=\!\fv.\]

Less formally, we will write

\[ M|_\fv \cong \bigoplus_{i\ge 0} n_i(M,\fv) P_i(2)\]

to indicate that the above isomorphism holds for $\alpha^\ast(M)$, where $\alpha \in \Inj_\KK(A_2,A_r)$ is such that $\im\alpha\!=\!\fv$.

Given $X \in \rep(K_2;V_1,V_2)$, we write

\[ \dim_\KK\Hom_{K_2}(X,M|_\fv) := \dim_\KK\Hom_{K_2}(X,\alpha^\ast(M)).\]

for some $\alpha \in \Inj_\KK(A_2,A_r)$ with $\im\alpha\!=\!\fv$.

Let $r\!\ge\!2$, $M \in \EKP(K_r)\cap \rep(K_r;V_1,V_2)$. Then the following statements hold:

* The map

\[ \res_M : \Gr_2(A_r) \lra \Iso(K_2;V_1,V_2) \ \ ; \ \  \fv \mapsto [M|_\fv]\]

is continuous.

* If $X \in \rep(K_2;U_1,U_2)$, then the map

\[ d_{X,M} : \Gr_2(A_r) \lra \NN_0 \ \ ; \ \ \fv \mapsto \dim_\KK\Hom_{K_2}(X,M|_\fv)\]

is upper semicontinuous.

(1) It follows from  <cit.> that the map

\[ \msim :  \Inj_\KK(A_2,A_r) \lra \Gr_2(A_r) \ \ ; \ \ \alpha \mapsto \im\alpha\]

is a principal $\GL(A_2)$-bundle (a $\GL(A_2)$-torsor). In particular,

\[ \overline{\msim} : \Inj_\KK(A_2,A_r)/\GL(A_2) \lra \Gr_2(A_r) \ \ ; \ \ \bar{\alpha} \mapsto \im\alpha\]

is a homeomorphism.

Consider the map

\[ \widetilde{\res}_M : \Inj_\KK(A_2,A_r) \lra  \rep(K_2;V_1,V_2) \ \ ; \ \ \alpha \mapsto \alpha^\ast(M).\]

We identify $\rep(K_2;V_1,V_2)$ with $\Hom_\KK(V_1,V_2)^2$ and let $\{\gamma_1,\gamma_2\}$ be a basis of $A_2$. Then

\[ \widehat{\res}_M : \Hom_\KK(A_2,A_r) \lra \Hom_\KK(V_1,V_2)^2 \ \ ; \ \ \alpha \mapsto (M(\alpha(\gamma_1)),M(\alpha(\gamma_2)))\]

is a linear map and hence in particular a morphism. (Here we have written $M(\alpha(\gamma_i))\!:=\!\psi_M(\alpha(\gamma_i)\!\otimes -)$.) Consequently,
$\widetilde{\res}_M\!=\!\widehat{\res}_M|_{\Inj_\KK(A_2,A_r)}$ is also a morphism.

In view of Lemma <ref>, $\widetilde{\res}_M$ gives rise to  a continuous map

\[ \iso_M : \Inj_\KK(A_2,A_r)/\GL(A_2) \lra \Iso(K_2;V_1,V_2) \ \ ; \ \ \bar{\alpha} \mapsto [\alpha^\ast(M)].\]

\[ \res_M \circ \overline{\msim} = \iso_M,\]

we obtain (1).

(2) Thanks to Lemma <ref>, the map

\[ d_X : \rep(K_2;V_1,V_2) \lra \NN_0 \ \ ; \ \ Y \mapsto \dim_\KK\Hom_{K_2}(X,Y)\]

is upper semicontinuous. Since images of $(\GL(V_2)\!\times\!\GL(V_1))$-stable open sets under $\pi : \rep(K_2;V_2,V_2)$ $ \lra \Iso(K_2;V_1,V_2)$ are open, the map

\[ \bar{d}_X : \Iso(K_2;V_1,V_2) \lra \NN_0 \ \ ; \ \ [Y] \mapsto \dim_\KK\Hom_{K_2}(X,Y)\]

enjoys the same property. In view of (1), $d_{X,M}\!=\! \bar{d}_X\circ \res_M$ is upper semicontinuous as well.

Let $M \in \EKP(K_r)$. Then there exists a sequence $(n_i(M))_{i \in \NN_0} \in \NN_0^{\NN_0}$ such that

\[ O_M := \{ \fv \in \Gr_2(A_r) \ ; \  n_i(M,\fv)\!=\!n_i(M) \ \ \ \forall \ i \in \NN_0\}\]

is a dense open subset of $\Gr_2(A_r)$.

For $i \in \NN_0$ we consider the map $d_i\!:=\!d_{P_i(2),M}$. Proposition <ref> provides $c_i \in \NN_0$ such that

\[ O_i := \{ \fv \in \Gr_2(A_r) \ ; \ d_i(\fv) = c_i\}\]

is a dense open subset of $\Gr_2(A_r)$. Since $\dim_\KK\Hom_{K_2}(P_i(2),P_j(2))\!=\!\max\{j\!-\!i\!+\!1,0\}$, it follows that

\[ n_i(M,\fv) = d_i(\fv)\!-\!2d_{i+1}(\fv)\!+\!d_{i+2}(\fv)\]

for every $\fv \in \Gr_2(A_r)$ and $i \in \NN_0$. Consequently,

\[ n_i(M,\fv) = c_i\!-\!2c_{i+1}\!+\!c_{i+2}\]

for every $\fv \in O_i\cap O_{i+1}\cap O_{i+2}$. There exists $\ell \in \NN_0$ such that $n_i(M,\fv)\!=\!0\!=\! d_i(\fv)$ for all $i\!\ge\!\ell$ and $\fv \in \Gr_2(A_r)$. We define

\[ n_i(M):= c_i\!-\!2c_{i+1}\!+\!c_{i+2}.\]

for every $i \in \NN_0$ and obtain $n_i(M)\!=\!0\!=n_i(M,\fv)$ for all $i\!\ge\!\ell$. Consequently, $\bigcap_{i=0}^{\ell-1} O_i \subseteq O_M$.

Suppose that $\fv \in O_M$. Then we have

\[ d_i(\fv)\!-\!2d_{i+1}(\fv)\!+\!d_{i+2}(\fv) = n_i(M,\fv) =  c_i\!-\!2c_{i+1}\!+\!c_{i+2} \ \ \ \ 0\!\le\!i\!\le\!\ell\!-\!1.\]

Since the $(\ell\!\times\!\ell)$-matrix

\[ A = \begin{pmatrix} 1 & -2 & 1 & 0 & \cdots & 0 \\
0 &  1 & -2 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\
0 & 0 &  \cdots & 1 & -2 & 1 \\
0 & 0 & \cdots  & 0 & 1 & -2\\
0 & 0 & \cdots & 0 & 0 & 1 \end{pmatrix}\]

is invertible, it follows that $d_i(\fv)\!=\!c_i$ for all $i \in \NN_0$. Consequently, $\fv \in \bigcap_{i=0}^{\ell-1}O_i$, so that $O_M\!=\!\bigcap_{i=0}^{\ell-1}O_i$ is a dense, open subset of $\Gr_2(A_r)$.

Let $M \in \EKP(K_r)\!\smallsetminus\!\{(0)\}$. We call

\[ M_{\gen} :=  \bigoplus_{i\ge 0} n_i(M)P_i\]

the generic decomposition of $M$.

We let $\PP(A_r)$ be the projective space of lines in $A_r$. The basis $\{\gamma_1,\ldots, \gamma_r\}$ defines an isomorphism $\PP(A_r) \cong \PP^{r-1}$ of projective varieties.  For $x \in \PP(A_r)$ the number
$\rk(x_M)\!=\! \rk(a_M) \ \text{for some} \ a \in x\!\smallsetminus\!\{0\}$ is well-defined and we refer to

\[ \rk(M) := \max\{\rk(x_M) \ ; \ x \in \PP(A_r)\}\]

as the generic rank of $M$. We say that $M$ has constant rank, provided $\rk(x_M)\!=\!\rk(M)$ for all $x \in \PP(A_r)$. The full subcategory of $\rep(K_r)$, whose  objects have constant rank, will be denoted
$\CR(K_r)$. It contains the full subcategory $\EIP(K_r)$, whose objects $M$ satisfy the condition $\rk(M)\!=\!\dim_\KK M_2$. We have a duality $D_{K_r} : \CR(K_r) \lra \CR(K_r)$ that sends $\EKP(K_r)$ to $\EIP(K_r)$ and
vice versa. The aforementioned classification of indecomposable $K_2$-representations implies that $\CR(K_2)\!=\!\add(\EKP(K_2)\cup\EIP(K_2))$, so that every $M \in \CR(K_r)$ affords a generic decomposition

\[ M_{\gen} = \bigoplus_{i\ge 0} n_i(M)P_i(2)\!\oplus\!\bigoplus_{j\ge 0} m_j(M) I_j(2),\]

where $I_j(2)\!:=\!D_{K_r}(P_j(2))$ is indecomposable preinjective.

§ CATEGORIES OF RELATIVE PROJECTIVE MODULES

Let $d \in \{1,\ldots, r\!-\!1\}$. In this section we introduce the full subcategory $\repp(K_r,d)$ of $\EKP(K_r)$, that will turn out to be equivalent to the category of Steiner bundles on the Grassmannian $\Gr_d(A_r)$.
The objects of $\repp(K_r,d)$ may be characterized via rank varieties, whose definition is analogous to those investigated in [16]. By contrast to the setting in [16], their complements may also be
defined as being the objects of the right $\Hom$-orthogonal category of a class of elementary $K_r$-modules.

§.§ Rank varieties and families of elementary modules

As noted above, the map

\[ \msim : \Inj_\KK(A_d,A_r) \lra \Gr_d(A_r) \  \  ;  \ \ \alpha \mapsto \im \alpha\]

is a $\GL(A_d)$-torsor with respect to the canonical right $\GL(A_d)$-action $\alpha.h\!:=\!\alpha\circ h$, cf. <cit.>.

Let $M \in \rep(K_r)$. Then the following statements hold:

* The subset

\[ \cP(K_r,d)_M := \{ \alpha \in \Inj_\KK(A_d,A_r) \ ; \ \alpha^\ast(M) \ \text{is not projective}\}\]

is closed and $\GL(A_d)$-stable.

* The set

\[\cV(K_r,d)_M := \{ \fv \in \Gr_d(A_r) \ ; \ \exists \ \alpha \in \msim^{-1}(\fv) \ \text{such that} \ \alpha^\ast(M) \ \text{is not projective}\}\]

is a closed subset of $\Gr_d(A_r)$.

(1) Recall that a $\KK K_d$-module $N$ is projective if and only if $\dim_\KK\Ext^1_{K_d}(N,S_2)\!=\!0$. In view of the upper semicontinuity of the map $\rep(K_d;M_1,M_2) \lra \NN_0 \ ; \ N \mapsto
\dim_\KK\Ext^1_{K_d}(N,S_2)$,[This follows for instance from Lemma <ref> and $\dim_\KK\Ext^1_{K_d}(N,S_2)\!=\!\dim_\KK \Hom_{K_d}(N,S_2)\!-\!\langle \udim N,\udim S_2\rangle_d$.] we
conclude that

\[ C_d := \{ X \in \rep(K_d;M_1,M_2) \ ; \ X \ \text{is not projective}\}\]

is a closed subset of $\rep(K_d;M_1,M_2)$. The arguments of the proof of Proposition <ref> ensure the continuity of the map

\[ \Inj_\KK(A_d,A_r) \lra \rep(K_d;M_1,M_2) \ \ ; \ \ \alpha \mapsto \alpha^\ast(M).\]

Consequently, $\cP(K_r,d)_M$ is a closed subset of $\Inj_\KK(A_d,A_r)$.

Recall that $P_1$ and $S_2$ are representatives of the isomorphism classes of the projective indecomposable $\KK K_d$-modules. Since their dimensions pairwise differ, it follows that $Q^{(h)} \cong Q$ for
$Q \in \{P_1,S_2\}$ and $h \in \GL(A_d)$ and hence $P^{(h)} \cong P$ for every projective $\KK K_d$-module $P$. Let $\alpha \not \in \cP(K_r,d)_M$ and $h \in \GL(A_d)$. Then

\[ (h\dact\alpha)^\ast(M) \cong (\alpha^\ast(M))^{(h)},\]

is projective, so that $h\dact\alpha \not\in \cP(K_r,d)_M$. As a result, the variety $\cP(K_r,d)_M$ is $\GL(A_d)$-stable.

(2) As before, we consider the space $\Inj_\KK(A_d,A_r)/\GL(A_d)$ together with the topology induced by the canonical projection

\[ \pi : \Inj_\KK(A_d,A_r) \lra \Inj_\KK(A_d,A_r)/\GL(A_d).\]

As observed in the proof of Proposition <ref>, the map

\[ \msim : \Inj_\KK(A_d,A_r) \lra \Gr_d(A_r) \ \ ; \ \ \alpha \mapsto \im \alpha \]

induces a homeomorphism

\[ \overline{\msim} :  \Inj_\KK(A_d,A_r)/\GL(A_d) \lra \Gr_d(A_r) \ \ ; \ \ [\alpha] \mapsto \im \alpha.\]

In view of (1), the open set $U_M\!:=\!\Inj_\KK(A_d,A_r)\!\smallsetminus\!\cP(K_r,d)_M$ is $\GL(A_d)$-stable. Thus, $\pi^{-1}(\pi(U_M))\!=\!U_M$, so that $\pi(U_M)\!=\!(\Inj_\KK(A_d,A_r)/\GL(A_d))\!\smallsetminus\!
\pi(\cP(K_r,d)_M)$ is an open subset of  $\Inj_\KK(A_d,A_r)/\GL(A_d)$. It follows that $\cV(K_r,d)_M\!=\!\msim(\cP(K_r,d)_M)\!=\!\overline{\msim}(\pi(\cP(K_r,d)_M))$ is closed.

Since $\GL(A_d)$ acts transitively on $\im^{-1}(\fv)$, it follows from (1) that

\[ \cV(K_r,d)_M = \{ \fv \in \Gr_d(A_r) \ ; \  \alpha^\ast(M) \ \text{is not projective for all}  \ \alpha \in \msim^{-1}(\fv) \}.\]

Let $(0) \lra M' \lra M \lra M'' \lra (0)$ be an exact sequence in $\rep(K_r)$.

* We have

\[ \cV(K_r,d)_{M'} \subseteq \cV(K_r,d)_M \subseteq \cV(K_r,d)_{M'}\cup\cV(K_r,d)_{M''}.\]

* If the sequence splits, then

\[  \cV(K_r,d)_M = \cV(K_r,d)_{M'}\cup\cV(K_r,d)_{M''}. \]

(1) Since the algebra $\KK K_d$ is hereditary, we have $\cV(K_r,d)_{M'} \subseteq \cV(K_r,d)_M$.

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

We consider the projective $K_r$-representations $(0,A_d)\cong P_0(r)^d$ and $(\KK,A_r)\cong P_1(r)$, where we have $\psi_{P_1(r)}(a\!\otimes\!t)\!=\!t a$. An element $\alpha \in \Hom_\KK(A_d,A_r)$ can be viewed as a

\[ \bar{\alpha} : (0,A_d) \lra (\KK,A_r) \ \ ; \ \  \bar{\alpha}_1\!:=\!0 \ \ \text{and} \ \ \bar{\alpha}_2\!:=\!\alpha.\]

Let $M \in \rep(K_r)$. As noted in Section <ref>, $M$ may be interpreted as a linear map

\[ \varphi_M : M_1 \lra \Hom_\KK(A_r,M_2) \ \ ; \ \  \varphi_M(m)(a)\!:=\!\psi_M(a\!\otimes m).\]

Given $\alpha \in \Hom_\KK(A_d,A_r)$, we obtain

\[ \varphi_{\alpha^\ast(M)} : M_1 \lra \Hom_\KK(A_d,M_2) \   \  ; \ \ m \mapsto \varphi_M(m)\circ \alpha.\]

A representation $M \in \rep(K_r)$ is called regular, if all its indecomposable constituents are regular. We recall the notion of elementary representations, which are just the simple objects in the full subcategory $\reg(K_r) \subseteq \rep(K_r)$ of regular representations.

A non-zero regular representation $E \in \rep(K_r)$ is referred to as elementary, provided there does not exist a short exact sequence

\[ (0) \lra A \lra E \lra B \lra (0)\]

with $A,B$ non-zero and regular.

Let $\alpha \in \Inj_\KK(A_d,A_r)$. Then the following statements hold:

* The module $\coker\bar{\alpha}$ is elementary with dimension vector $\udim \coker\bar{\alpha}\!=\!(1,r\!-\!d)$.

* Let $\beta \in \Inj_\KK(A_d,A_r)$. Then $\coker\bar{\alpha}\cong \coker\bar{\beta}$ if and only if $\im \alpha\!=\!\im\beta$.

* If $N\in \rep(K_r)$ is indecomposable with $\udim N\!=\!(1,r\!-\!d)$, then there is $\zeta \in \Inj_\KK(A_d,A_r)$ such that $N \cong \coker\bar{\zeta}$.

* Given $M \in \rep(K_r)$, we have an isomorphism of vector spaces $\coker\varphi_{\alpha^\ast(M)} \cong \Ext^1_{K_r}(\coker\bar{\alpha},M)$.

* Given $c \in \{1,\ldots, d\}$ and $\beta \in \Inj_\KK(A_c,A_d)$, there exists an epimorphism $\coker\overline{\alpha\circ \beta} \lra \coker\bar{\alpha}$.

(1)-(3) Recall that the projective cover $P_1(r)$ of the simple representation $S_1\!=\!(\KK,0)$ is given by $P_1(r)\!=\!(\KK,A_r,(P_1(r)(\gamma_i))_{1 \leq i \leq r})$ with $P_1(r)(\gamma_i)(\lambda)\!=\!\lambda
\gamma_i$ for all $i \in \{1,\ldots,r\}$. For $\alpha \in \Inj_\KK(A_d,A_r)$ we consider the monomorphism of representations $x_\alpha\!=\!(0,(x_\alpha)_2) \colon P_0(r)^d$ $\lra P_1(r)$, given by

\[(x_\alpha)_2 \colon \KK^d \lra A_r \ \ ; \ \ \lambda \mapsto \sum^d_{i=1}  \lambda_i \alpha(\gamma_i).\]

We get a commutative diagram

\[ \xymatrix{
(0) \ar[r] & P_0(r)^d \ar^{x_\alpha}[r] \ar^{\cong}[d] & P_1(r) \ar[r] \ar^{\cong}[d]& \coker x_\alpha \ar[r] \ar[d] & (0)  \\
(0) \ar[r] & (0,A_d) \ar^{\overline{\alpha}}[r] & (\KK,A_r) \ar[r] & \coker \overline{\alpha} \ar[r] & (0)

with exact rows. In particular,  $\udim\coker \overline{\alpha}\!=\!(1,r\!-\!d)$ and the 5-Lemma yields $\coker x_\alpha \cong \coker \overline{\alpha}$. Hence (1), (2) and (3) follow from  <cit.>,

(4) The statement follows from a small modification of the proof of <cit.>. For the benefit of the reader we outline the changes. Since $\Ext^1_{K_r}((\KK,A_r),-)\!=\!(0)$, application of $\Hom_{K_r}(-,M)$ to the
short exact sequence

\[(0) \lra (0,A_d) \stackrel{\overline{\alpha}}{\lra} (\KK,A_r) \lra \coker \overline{\alpha} \lra (0)\]

yields the following diagram with exact rows

\[
\xymatrix{
\Hom_{K_r}((\KK,A_r),M) \ar^{\overline{\alpha}^\ast}[r] \ar^{f}[d]& \Hom_{K_r}((0,A_d),M) \ar[r] \ar^{g}[d] & \Ext^1_{K_r}(\coker \overline{\alpha},M) \ar[r] & (0)  \\
M_1 \ar^{\varphi_{\alpha^{\ast}(M)}}[r] &  \Hom_\KK(A_d,M_2) \ar[r] & \coker \varphi_{\alpha^{\ast}(M)} \ar[r] & (0),

where $f(h)\!:=\!h_1(1_\KK)$ for all $h \in \Hom_{K_r}((\KK,A_r),M)$ and $g(\eta)\!:=\!\eta_2$ for all $\eta \colon (0,A_d) \lra M$. Let $(h_1,h_2)\!=\!h \in \Hom_{K_r}((\KK,A_r),M)$. We recall that $\psi_M \circ (\id_{A_r}
\otimes h_1)\! =\! h_2 \circ \psi_{(\KK,A_r)}$ and $\psi_{(\KK,A_r)}(a \otimes \lambda)\! =\! \lambda a$ for all $\lambda \in \KK$ and $a \in A_r$. For $a \in A_d$ we obtain

\begin{align*}
[(\varphi_{\alpha^\ast(M)} \circ f)(h)](a) &= [\varphi_M(h_1(1_\KK)) \circ \alpha](a) = [\psi_M \circ (\id_{A_r} \otimes h_1)](\alpha(a) \otimes 1_\KK)\\
&= [h_2 \circ \psi_{(\KK,A_r)}](\alpha(a) \otimes 1_\KK) = h_2(\alpha(a)) =  (h \circ \overline{\alpha})_2(a)  \\
&= [(g \circ \overline{\alpha}^{\ast})(h)](a).
\end{align*}

Hence the left-hand square of the diagram commutes and $f, g$ being isomorphisms gives us

\[\dim_\KK \coker \varphi_{\alpha^{\ast}(M)} = \dim_\KK \Ext^1_{K_r}(\coker \overline{\alpha},M).\]

(5) This follows from the proof of <cit.>.

Let $\fv \in \Gr_d(A_r)$. Thanks to Proposition <ref>(2), we may define

\[ \coker \fv := \coker \bar{\alpha} \ \ \ \ \ \ \alpha \in \msim^{-1}(\fv).\]

For $\fv \in \Gr_d(A_r)$, we put

\[ E(\fv) := D_{K_r}(\tau_{K_r}(\coker \bar{\alpha})) \ \ \ \ \ \ \alpha \in \msim^{-1}(\fv).\]

In view of <cit.> $E(\fv)$ is an elementary representation.

Given $M \in \rep(K_r)$ and $\fv \in \Gr_d(A_r)$, we recall the notation

\[ \psi_{M,\fv} := (\psi_M)|_{\fv\otimes_\KK M_1}.\]

Similarly, we define $\varphi_{M,\fv} : M_1\lra \Hom_\KK(\fv,M_2)$ via

\[ \varphi_{M,\fv}(m) := \varphi_M(m)|_\fv  \ \ \ \ \forall \ m \in M_1.\]

Let $M \in \rep(K_r)$, $\fv \in \Gr_d(A_r)$.

* Let $\alpha \in \msim^{-1}(\fv)$. Then

\[\dim_\KK M_2\!-\!\rk(\psi_{M,\fv}) = \dim_\KK M_2\!-\!\dim_\KK\Rad(\alpha^{\ast}(M))\]

is the multiplicity of $P_0(d)$ in the decomposition of $\alpha^{\ast}(M) \in \rep(K_d)$ into indecomposable direct summands.

* We have $\dim_\KK \ker \psi_{M,\fv}\!=\! \dim_\KK \Hom_{K_r}(E(\fv),M)$.

(1) This follows immediately from the definition.

(2) We consider the dual representation $D_{K_r}(M) = (M_2^{\ast},M_1^{\ast},\psi_{D_{K_r}(M)} )$ with structure map

\[\psi_{D_{K_r}(M)} \colon A_r\!\otimes_\KK\!M_2^{\ast} \lra M_1^{\ast} \ \ ; \ \  a \otimes f  \mapsto f \circ a_M.\]

There results a commutative diagram

\[\xymatrixcolsep{6pc} \xymatrix{
M_2^{\ast} \ar^{(\psi_{M,\fv})^{\ast}}[r] \ar@{=}[d]& (\fv\!\otimes_\KK\!M_1)^{\ast} \ar_{\cong}^{\eta}[d]\\
M_2^{\ast} \ar^{\varphi_{D_{K_r}(M),\fv}}[r] & \Hom_\KK(\fv,M_1^{\ast}),

with $\eta(f)(a)(m_1)\!=\!f(a \otimes m_1)$ for all $a \in \fv, m_1 \in M_1$ and $f \in (\fv\!\otimes_\KK\!M_1)^{\ast}$: For $h \in M_2^\ast, a \in \fv$ and $m \in M_1$ we have

\begin{align*}
[ \varphi_{D_{K_r}(M),\fv}(h)(a)](m) &= [ \psi_{D_{K_r}(M)}(a \otimes h)](m) = (h \circ a_M)(m)\\
&= (h \circ \psi_M)(a \otimes m) = (h \circ \psi_{M,\fv})(a \otimes m) \\
&= [\eta(h \circ \psi_{M,\fv})(a)](m) = [(\eta \circ (\psi_{M,\fv})^{\ast})(h)(a)](m).
\end{align*}

Hence $\varphi_{D_{K_r}(M),\fv}\!=\!\eta \circ (\psi_{M,\fv})^{\ast}$. Consequently, $\coker \varphi_{D_{K_r}(M),\fv} \cong \ker \psi_{M,\fv}$.

Let $\alpha \in \im^{-1}(\fv)$. We conclude with Proposition <ref>(2),(4) that

\[\dim_\KK\coker\varphi_{\alpha^\ast(D_{K_r}(M))} =\dim_\KK\coker\varphi_{D_{K_r}(M),\fv} = \dim_\KK\ker\psi_{M,\fv}.\]

Since $\coker\overline{\alpha}$ is regular, the Auslander-Reiten formula <cit.> in conjunction with Proposition <ref>(4) now yields

\begin{align*}
\dim_\KK\ker\psi_{M,\fv} & = \dim_\KK \coker \varphi_{\alpha^{\ast}(D_{K_r}(M))} = \dim_\KK \Ext^1_{K_r}(\coker \overline{\alpha},D_{K_r}(M)) \\
&= \dim_\KK \Hom_{K_r}(D_{K_r}(M),\tau_{K_r}(\coker \overline{\alpha})) \\
&=\dim_\KK \Hom_{K_r}(E(\fv),M) \hfill \qedhere.
\end{align*}

For a pair  $(V_1,V_2)$ of $\KK$-vector spaces and $d\in \{1,\ldots,r\}$, we set

\[ \Delta_{(V_1,V_2)}(d) := \dim_\KK V_2\!-\!d\dim_\KK V_1.\]

If $M \in \rep(K_r)$, we write

\[\Delta_M(d) := \Delta_{(M_1,M_2)}(d).\]

Let $M \in \rep(K_r)$. Given $\fv \in \Gr_d(A_r)$, the following statements are equivalent:

* $\Hom_{K_r}(E(\fv),M)\!=\!(0)$.

* For every $\alpha \in \msim^{-1}(\fv)$, the map $\psi_{\alpha^\ast(M)}$ is injective.

* There is $\alpha \in \msim^{-1}(\fv)$ such that $\psi_{\alpha^\ast(M)}$ is injective.

* There is $\alpha \in \msim^{-1}(\fv)$ such that $\alpha^\ast(M) \cong \Delta_M(d)P_0(d)\!\oplus\!(\dim_kM_1)P_1(d)$.

* $\fv \not\in \cV(K_r,d)_M$.

(1) $\Rightarrow$ (2). Let $\alpha \in \im^{-1}(\fv)$. Proposition <ref>(2) yields $0\!=\!\dim_\KK\Hom_{K_r}(E(\fv),M)\!=\!\dim_\KK \ker \psi_{\alpha^\ast(M)}$.

(2) $\Rightarrow$ (3). Trivial.

(3) $\Rightarrow$ (4). Let $\alpha \in \msim^{-1}(\fv)$ be such that $\psi_{\alpha^\ast(M)}$ is injective. Then we have

\begin{align*}
\dim_\KK M_2\! -\! \rk(\psi_{M,\fv}) & = \dim_\KK M_2\!-\!\dim_\KK (\fv\!\otimes_\KK\!M_1) \\
&= \dim_\KK M_2\! -\! d \dim_\KK M_1 = \Delta_M(d).
\end{align*}

By Proposition <ref>(1), we may write $\alpha^{\ast}(M)\!=\!{\Delta_M(d)} P_0(d)\!\oplus\!N$, where $N \in \rep(K_d)$ does not have $P_0(d)$ as a direct summand. Therefore, a projective cover of $N$ is given by
$(\dim_\KK N_1)P_1(d) \twoheadrightarrow N$. Since $\dim_\KK N_1\!=\!\dim_\KK M_1$ and

\[ \dim_\KK N_2 = \dim_\KK M_2\!-\! \Delta_M(d) = d \dim_\KK M_1, \]

we conclude $\udim N\! =\! \udim [(\dim_\KK N_1) P_1(d)]$. Hence $\alpha^{\ast}(M) \cong \Delta_M(d) P_0(d)\!\oplus\! (\dim_\KK M_1)\!P_1(d)$.

(4) $\Rightarrow$ (5). This is a direct consequence of the remark following Lemma <ref>.

(5) $\Rightarrow$ (1). Let $\alpha \in \msim^{-1}(\fv)$. We write $\alpha^{\ast}(M) \cong a P_0(d)\! \oplus\!b P_1(d)$, so that

\[ \dim_\KK \Rad(\alpha^{\ast}(M)) = \dim_\KK \Rad(a P_0(d)\!\oplus\! b P_1(d)) = b \dim_\KK \Rad(P_1(d)) = b d.\]

Since $\udim P_0(d)$ and $\udim P_1(d)$ are linearly independent, it follows that $a\!=\!\Delta_M(d)$ and $b\!=\! \dim_\KK M_1$. We obtain
$\dim_\KK \ker \psi_{M,\fv}\! = \! d \dim_\KK M_1\! -\! \rk(\psi_{M,\fv})\! =\! d \dim_\KK M_1\!  -\! \dim_\KK\Rad(\alpha^{\ast}(M))\!=\!0$, so that Proposition <ref> yields (1).

Let $M \in \rep(K_r)$. Theorem <ref> implies

\[ \cV(K_r,d)_M = \{\fv \in \Gr_d(A_r) \ ; \  \rk(\psi_{M,\fv})\!<\!d\dim_\KK M_1\},\]

so that we refer to $\cV(K_r,d)_M$ as the $d$-th rank variety of $M$.

(1) For $d\!=\!1$, we have $\Gr_1(A_r)\!=\!\PP(A_r)$ and

\[ \cV(K_r,1)_M := \{ x \in \PP(A_r) \ ; \ \rk(x_M)\!<\!\dim_\KK M_1\}.\]

In particular, $\cV(K_r,1)_M\!=\!\emptyset$ if and only if $M \in \EKP(K_r)$.

(2) Let $\alpha \in \Inj_\KK(A_d,A_r)$. The pull-back functor $\alpha^\ast : \rep(K_r) \lra \rep(K_d)$ takes projectives to projectives. Theorem <ref> implies that
$\beta \in \GL(A_d).\alpha$ if and only if for every $M \in \rep(K_r)$ we have $\alpha^\ast(M) \ \text{projective} \ \Leftrightarrow \ \beta^\ast(M) \ \text{projective}$. Hence
$\Inj_\KK(A_d,A_r)/\GL(A_d)$ may be viewed as an analogue of the space of equivalence classes of $p$-points (in the context of abelian unipotent group schemes), cf. [31].

§.§ The categories $\repp(K_r,d)$

As before, we fix $d \in \{1,\ldots, r\!-\!1\}$. In this section, we introduce the subcategory $\repp(K_r,d)$ of $\rep(K_r)$ that will be instrumental in our study of Steiner bundles on the Grassmannian $\Gr_d(A_r)$.

We let $\repp(K_r,d)$ be the full subcategory of $\rep(K_r)$, whose objects $M$ are given by $\cV(K_r,d)_M\!=\!\emptyset$.

(1) Note that $\repp(K_r,1)\!=\!\EKP(K_r)$ is just the category of equal kernels representations, which usually has wild representation type, cf. <cit.>. The roughly analogous concept of shifted
cyclic subgroups for elementary abelian $p$-groups yields projective modules, see <cit.>.

(2) The category $\repp(K_r,d)$ is closed under taking submodules, cf. Lemma <ref>.

(3) Suppose that $M \in \repp(K_r,d)$ is such that $\dim_\KK M_1\!\le\!d$. Then $A_r\!\otimes_\KK\!M_1\!=\!\bigcup_{\fv \in \Gr_d(A_r)}\fv\otimes_\KK\!M_1$, so that $\psi_M$ is injective. Consequently, the direct
summand $(M_1,\psi_M(M_1),\psi_M)$ of $M$ is isomorphic to $(\dim_\KK M_1)P_1(r)$ and $M\cong \Delta_M(r)P_0(r)\!\oplus\!(\dim_\KK M_1)P_1(r)$ is projective.

(4) An object $M \in \repp(K_r,d)$ can be characterized by saying that $\alpha^\ast(M)$ is projective for every $\alpha \in \Inj_\KK(A_d,A_r)$. Equivalently, for $M \in \modd \KK K_r$ the restrictions
$M|_{\KK .\fv} \in \modd \KK .\fv$ are projective for every $\fv \in \Gr_d(A_r)$. The latter condition can be shown to be equivalent to $M$ being $(\KK K_r, \KK .\fv)$-projective in the sense of relative homological
algebra, cf. [38].

The following statements hold:

* $\repp(K_r,d) \subseteq \EKP(K_r)$ is a torsion free class that is closed under $\tau^{-1}_{K_r}$.

* $\repp(K_r,d)$ contains all preprojective representations.

* We have $\repp(K_r,r\!-\!1) \subseteq \repp(K_r,r\!-\!2) \subseteq \cdots \subseteq \repp(K_r,1) = \EKP(K_r)$.

(1) and (2) follow from Theorem <ref> and <cit.>.

(3) Let $M \in \repp(K_r,b)$ for some $1\! <\! b\!<\! r$ and consider $\fv \in \Gr_{b-1}(A_r)$. Then there is $\fw \in \Gr_b(A_r)$ such that $\fv \subseteq \fw$. Theorem <ref> yields

\[ \ker\psi_{M,\fv} \subseteq \ker\psi_{M,\fw} = (0),\]

so that $M \in \repp(K_r,b\!-\!1)$.

Let $X \subseteq \Gr_d(A_r)$ be a subset. Using Theorem <ref> and <cit.>, one can show that (1) and (2) of the foregoing result hold for the full subcategory $\rep_X(K_r,d)$, whose objects
$M$ satisfy $\cV(K_r,d)_M \subseteq X$.

In the sequel the shift functors $\sigma_{K_r},\sigma^{-1}_{K_r}: \rep(K_r) \lra \rep(K_r)$ will be of major importance. These functors correspond to the BGP-reflection functors but take into account that the opposite
quiver of $K_r$ is isomorphic to $K_r$, i.e. $D_{K_r} \circ \sigma_{K_r} \cong \sigma^{-1}_{K_r} \circ D_{K_r}$, where $D_{K_r} \colon \rep(K_r) \lra \rep(K_r)$ denotes the standard duality.

Given a representation $M \in \rep(K_r)$, $\sigma_{K_r}(M)$ is by definition the representation

\[ (\sigma_{K_r}(M)_1,\sigma_{K_r}(M)_2) = (\ker \psi_M,M_1), \]

where we identify $\psi_M$ by means of the basis $\{\gamma_1,\ldots, \gamma_r\}$ with the map $\psi_M : (M_1)^r \lra M_2, (m_i) \mapsto \sum^{r}_{i=1} M(\gamma_i)(m_i)$. By definition, $[\sigma_{K_r}(M)](\gamma_i)
\colon \sigma_{K_r}(M)_1 \lra \sigma_{K_r}(M)_2 = \pi_{i}|_{\ker \psi_M}$, with $\pi_i \colon (M_1)^r \lra M_1$ being the projection onto the $i$-th component. If $f \in \Hom_{K_r}(M,N)$, then $\sigma_{K_r}(f)_1 : \sigma_{K_r}(M)_1 \lra \sigma_{K_r}(N)_1$ is the restriction to $\ker\psi_M$ of the map

\[ (m_i)_{i\le i\le r} \mapsto (f_1(m_i))_{1\le i \le r},\]

while $\sigma_{K_r}(f)_2\!:=\!f_1$.

According to <cit.>, $\sigma_{K_r}$ is left exact, while $\sigma_{K_r}^{-1}$ is right exact and the functor $\sigma_{K_r}$ induces an equivalence

\[ \sigma_{K_r} : \rep_2(K_r) \lra \rep_1(K_r)\]

between the full subcategories $\rep_i(K_r)$ of $\rep(K_r)$, whose objects don't have any direct summands isomorphic to $S_i$. By the same token, $\sigma_{K_r}^{-1}$ is a quasi-inverse of $\sigma_{K_r}$.
Moreover, $\sigma_{K_r}$ and $\sigma^{-1}_{K_r}$ induce quasi-inverse equivalences on the full subcategory $\reg(K_r) \subseteq \rep(K_r)$ of regular representations. We have $\sigma_{K_r} \circ \sigma_{K_r} \cong
\tau_{K_r}$ as well as $\sigma_{K_r}(P_{i+1}(r)) \cong  P_{i}(r)$ for all $i\!\geq\!0$.

Recall that $\{ \udim P_i(r) \ ; \  i \in \NN_0 \}$ consists exactly of those tuples $(a,b) \in \NN^2_0$ that satisfy $a\!<\!b$ and $q_r(a,b)\!=\!1$. Since all irreducible morphisms between preprojective representations in
$\rep(K_r)$ are injective, it follows that $P_i(r)$ is isomorphic to a subrepresentation of $P_j(r)$ if and only if $i\! \leq\!j$. We
also note that

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

for $M$ indecomposable and not isomorphic to $P_0(r)$, while $\sigma_{K_r}(P_0(r))\!=\!(0)$.  In conjunction with the left exactness of $\sigma_{K_r}$ this implies that $\sigma_{K_r} \colon \rep_2(K_r) \lra \rep_1(K_r)$ is exact. By the same token, $\sigma_{K_r}^{-1} \colon \rep_1(K_r) \lra \rep_2(K_r)$ is exact.

Let $1\! \leq \!d \!<\! r$. We have

\[\{ [\sigma_{K_r}(E(\fv))] \ ; \  \fv \in \Gr_d(A_r) \} = \{  [\coker \fw] \ ;  \ \fw \in \Gr_{r-d}(A_r) \}.\]

Let $\fv \in \Gr_d(A_r)$. Since $E(\fv)$ is regular indecomposable, the same holds for $\sigma_{K_r}(E(\fv))$. As $\udim \sigma_{K_r}(E(\fv))\!=\! \udim \sigma^{-1}_{K_r}(D_{K_r}(\coker \fv))\!=\!(1,d)$, it
follows from Proposition <ref>(3) that $\sigma_{K_r}(E(\fv))$ $\cong \coker \fw$ for some $\fw \in \Gr_{r-d}(A_r)$.

If $\fw \in \Gr_{r-d}(A_r)$, then the preceding argument provides $\fv \in \Gr_d(A_r)$ such that $\sigma_{K_r}(E(\fw)) \cong \coker \fv$. We therefore obtain

\[ \sigma_{K_r}(E(\fv)) \cong \sigma_{K_r}(D_{K_r}(\tau_{K_r}(\coker \fv))) \cong D_{K_r}(\sigma_{K_r}(\coker\fv)) \cong D_{K_r}(\tau_{K_r}(E(\fw))) \cong \coker \fw. \qedhere\]

Let $M,E \in \rep(K_r)$ with $E$ regular. We claim that

(i)  $\Hom_{K_r}(E,M) \cong \Hom_{K_r}(\sigma_{K_r}(E),\sigma_{K_r}(M))$, and

(ii) $\Hom_{K_r}(E,M) \cong \Hom_{K_r}(\tau_{K_r}(E),\tau_{K_r}(M))$.
In order to verify (i), we write $M\!=\!S_2^l\!\oplus\!N$, with $S_2$ not being a direct summand of $N$. Since there are no non-zero homomorphisms from non-zero regular representations to
projective representations (see <cit.>), we have $\Hom_{K_r}(E,S_2)\!=\!(0)$. Moreover, $\sigma_{K_r}(S_2)\!=\!(0)$ implies  $\sigma_{K_r}(M) \cong \sigma_{K_r}(N)$ and we conclude

\begin{eqnarray*}  \Hom_{K_r}(E,M)  & \cong   & \Hom_{K_r}(E,S_2^l)\!\oplus\! \Hom_{K_r}(E,N) \cong  \Hom_{K_r}(\sigma_{K_r}(E),\sigma_{K_r}(N))\\
& \cong & \Hom_{K_r}(\sigma_{K_r}(E),\sigma_{K_r}(M)).
\end{eqnarray*}

Since $\sigma_{K_r}^2 \cong \tau_{K_r}$, we also get (ii).

Let $1\! \leq\! d\! <\! r$ and $M \in \rep(K_r)$.

* $M \in \repp(K_r,d)$ if and only if $\Hom_{K_r}(\coker\fw,\sigma_{K_r}(M))\!=\!(0)$ for all $\fw \in \Gr_{r-d}(A_r)$.

* $M \in \repp(K_r,r\!-\!1)$ if and only if $\sigma_{K_r}(M) \in \EKP(K_r)$.

* We have $\sigma_{K_r}^{-1}(\repp(K_r,d)) \subseteq \repp(K_r,r\!-\!1)$. In particular, $\repp(K_r,d)$ is $\sigma^{-1}_{K_r}$-stable.

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

(2) In view of (i), we have $M \in \rep(K_r,r\!-\!1)$ if and only if

\[ (0) = \Hom_{K_r}(E(\fv),M) \cong \Hom_{K_r}(\sigma_{K_r}(E(\fv)),\sigma_{K_r}(M)) \cong  \Hom_{K_r}(D_{K_r}(\sigma_{K_r}(\coker \fv)),\sigma_{K_r}(M))\]

for all $\fv \in \Gr_{r-1}(A_r)$. Since $\dim_\KK D_{K_r}(\sigma_{K_r}(\coker \fv)) = (1,r\!-\!1)$ for all $\fv \in \Gr_{r-1}(A_r)$, the statement follows from Lemma <ref> and Proposition <ref>(3).

(3) Let $M\!=\!\sigma_{K_r}^{-1}(N)$ for some $N \in \repp(K_r,d)$. Let $\fv \in \Gr_d(A_r)$. Since $\dim_\KK \Hom_{K_r}(E(\fv),S_1)\!=\!\dim_\KK E(\fv)_1\!\neq\!0$, it follows that $N \in \rep_1(K_r)$.
Then Proposition <ref>(3) in conjunction with the equivalence $\rep_2(K_r) \lra \rep_1(K_r)$ yields $\sigma_{K_r}(M) \cong N \in \EKP(K_r)$, so that (2) ensures that $M \in \rep(K_r,r\!-\!1)$.
By Proposition  <ref>(3), this also implies that $\repp(K_r,d)$ is $\sigma^{-1}_{K_r}$-stable.

In view of $\sigma_{K_r}(P_{i+1}(r)) \cong  P_{i}(r)$ for all $i\!\geq\!0$, parts (2) and (3) of Theorem <ref> imply inductively that $P_i(r) \in \repp(K_r,r\!-\!1)$ for all $i\!\ge\!0$.

We finally record a topological property of the set of relative projective modules of fixed dimension vector. The relevance of the technical condition (2) will be clarified in the following section.

Let $V_1,V_2$ be $\KK$-vector spaces, $d\in \{1,\ldots,r\!-\!1\}$. Then the following statements hold:

* The set $\repp(K_r,d)\cap\rep(K_r;V_1,V_2) \!=\!\{M \in \rep(K_r;V_1,V_2) \ ; \ \cV(K_r,d)_M\!=\!\emptyset\}$ is open.

* If $\Delta_{(V_1,V_2)}(d)\!\ge\!d(r\!-\!d)$, then $\repp(K_r,d)\cap\rep(K_r;V_1,V_2)$ lies dense in $\rep(K_r;V_1,V_2)$.

(1) We interpret $\rep(K_r;V_1,V_2)$ as $\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)$. In view of Theorem <ref>, the relevant set is given by

\[ \cO_{(r,d)} := \{ \psi \in \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2) \ ; \ \psi\circ (\alpha\!\otimes \id_{V_1}) \in \Inj_\KK(A_d\!\otimes_\KK\!V_1,V_2) \ \ \ \forall \ \alpha \in \Inj_\KK(A_d,A_r)\} .\]

We consider the canonical map

\[ \msim : \Inj_\KK(A_d,A_r) \lra \Gr_d(A_r) \ \ ; \ \ \alpha \mapsto \im\alpha.\]

As noted before, this map defines a principal $\GL(A_d)$-bundle, and hence so does

\[ \kappa : \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Inj_\KK(A_d,A_r) \lra \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r) \ \ ; \ \ (\psi, \alpha) \mapsto (\psi,\im\alpha).\]

In particular, $\kappa$ is an open morphism.

For $\psi \in \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)$ and $\fv \in \Gr_d(A_r)$, we put $\psi_\fv\!:=\!\psi|_{\fv\otimes_\KK V_1}$. We consider the sets

\[ \cO_1\!:=\!\{(\psi, \alpha) \in \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Inj_\KK(A_d,A_r) \ ; \ \rk(\psi\circ (\alpha\otimes\id_{V_1}))\!=\!d\dim_\KK V_1\}\]

\[ \cO_2 \!:= \! \{(\psi,\fv) \in \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r) \ ; \ \rk(\psi_\fv)\!=\!d\dim_\KK V_1\}.\]

Since $\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Inj_\KK(A_d,A_r) \lra \Hom_\KK(A_d\!\otimes_\KK\!V_1,V_2) \ ; \ (\psi,\alpha) \mapsto \psi\circ(\alpha\otimes\id_{V_1})$ is a morphism, lower semicontinuity of ranks
ensures that $\cO_1$ is an open subset, so that

\[\cO_2 = \kappa(\cO_1)\]

is open as well. As a result,

\[ \cC_{(r,d)}\!:=\!\{(\psi,\fv) \in \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r) \ ; \ \rk(\psi_\fv)\!<\!d\dim_\KK V_1\}.\]

is closed.

As the projective variety $\Gr_d(A_r)$ is complete, the morphism $\pr : \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r) \lra \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2) \ ; \ (\psi,\fv) \mapsto \psi$ is closed.

\[\cX_{(r,d)}:=\pr(\cC_{(r,d)})\]

is a closed subset of the affine space $\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)$.

Suppose that $\psi \not \in \cO_{(r,d)}$. Then there is $\fv \in \Gr_d(A_r)$ such that $\rk(\psi_\fv)\!<\!d\dim_\KK V_1$, so that $(\psi,\fv) \in \cC_{(r,d)}$. It follows that $\psi \in \cX_{(r,d)}$. Conversely, if $\psi \in \cX_{(r,d)}$,
then there is $\fv \in \Gr_d(A_r)$ such that $(\psi,\fv) \in \cC_{(r,d)}$. Thus, there is $\alpha \in \Inj_\KK(A_d,A_r)$ such that $\rk(\psi \circ (\alpha\otimes\id_{V_1}))\!<\!d\dim_\KK V_1$, whence $\psi\not \in \cO_{(r,d)}$.

As an upshot of the above, we obtain that

\[ \cO_{(r,d)} = \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\smallsetminus\!\cX_{(r,d)}\]

is open.

(2) In view of (1), it suffices to show that $\repp(K_r,d)\cap\rep(K_r;V_1,V_2)\!\ne\!\emptyset$. Setting $n\!:=\!\dim_\KK V_1$ and $m\!:=\!\dim_\KK V_2$, we define for $\ell \in \{0,\ldots,n\}$

\[ \Hom_\KK(V_1,V_2)_{ \le \ell} :=  \{ f \in \Hom_\KK(V_1,V_2) \ ; \ \rk(f)\!\le\!\ell\}.\]

We proceed in several steps, beginning by recalling that (see <cit.>)

(i) $\Hom_\KK (V_1,V_2)_{\le \ell}$ is a closed, irreducible subspace of dimension $\ell(n\!+\!m\!-\!\ell)$.

Next, we verify the following claim

(ii) The closed subset $\cC_{(r,d)}$ defined in (1) is an irreducible variety of dimension

\[ \dim \cC_{(r,d)} = rmn\!-\!1\!-\!\Delta_{(V_1,V_2)}(d)\!+\!d(r\!-\!d).\]

In what follows, we let $\ell\!:=\!dn\!-\!1$. The algebraic group $\KK^\times$ acts on $\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r)$ via

\[ \alpha\dact(\psi,\fv) := (\alpha\psi,\fv) \ \ \ \ \forall \ (\psi,\fv) \in \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r).\]

Note that $\cC_{(r,d)}$ is $\KK^\times$-stable, so that every irreducible component $Z$ of $\cC_{(r,d)}$ enjoys the same property.

We consider the surjective morphism

\[ q : \cC_{(r,d)} \lra \Gr_d(A_r) \  \  ; \  \  (\psi,\fv) \mapsto \fv\]

as well as

\[ \iota_0 : \Gr_d(A_r) \lra \Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r) \ \ ; \ \  \fv \mapsto (0,\fv) \]

Let $Z$ be an irreducible component of $\cC_{(r,d)}$. Then $Z$ is a closed subset of $\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)\!\times\!\Gr_d(A_r)$. If $\fv \in q(Z)$, then there is $\psi \in
\Hom_\KK(A_r\!\otimes_\KK\!V_1,V_2)$ such that $(\psi,\fv) \in Z$, and the $\KK^\times$-stability mentioned above implies $\iota_0(\fv)\!=\!(0,\fv) \in \overline{\KK^\times\dact(\psi,\fv)} \subseteq Z$. Consequently, $q(Z)\!=\!
\iota_0^{-1}(Z)$ is closed.

Let $\fv \in \Gr_d(A_r)$. Writing $A_r\!=\!\fv\!\oplus\!\fw$, we obtain

\[ q^{-1}(\fv) \cong\Hom_\KK(\fv\!\otimes_\KK\!V_1,V_2)_{\le \ell}\!\times\! \Hom_\KK(\fw\!\otimes_\KK\!V_1,V_2),\]

so that (i) ensures that $q^{-1}(\fv)$ is irreducible of dimension

\[ \dim q^{-1}(\fv) = \ell(m\!+\!dn\!-\!\ell)\!+\!mn(r\!-\!d) = \ell(m\!+\!1)\!+\!mn(r\!-\!d).\]

In view of <cit.>, the variety $\cC_{(r,d)}$ is irreducible and the fiber dimension theorem <cit.> yields

\begin{eqnarray*}
\dim \cC_{(r,d)} & = & \ell(m\!+\!1)\!+\!mn(r\!-\!d) \!+\!d(r\!-\!d)\\
& = & (dn\!-\!1)(m\!+\!1)\!+\!mn(r\!-\!d)\!+\!d(r\!-\!d)\\
& = & dmn\!+\!\!dn\!-\!m\!-\!1\!+\!mn(r\!-\!d)\!+\!d(r\!-\!d)\\
& = & rmn\!-\!1\!-\!\Delta_{(V_1,V_2)}(d)\!+\!d(r\!-\!d),
\end{eqnarray*}

as desired. $\diamond$

By virtue of (ii) and our current assumption, the fiber dimension theorem implies

\[ \dim \cX_{(r,d)} \le \dim \cC_{(r,d)}  \le rmn\!-\!1 < \dim \Hom_\KK(A_r\!\otimes_\KK V_1,V_2),\]

so that $\cO_{(r,d)}\!=\!\Hom_\KK(A_r\!\otimes_\KK\! V_1,V_2)\!\smallsetminus\!\cX_{(r,d)}\!\ne\!\emptyset$. Hence there is $M \in \rep(K_r;V_1,V_2)$ such that $\cV(K_r,d)_M\!=\!\emptyset$.

§.§ Modules of minimal type and applications

It is a consequence of work of Westwick [63] that  for “most" objects $M \in \EKP(K_r)$ the inequality

\[ \Delta_M(1) \ge r\!-\!1\]

holds. As we shall show below, the following definition naturally generalizes those $\EKP$-representations for which we have equality.

Let $d \in \{1,\ldots, r\!-\!1\}$. We say that $M \in \repp(K_r,d)$ has minimal type, provided $\Delta_M(d)\!=\!d(r\!-\!d)$.

Our approach rests on the following technical Lemma:

Let $M \in \repp(K_r,d)$. Then

\[ C_M :=\{(m,\fv) \in M_2\!\times\!\Gr_d(A_r) \ ; \ m \in \im \psi_{M,\fv}\} \subseteq M_2\!\times\!\Gr_d(A_r)\]

is a closed, irreducible subset of dimension $\dim C_M\!=\!d(r\!-\!d)\!+\!d\dim_\KK M_1$.

We proceed in several steps.

(a)  Let $f : V \lra W$ be a $\KK$-linear map, $C \subseteq \Gr_q(V)$ be closed such that $\dim_\KK f(\fv)\!=\!q$ for all $\fv \in C$. Then

\[ \msim_{f,C} : C \lra \Gr_q(W) \ \ ; \   \   \fv \mapsto f(\fv)\]

is a morphism of projective varieties.

We denote by $\mspl_W : \Gr_q(W) \lra \PP(\bigwedge^q(W))$ the Plücker embedding and observe that $(\mspl_W\circ\msim_{f,C})(\fv)\!=\!\bigwedge^q(f(\fv))\!=\!\bigwedge^q(f)(\bigwedge^q(\fv))$ for every $\fv \in C$. The
map $\bigwedge^q(f) : \bigwedge^q(V) \lra \bigwedge^q(W)$ is $\KK$-linear and
$O_f\!:=\!\{ [x] \in \PP(\bigwedge^q(V)) \ ; \ \bigwedge^q(f)(x)\!\ne\!0\}$ is open and such that

\[ \varphi : O_f \lra \PP(\bigwedge^q(W)) \  ;  \ [x] \mapsto [\bigwedge^q(f)(x)]\]

is a morphism. Since $\mspl_V(C) \subseteq O_f$, it follows that $\mspl_W\circ\msim_{f,C}\!=\!\varphi\circ \mspl_V|_C$ is a morphism. Hence the same holds for $\msim_{f,C}$. $\diamond$

(b)  The map

\[ \zeta_M : \Gr_d(A_r) \lra \Gr_{d\dim_\KK M_1}(M_2) \  \  ;   \  \   \fv \mapsto \psi_M(\fv\!\otimes_\KK\!M_1)\]

is a morphism.

We first consider the map

\[ \eta_M : \Gr_d(A_r) \lra \Gr_{d\dim_\KK M_1}(A_r\!\otimes_\KK\!M_1) \ \ ; \ \ \fv \mapsto \fv\!\otimes_\KK\!M_1.\]

Then we have

\[ (\mspl_{A_r\otimes_\KK M_1}\circ \eta_M)(\fv) = \bigwedge^{d\dim_\KK M_1}(\fv\!\otimes_\KK\!M_1) \cong \bigwedge^d(\fv)\!\otimes_\KK\!\bigwedge^{\dim_\KK M_1}(M_1) \cong \mspl_{A_r}(\fv)\!\otimes_\KK\!
\bigwedge^{\dim_\KK M_1}(M_1),\]

where the second to last isomorphism holds for dimension reasons. Hence $\mspl_{A_r\otimes_\KK M_1}\circ \eta_M$ is a morphism, so that $\eta_M$ enjoys the same property.

Consequently, $C\!:=\!\im \eta_M$ is closed. Since $\zeta_M\!=\!\msim_{\psi_M,C}\circ \eta_M$, our assertion follows from (a). $\diamond$

The incidence variety $\cI_M\!:=\!\{(m,\fw) \in M_2\!\times\!\Gr_{d\dim_\KK M_1}(M_2) \ ; \ m \in \fw\}$ is known to be closed. By (b), the map

\[ \id_{M_2}\!\times\zeta_M : M_2\!\times\!\Gr_d(A_r) \lra M_2\!\times\!\Gr_{d\dim_\KK M_1}(M_2)\]

is a morphism. Hence $C_M\!=\!(\id_{M_2}\!\times \zeta_M)^{-1}(\cI_M)$ is closed as well.

We consider the projection

\[ \pr_2 : C_M \lra \Gr_d(A_r) \ \ ; \ \ (m,\fv) \mapsto \fv.\]

For every $\fv \in \Gr_d(A_r)$, the fiber $\pr_2^{-1}(\fv)\!\cong\!\im \psi_{M,v}$ is irreducible of dimension $d\dim_\KK M_1$. Note that $\KK^\times$ acts on $C_M$ via

\[ \alpha. (m,\fv) = (\alpha m,\fv) \  \  \  \  \forall \ \alpha \in \KK^\times, (m,\fv) \in C_M.\]

It follows that every irreducible component $C \subseteq C_M$ is $\KK^\times$-stable, so that $(m,\fv) \in C$ implies $(0,\fv) \in C$. The morphism

\[ \iota : \Gr_d(A_r) \lra C_M \  \  ;  \  \  \fv \mapsto (0,\fv)\]

thus yields $\pr_2(C)\!=\!\iota^{-1}(C)$, showing that $\pr_2(C)$ is closed. The irreducibility of $C_M$ now follows from <cit.>, while its dimension is given by the fiber dimension theorem.

For $M \in \rep(K_r)$ we put

\[ I_M := \bigcup_{\fv \in \Gr_d(A_r)}\im\psi_{M,\fv} \subseteq M_2.\]

Let $M \in \repp(K_r,d)$. Then we have

\[ \dim_\KK \ker\psi_M \le (r\!-\!d)(\dim_\KK M_1\!-\!\min\{\dim_\KK M_1,d\}).\]

In particular,

\[ \Delta_M(d) \ge (r\!-\!d)\min\{\dim_\KK M_1,d\}.\]

We consider the representation

\[ P(M)\!:=\!(M_1,A_r\!\otimes_\KK\!M_1,\id_{A_r\otimes_\KK M_1}).\]

Since $P(M)\cong (\dim_\KK M_1)P_1(r)$ is projective, Lemma <ref> shows that the set

\[ C_{P(M)} = \{ (x,\fv) \in (A_r\!\otimes_\KK\!M_1)\!\times\!\Gr_d(A_r) \ ; \ x \in \fv\!\otimes_\KK\!M_1\}\]

is closed, irreducible and of dimension $\dim C_{P(M)}\!=\!d\dim_\KK M_1\!+\!d(r\!-\!d)$. The projection

\[ \pr_1 : C_{P(M)} \lra A_r\!\otimes_\KK\!M_1\]

has image $I_{P(M)}\!=\!\bigcup_{\fv \in \Gr_d(A_r)}\fv\!\otimes_\KK\!M_1 \subseteq A_r\!\otimes_\KK\!M_1$. Since $\Gr_d(A_r)$ is complete and $C_{P(M)}$ is closed and irreducible, $I_{P(M)}$ is a closed, irreducible
subset of $A_r\!\otimes_\KK\!M$. Moreover, given $x \in I_{P(M)}$, we have $\pr_1^{-1}(x)\cong \{\fv \in \Gr_d(A_r) \ ; \ x \in \fv\!\otimes_\KK\!M_1\}$.

Let $\{m_1,\ldots, m_s\}$ be a basis of $M_1$, so that $s\!=\!\dim_\KK M_1$. Every $x \in A_r\!\otimes_\KK\!M_1$ is of the form $x\!=\!\sum_{i=1}^s v_i(x)\otimes m_i$. We let $\fc(x)\!:=\!\langle v_i(x) \ , \ i \in \{1,\ldots, s\}
\rangle \subseteq A_r$ be the coefficient space of $x$ and put $\ell(x)\!=\!\dim_\KK\fc(x)$. Writing $v_i(x)\!:=\!\sum_{j=1}^ra_{ij}(x)\gamma_j$ and $A(x)\!:=\!(a_{ij}(x)) \in \Mat_{s\times r}(k)$, we obtain $\dim_\KK\fc(x)\!=\!
\rk(A(x))$. Since $\rk(A(x))\!\le\!\min\{s,d\}$ for all $x \in I_{P(M)}$, it follows from $\sum_{i=1}^{\min\{s,d\}}\gamma_i\otimes m_i \in I_{P(M)}$ and the lower semicontinuity of $x \mapsto \rk(A(x))$ that

\[ O_M := \{ x \in I_{P(M)} \ ; \ \ell(x)\!=\!\min\{s,d\}\}\]

is a dense open subset of $I_{P(M)}$.

Let $x \in O_M$. Then $(x,\fv )\in \pr_1^{-1}(x)$ if and only if $\fc(x) \subseteq \fv$. If $s\!\ge\!d$, then $\pr^{-1}_1(x)\!=\!\{(x\,\fc(x))\}$. Alternatively, $\pr_1^{-1}(x) \cong \Gr_{d-s}(A_r/\fc(x))$,
so that $\dim \pr^{-1}_1(x)\!=\!(d\!-\!s)(r\!-\!d)$. The fiber dimension theorem now shows that

\[\dim I_{P(M)} = ds\!+\!(r\!-\!d)\min\{s,d\}.\]

Note that $M \in \repp(K_r,d)$ implies that $\ker\psi_M\cap I_{P(M)}\!= \{0\}$.  The affine dimension theorem <cit.> now ensures that

\[ 0 = \dim \ker\psi_M\cap I_{P(M)} \ge \dim_\KK \ker \psi_M\!+\!\dim I_{P(M)}\!-\!\dim_\KK(A_r\!\otimes_\KK\!M_1),\]

\[ \dim_\KK\ker\psi_M\le rs\!-\!\dim I_{P(M)} = (r\!-\!d)(s\!-\!\min\{s,d\}).\]

\[ \dim_\KK M_2 \ge \rk(\psi_M) = rs\!-\!\dim_\KK\ker \psi_M \ge rs\!-\!(r\!-\!d)(s\!-\!\min\{s,d\}) = ds\!+\!(r\!-\!d)\min\{s,d\},\]

so that $\Delta_M(d)\!\ge\!(r\!-\!d)\min\{\dim_\KK M_1,d\}$.

Theorem <ref> has been used by the first named author [10] to determine the orbit representatives of the dimension vectors of the elementary $K_r$-modules relative to the actions given by
$\sigma_{K_r}$ and the duality. This extends work by Ringel [57] concerning $r\!=\!3$, where module representatives were also identified. In this regard, the case $r\!=\!3$ appears to be special.

We shall see later that the following result reflects the scarcity of indecomposable vector bundles $\cF$ on $\Gr_d(A_r)$ of rank $\rk(\cF)\!<\!d(r\!-\!d)$.

Let $M,N \in \repp(K_r,d)$ be representations.

* Suppose that $\Delta_M(d)\!<\!d(r\!-\!d)$.

* If $M$ is indecomposable, then $M\cong P_0(r)$, or $d\!\ne\!1$ and $M\!\cong P_1(r)$.

* $M \cong \Delta_M(r)P_0(r)\!\oplus\!(\dim_\KK M_1)P_1(r)$ is projective.

* Suppose that $\Delta_M(d)\!=\!d(r\!-\!d)$.

* If $M$ is not projective, then every $f \in \Hom_{K_r}(M,N)\!\smallsetminus\!\{0\}$ is injective.

* $M$ is a brick or projective.

* If $M$ is not projective, then $\dim_\KK M_1\!\ge\!d\!+\!1$.

* If $\dim_\KK M_1\!\ge\!d\!+\!1$, then $M$ is not projective.

(1a) Suppose first that $M\not\cong P_0(r)$ is indecomposable. Then $\Delta_M(r)\!\le\!0$ and

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

is a projective resolution of $M$. In view of

\[ (\dim_\KK M_1)(r\!-\!d) = \Delta_M(d)\!-\!\Delta_M(r),\]

our current assumption in conjunction with Theorem <ref> implies $d\!>\!\dim_\KK M_1$ and $\Delta_M(d)\!\ge\!(\dim_\KK M_1)(r\!-\!d)$, whence $\Delta_M(r)\!\ge\!0$ and $d\!\ne\!1$. Consequently,
$\Delta_M(r)\!=\!0$, so that ($\ast$) and the indecomposability of $M$ show that $M\cong P_1(r)$.

(1b) Since $\repp(K_r,d)$ is closed under taking subrepresentations, the assertion follows from the Theorem of Krull-Remak-Schmidt.

(2a) Let $f \in \Hom_{K_r}(M,N)\!\smallsetminus\!\{0\}$. Then the terms of the short exact sequence

\[ (0) \lra \ker f \lra M \stackrel{f}{\lra} \im f \lra (0)\]

belong to $\repp(K_r,d)$ and

\[ \Delta_M(d) = \Delta_{\ker f}(d)\!+\!\Delta_{\im f}(d).\]

If $\Delta_{\ker f}(d)\!\ne\!0$, then $\Delta_{\im f}(d)\!<\!d(r\!-\!d)$, so that (1) shows that $\im f$ is projective. In view of Theorem <ref>, the assumption $\Delta_{\im f}(d)\!=\!0$ implies $(\im f)_1\!=\!(0)$, whence
$0\!=\!-d\dim_\KK (\im f)_2$, which contradicts $f\!\ne\!0$. Hence $\ker f$ is also projective, and so is $M\cong \ker f\!\oplus\!\im f$. As $M$ is not projective, we conclude that $\Delta_{\ker f}(d)\!=\!0$, whence
$\ker f\!=\!(0)$.

(2b) Suppose that $M$ is not projective and let $f \in \End_{K_r}(M)\!\smallsetminus\!\{0\}$. In view of (2a), $f$ is bijective. Hence there is a non-zero eigenvalue $\alpha \in \KK$ and another application of (2a) yields

(2c) As $M$ is a non-projective brick, the exact sequence ($\ast$) yields $\Delta_M(r)\!<\!0$ and

\[ (\dim_\KK M_1)(r\!-\! d) = \Delta_{(\dim_\KK M_1) P_1(r)}(d) > \Delta_M(d) = d(r\!-\!d),\]

so that $\dim_\KK M_1\!\ge\!d\!+\!1$.

(2d) Suppose that $M$ is projective. Then $M \cong \Delta_M(r)P_0(r)\!\oplus\!(\dim_\KK M_1)P_1(r)$, and we have

\[ 0 \le \Delta_M(r) = \Delta_M(d)\!-\!(\dim_\KK M_1)(r\!-\!d) \le -(r\!-\!d),\]

a contradiction. Hence $M$ is not projective.

Suppose that $M \in \repp(K_r,d)$ is non-projective and of minimal type. Then $M \not \in \repp(K_r,d\!+\!1)$.

Since $M$ is non-projective and of minimal type, Corollary <ref>(2b),(2c) provides $x\!\ge\!d\!+\!1$ such that $\udim M\!=\!(x,d(r\!-\!d\!+\!x))$. We thus have

\[ (\ast) \ \ \ \ \Delta_M(d\!+\!1) = d(r\!-\!d)\!-\!x.\]

Suppose that $M \in \repp(K_r,d\!+\!1)$. Since $x\!\ge\!d\!+\!1$, Theorem <ref> in conjunction with ($\ast$) gives $d(r\!-\!d)\!-\!x\!\ge\!(d\!+\!1)(r\!-\!d\!-\!1)$, whence $d\!+\!1\! \le\! x\!\le\! 2d\!+\!1\!-\!r$. As this
contradicts $d\!\le\!r\!-\!1$, we conclude that $M \not \in \rep(K_r,d\!+\!1)$.

(1) It follows that every regular $M \in \repp(K_r,d)$ of minimal type is a brick.

(2) Let $(V_1,V_2)$ be a pair of $\KK$-vector spaces. If $\Delta_{(V_1,V_2)}(d)\!\ge\!d(r\!-\!d)$, then Proposition <ref> ensures that $\repp(K_r,d)\cap\rep(K_r;V_1,V_2)$ is not empty. Alternatively, this set is either empty
(namely, if and only if $\Delta_M(r)\!<\!0$), or it consists of one $(\GL(V_2)\!\times\!\GL(V_1))$-orbit of projective representations.

Since the category $\repp(K_r,d)$ is closed under taking extensions (see Theorem <ref>), our next result provides a first characterization of objects in $\repp(K_r,d)$ in terms of those of minimal type.

Let $M \in \repp(K_r,d)$ be such that $\Delta_M(d)\!>\!d(r\!-\!d)$. There exists an exact sequence

\[ (0) \lra a P_0(r) \lra M \stackrel{\pi}{\lra} M_{\min} \lra (0),\]

where $M_{\min}$ has minimal type and $a \in \NN$.

By Lemma <ref>, the set

\[ C_M =\{(m,\fv) \in M_2\!\times\!\Gr_d(A_r) \ ; \ m \in \im \psi_{M,\fv}\} \subseteq M_2\!\times\!\Gr_d(A_r)\]

is a closed, irreducible subset of dimension $\dim C_M\!=\!d(r\!-\!d)\!+\!d\dim_\KK M_1$. Note that the morphism

\[ \pr_1 : C_M \lra M_2 \ \ ; \  \  (m,\fv) \mapsto m\]

has image $I_M\! =\!\bigcup_{\fv \in \Gr_d(A_r)}\im\psi_{M,\fv} \subseteq M_2$, so that $\Gr_d(A_r)$ being complete implies that $I_M$ is closed.[Observe that $\pr_1$ is the restriction of the projection
$M_2\!\times\!\Gr_d(A_r)\lra M_2$, which is closed.] Moreover, $I_M$ is irreducible, conical and of dimension

\[ \dim I_M \le d(r\!-\!d)\!+\!d\dim_\KK M_1\!<\!\dim_\KK M_2.\]

The Noether Normalization Theorem thus implies the existence of a linear subspace $X \subseteq M_2$ of dimension $\dim_\KK M_2\!-\!\dim I_M$ and such that

\[ X\cap I_M = \{0\},\]

cf. <cit.>. Since $\dim_\KK X\!\ge\!\dim_\KK M_2\!-\!d\dim_\KK M_1\!-\!d(r\!-\!d)\!=\!\Delta_M(d)\!-\!d(r\!-\!d)\!>\!0$, we can find a subspace $(0) \subsetneq Y \subseteq X$ such that

(i) $\dim_\KK Y\!=\!\Delta_M(d)\!-\!d(r\!-\!d)$, and

(ii) $Y\cap I_M\!=\!\{0\}$.
We now consider $M_{\min}\!:=\!N\!:=\!(M_1,M_2/Y)$ together with the canonical projection $\pi_2 : M_2 \lra N_2$ and put $\psi_N\!:=\!\pi_2\circ \psi_M$. Then $\pi\!:=\!(\id_{M_1},\pi_2)$ defines a surjective morphism
$\pi : M \lra N$ such that $\ker\pi \cong (\dim_\KK Y)P_0(r)$. Given $\fv \in \Gr_d(A_r)$, (ii) yields $\im\psi_{M,\fv}\cap Y\!=\!(0)$, whence

\[ \ker \psi_{N,\fv} = \psi_{M,\fv}^{-1}(Y) = (0).\]

Consequently, $M_{\min} \in \repp(K_r,d)$, while (i) yields $\Delta_{M_{\min}}(d)\!=\!\Delta_M(d)\!-\!\Delta_{(\dim_\KK Y)P_0(r)}(d)\!=\!\Delta_M(d)\!-\!\dim_\KK Y\!=\!d(r\!-\!d)$.

We turn to the problem of embedding $K_r$-representations of minimal type into $K_{r+s}$-representations of the same type.

Given $M \in \rep(K_r)$ such that $\dim_\KK M_1\!\ge\!d$, we have

\[ I_M = \bigcup_{W \in \Gr_d(M_1)} \psi_M(A_r\!\otimes_\KK\!W)\!=:\!J_M.\]

Recall that $P(M)\!=\!(M_1,A_r\!\otimes_\KK\!M_1,\id_{A_r\!\otimes_\KK M_1})$. We first show that $I_{P(M)}\!=\!J_{P(M)}$. Let $x \in I_{P(M)}$. Then there is $\fv \in \Gr_d(A_r)$ such that
$x \in \fv\!\otimes_\KK\!M_1$. Given a basis $\{a_1,\ldots, a_d\} \subseteq \fv$, we write $x\!=\!\sum_{i=1}^d a_i\otimes m_i$, so that there is $W \in \Gr_d(M_1)$ such that $\langle\{m_1,\ldots, m_d\}\rangle \subseteq W$.
Consequently, $x \in A_r\!\otimes_\KK\!W$. As a result, $I_{P(M)} \subseteq J_{P(M)}$. The proof of the reverse inclusion is analogous.

For general $M$, we note that $I_M\!=\!\psi_M(I_{P(M)})\!=\!\psi_M(J_{P(M)})\!=\!J_M$.

Let $M \in \repp(K_r,d)$ be of minimal type such that $\dim_\KK M_1\!\ge\!d\!+\!1$. Then there exists $N \in \repp(K_{r+1},d)$ of minimal type such that

\[ N|_{K_r} \cong M\!\oplus\!dP_0(r).\]

Writing $A_{r+1}\!=\!A_r\!\oplus\!\KK a$, we first construct $Q \in \repp(K_{r+1},d)$ such that

\[ Q|_{K_r} \cong M\!\oplus (\dim_\KK M_1)P_0(r).\]

We put $Q_1\!:=\!M_1$, $Q_2\!:=\!M_2\!\oplus\! Y_2$, with $\dim_\KK Y_2\!=\!\dim_\KK M_1$ and pick an isomorphism

\[ \lambda : \KK a\!\otimes_\KK\!M_1 \stackrel{\sim}{\lra} Y_2.\]

Interpreting $\psi_M$ and $\lambda$ as elements of $\Hom_\KK(A_{r+1}\!\otimes_\KK\!Q_1,Q_2)$, we have

\[ (\ast) \ \ \ \ \ \ \ \ \  \psi_M(A_r\!\otimes_\KK\!W)\cap \lambda(\KK a\!\otimes_\KK\!W) = (0) \ \  \text{for all} \ \  W \in \Gr_d(M_1).\]

We consider the representation $Q\!:=\!(M_1,Q_2,\psi_M\!+\!\lambda) \in \rep(K_{r+1})$.  Let $x \in \ker\psi_Q\cap I_{P(Q)}$. Lemma <ref> provides $W \in \Gr_d(M_1)$ such that
$x \in \ker\psi_Q\cap (A_{r+1}\!\otimes_\KK W)$. Writing $x\!=\!y\!+\!z$, with $y \in A_r\!\otimes_\KK\!W$ and $z \in \KK a\!\otimes_\KK\!W$ we obtain $\lambda(z)\!=\!-\psi_M(y) \in \psi_M(A_r\!\otimes_\KK\!W)\cap
\lambda(ka\!\otimes_\KK\!W)$, so that ($\ast$) implies $\lambda(z)\!=\!0\!=\!\psi_M(y)$. As $\lambda$ is injective, we conclude that $z\!=\!0$, while Lemma <ref> in conjunction with $M \in \repp(K_r,d)$ yields
$y \in \ker\psi_M\cap J_{P(M)}\!=\!\ker\psi_M\cap I_{P(M)}\!=\!(0)$. As a result, $x\!=\!0$, so that $Q \in \repp(K_{r+1},d)$. By construction, we have $Q|_{K_r} \cong M\!\oplus (\dim_\KK M_1)P_0(r)$.

Since $\Delta_Q(d)\!=\!\Delta_M(d)\!+\!\dim_\KK M_1\!=\!\dim_\KK M_1\!-\!d\!+\!d(r\!+\!1\!-\!d)$ and $\dim_\KK M_1\!>\!d$, Proposition <ref> provides a short exact sequence

\[ (0) \lra (\dim_\KK M_1\!-\!d)P_0(r\!+\!1) \lra Q \stackrel{\pi}{\lra} Q_{\min} \lra (0),\]

where $N\!:=\!Q_{\min}$ has minimal type. Let $\iota : M \lra Q|_{K_r}$ be the given injection. Since $\dim_\KK M\!\ge\!d\!+\!1$, Corollary <ref>(2d) ensures that $M$ is not projective. As $\ker \pi|_{K_r} \cong
(\dim_\KK M_1\!-\!d)P_0(r)$, the map $\pi \circ \iota : M \lra  N|_{K_r}$ is not zero, and hence by Corollary <ref>(2a) injective. The restriction of the sequence above yields

\[ (0) \lra (\dim_\KK M_1\!-\!d)P_0(r) \stackrel{\binom{\alpha}{\beta}}{\lra} (\dim_\KK M_1)P_0(r)\!\oplus\!M \stackrel{(\zeta,\pi\circ \iota)}{\lra} N|_{K_r} \lra (0),\]

and thanks to  <cit.> there results a push-out and pull-back diagram

\[ \begin{tikzcd}   (\dim_\KK M_1\!-\!d)P_0(r)  \arrow[r, "\alpha"] \arrow[d,"\beta"] & (\dim_\KK M_1)P_0(r) \arrow[d,"-\zeta"] \\
M \arrow[r,"\pi\circ \iota"] & N|_{K_r}.
\end{tikzcd} \]

Since $\pi\circ\iota$ is injective, so is $\alpha$. It follows that $dP_0(r) \cong \coker\alpha \cong \coker(\pi\circ \iota)$ (cf. <cit.>), so that there is an exact sequence

\[ (0) \lra M \stackrel{\pi\circ \iota}{\lra} N|_{K_r} \lra dP_0(r) \lra (0).\]

Consequently, $N|_{K_r} \cong M\!\oplus\! dP_0(r)$.

Suppose that $M \in \repp(K_r,d)$ is projective and of minimal type. Since $\Delta_{P_1(r)}(d)\!=\!(r\!-\!d)$, there is $\ell \in \{0,\ldots, d\}$ such that $M \cong \ell P_1(r)\!\oplus\!(d\!-\!\ell)(r\!-\!d)P_0(r)$.
We consider $N\!:= \ell P_1(r\!+\!1)\!\oplus\!(d\!-\!\ell)(r\!+\!1\!-\!d)P_0(r\!+\!1) \in \repp(K_{r+1},d)$. Then we have

\[ \Delta_N(d) = \ell (r\!+\!1\!-\!d)\!+\!(d\!-\!\ell)(r\!+\!1\!-\!d) = d(r\!+\!1\!-\!d),\]

so that $N$ has minimal type. Moreover,

\[ N|_{K_r} \cong \ell P_1(r)\!\oplus\!\ell P_0(r)\!\oplus\!(d\!-\!\ell)(r\!+\!1\!-\!d)P_0(r),\]

showing that $N|_{K_r} \cong M\!\oplus\! dP_0(r)$. By the remark at the beginning of Section <ref>, Proposition <ref> thus also holds in case $\dim_\KK M_1\!\le\!d$.

§.§ Examples: Representations and hyperplane arrangements
The representations of $\EKP(K_r)\!=\!\repp(K_r,1)$ discussed in this section correspond to the logarithmic bundles [23] and
Schwarzenberger bundles [60]. Our notation is meant to reflect this relationship

Let $r\!\ge\!2$, $V$ be an $r$-dimensional vector space and $m\!\ge\!r$. We say that $\ff\!:=\!(f_1,\ldots, f_m) \in (V^\ast)^m$ is in general position, provided $V^\ast\!=\!\langle f_i \ ; \ i \in J \rangle$ for every subset $J
\subseteq \{1,\ldots, m\}$ such that $|J|\!=\!r$. We consider the projective space $\PP^{r-1}\!=\!\PP(V)$. A hyperplane arrangement of $V$ is an $m$-tuple $\cH\!:=\!(H_1,\ldots, H_m)$ of linear hyperplanes of
$\PP(V)$. We say that $\cH$ is in general position, provided for every $\ell\! \le\! r$ and every $\ell$-element subset $J_\ell\subseteq \{1,\ldots, m\}$, we have $\dim_\KK \bigcap_{i\in J_\ell}H_i\!=\!r\!-\!1\!-\!\ell$.
(Here we put $\dim \emptyset\!=\!-1$.) By definition, each $H_i\:=\!Z(f_i)$ is the set of zeros for some $f_i \in V^\ast\!\smallsetminus\!\{0\}$. It follows that $\cH$ is in general position if and only if $(f_1,\ldots, f_m)$ enjoys this

Let $\ff\!:=\!(f_1,\ldots, f_m) \in (V^\ast)^m$. Following <cit.>, we set

\[ I_\ff\!:=\{\lambda \in \KK^m \ ; \ \sum_{i=1}^m\lambda_if_i\!=\!0\} \ \  \text{and} \ \ W_m\!:=\!\{\lambda \in \KK^m \ ; \ \sum_{i=1}^m\lambda_i\!=\!0\},\]

and refer to the linear map

\[ t_\ff : V\!\otimes_\KK\!I_\ff  \lra W_m \ \ ; \  \  v \otimes \lambda \mapsto (\lambda_1f_1(v), \ldots, \lambda_mf_m(v))\]

as the fundamental tensor of $\ff$ (viewed as an element of $V^\ast\!\otimes_\KK\!I_\ff^\ast\!\otimes_\KK\!W_m$).

Given $\ff\!:=\!(f_1,\ldots, f_m) \in (A_r^\ast)^m$, we define $Q_\ff\!:=\!(\KK^m,\KK^m,(Q_\ff(\gamma_i))_{1\le i \le r}) \in \rep(K_r)$ via

\[ Q_\ff(\gamma_i)(\lambda) = (\lambda_1f_1(\gamma_i), \ldots, \lambda_mf_m(\gamma_i)) \ \ \ \ \ \forall \ i \in \{1,\ldots, r\}, \lambda \in \KK^m.\]

Then, setting $V\!:=\!A_r$, we see that

\[ M_\ff\!:=\!(I_\ff,W_m, (Q_\ff(\gamma_i)|_{I_\ff})_{1\le i \le r})\]

is a subrepresentation of $Q_\ff$ such that $\psi_{M_\ff}\!=\!t_\ff$. If $\cH$ is a hyperplane arrangement given by $\ff \in ((A_r)^\ast\!\smallsetminus\!\{0\})^m$, we abuse notation and  write $M_\cH\!=\!M_\ff$
as well as $t_\cH\!=\!t_\ff$.

Suppose that $m\!\ge\!r\!\ge\!3$, and let $\cH\!=\!(H_1,\ldots, H_m)$ be a hyperplane arrangement in general position.

* $M_\cH \in \EKP(K_r)$ has dimension vector $\udim M_\cH\!=\!(m\!-\!r,m\!-\!1)$. In particular, $M_\cH$ has minimal type.

* If $m\!\ge\! r\!+\!2$, then $M_\cH \not \in \repp(K_r,2)$.

(1) Let $a \in A_r\!\smallsetminus\!\{0\}$. Then we have

\[ a_{M_\cH}(\lambda) = \psi_{M_\cH}(a\otimes \lambda) = t_\cH(a\!\otimes \lambda)\]

for all $\lambda \in I_\ff$. We may now apply <cit.> (which holds for arbitrary ground fields) to see that the linear map $a_{M_\cH}$ is injective. Consequently, $M_\cH \in \EKP(K_r)$.

We write $\cH\!:=\!(Z(f_1),\ldots, Z(f_m))$, $\ff\!:=\!(f_1,\ldots, f_m)$. By construction, we have $\dim_\KK(M_\cH)_2\!=\!\dim_\KK W_m\!=\!m\!-\!1$. Since $m\!\ge\!r$ and $\cH$ is in general position, we have
$A_r^\ast\!=\!\langle f_1,\ldots, f_r\rangle$, so that $\{f_1,\ldots, f_r\} \subseteq A_r^\ast$ is a basis of $A_r^\ast$. Hence the linear map

\[\alpha_\ff : \KK^m \lra A_r^\ast \ \ ; \  \  \lambda \mapsto  \sum_{i=1}^m\lambda_if_i\]

is surjective, whence $\dim_\KK I_\ff\!=\!\dim_\KK\ker\alpha_\ff\!=\!m\!-\!r$.

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

A subclass of the modules $M_\cH$ considered above is given by the family $(M_\cS[m])_{m\ge r+1}$ of Schwarzenberger modules. Let $X_1,X_2$ be indeterminates over $\KK$. For $m\!\ge\!r\!+\!1$, we consider the
representation $M_\cS[m] \in \rep(K_r)$, defined via

\[ M_\cS[m]_1 := \KK[X_1,X_2]_{m-r-1} \ \ ; \ \ M_\cS[m]_2 := \KK[X_1,X_2]_{m-2} \ \ ; \ \ M_\cS[m](\gamma_i)(a) := X_1^{i-1}X_2^{r-i}a \ \ \ \forall \ a \in M[m]_1\]

for $1\!\le\!i\!\le\!r$. Then we have $M_\cS[m] \in \EKP(K_r)$, while $\udim M_\cS[m]\!=\!(m\!-\!r,m\!-\!1)$. The modules $M_\cS[m]$ turn out to correspond to vector bundles on $\PP^{r-1}$ that were introduced by
Schwarzenberger, cf. [60].

§.§ The generic canonical decomposition
Our first result provides the canonical decomposition <cit.> of certain dimension vectors. For ease of notation, we write

Suppose that $V_1,V_2$ are vector spaces such that $\repp(K_r,d)\cap\rep(K_r;V_1,V_2)\!\ne\!\emptyset$. According to Theorem <ref>, the assumption $\Delta_{(V_1,V_2)}(d)\!=\!0$ implies
$\dim_\KK V_1\!=\!0$ and hence $\dim_\KK V_2\!=\!\Delta_{(V_1,V_2)}(d)\!=\!0$. Hence the technical condition $\Delta_{(V_1,V_2)}\!\ge\!1$ of the ensuing results only excludes the trivial case.

Let $V_1,V_2$ be vector spaces such that $\Delta_{(V_1,V_2)}\!\ge\!1$ and write $\dim_\KK V_1\!=\!j\Delta_{(V_1,V_2)}\!+\!b$ for $b \in \{0,\ldots,\Delta_{(V_1,V_2)}\!-\!1\}$.
Then the set

\[ \rep(K_2;V_1,V_2)_0 := \{M \in \rep(K_2;V_1,V_2) \ ; \ M \cong (\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\! bP_{j+1}(2)\}\]

is $(\GL(V_2)\!\times\!\GL(V_1))$-stable, dense and open.

By definition, $ \rep(K_2;V_1,V_2)_0$ is $(\GL(V_2)\!\times\!\GL(V_1))$-stable.

Since $P_j(2),P_{j+1}(2)$ are bricks, Corollary <ref> implies that these representations belong to their respective open sheets. By virtue of $\dim_\KK V_i\!=\!(j\!+\!i\!-\!1)\Delta_{(V_1,V_2)}\!+\!b$ and
$\Ext^1_{K_2}(P_\ell(2),P_k(2))\!=\!(0)$ for $\{\ell,k\}\!=\!\{j,j\!+\!1\}$, the conditions of <cit.> are met, so that our assertion follows from Kac's Theorem.

Let $r\!\ge\!3$. We consider the map

\[ \Res : \rep(K_r;V_1,V_2)\!\times\!\Inj_\KK(A_2,A_r) \lra \rep(K_2;V_1,V_2) \ \ ; \ \ (M,\alpha) \mapsto \alpha^\ast(M).\]

Let $V_1,V_2$ be $\KK$-vector spaces such that $\Delta_{(V_1,V_2)}\!\ge\!1$, and write

\[\dim_\KK V_1= j\Delta_{(V_1,V_2)}\!+\!b\]

for some $b \in \{0,\ldots,\Delta_{(V_1,V_2)}\!-\!1\}$. The following statements hold:

* $\Res$ is a morphism.

* The set $\Res((\EKP(K_r)\cap\rep(K_r;V_1,V_2))\!\times\!\Inj_\KK(A_2,A_r))$ is open.

* Suppose that $\EKP(K_r)\cap \rep(K_r;V_1,V_2)\!\ne\!\emptyset$. Then

\[ U := \{(M,\alpha) \in \rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r) \ ; \  \alpha^\ast(M) \cong (\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\!bP_{j+1}(2)\}\]

is a dense, open subset of $\rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r)$.

(1) We interpret $\rep(K_r;V_1,V_2)$ as $\Hom_\KK(A_r, \Hom_\KK(V_1,V_2))$, so that

\[ \Res(\varrho,\alpha) = \varrho\circ \alpha\]

corresponds to matrix multiplication. Consequently, $\Res$ is a morphism.

(2) Note that

\[ \Res((\EKP(K_r)\cap\rep(K_r;V_1,V_2))\!\times\!\Inj_\KK(A_2,A_r)) = \bigcup_{\alpha \in \Inj_\KK(A_2,A_r)} \alpha^\ast(\EKP(K_r)\cap \rep(K_r;V_1,V_2)).\]

Given $\alpha \in \Inj_\KK(A_2,A_r)$, we write $A_r\!=\!\im\alpha\!\oplus\!\fw \cong  A_2\!\oplus\!\fw$ for some $\fw \in \Gr_{r-2}(A_r)$. Then we have

\[ \Hom_\KK(A_r,\Hom_\KK(V_1,V_2)) \cong \Hom_\KK(A_2,\Hom_\KK(V_1,V_2))\!\oplus \Hom_\KK(\fw,\Hom_\KK(V_1,V_2)),\]

and $\alpha^\ast$ is the projection onto the first summand.

For varieties $X,Y,$ the projection $\pr_X : X\!\times\!Y \lra X$ is easily seen to be open. We apply this to the situation above to see that $\alpha^\ast : \rep(K_r;V_1,V_2) \lra \rep(K_2;V_1,V_2)$ is open.
Since Proposition <ref> ensures that $\EKP(K_r)\cap \rep(K_r;V_1,V_2)$ is open, our assertion follows.

(3) In view of (2) and Lemma <ref>, our current assumption implies that

\[ \Res((\EKP(K_r)\cap\rep(K_r;V_1,V_2))\!\times\!\Inj_\KK(A_2,A_r))\cap \rep(K_2;V_1,V_2)_0\]

is a non-empty, open subset of $\rep(K_2;V_1,V_2)$. By (1),

\[  U = \Res^{-1}(\rep(K_2;V_1,V_2)_0)\]

is thus a non-empty, open subset of $\rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r)$. Being an open subset of the irreducible variety $\rep(K_r;V_1,V_2)\!\times\! \Hom_\KK(A_2,A_r)$, the variety $\rep(K_r;V_1,V_2)\!\times\!
\Inj_\KK(A_2,A_r)$ is irreducible, so that $U$ lies dense in $\rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r)$.

Recall the notion of the generic decomposition $M_{\gen}$ of $M \in \repp(K_r,d)$, which we have discussed at the end of Section <ref>.

Suppose that $V_1,V_2$ are $\KK$-vector spaces such that $\Delta_{(V_1,V_2)}\!\ge\!1$ and $\EKP(K_r)\cap\rep(K_r;V_1,V_2)\!\ne\!\emptyset$. Let $(j,b) \in \NN_0\!\times\!\{0,\ldots,
\Delta_{(V_1,V_2)}\!-\!1\}$ be given by $\dim_\KK V_1\!=\!j\Delta_{(V_1,V_2)}\!+\!b$. Then

\[ O_{\gen} := \{ M \in \EKP(K_r)\cap\rep(K_r;V_1,V_2) \ ; \ M_{\gen}\! =\! (\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\!bP_{j+1}(2)\}\]

is a dense open subset of $\rep(K_r;V_1,V_2)$.

According to Proposition <ref>

\begin{eqnarray*}
\{(M,\alpha) \in \rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r) \ ; \  \alpha^\ast(M)\! & \cong & \!(\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\!bP_{j+1}(2)\} \\
& = & U= \Res^{-1}(\rep(K_2;V_1,V_2)_0)
\end{eqnarray*}

is a dense, open subset of $\rep(K_r;V_1,V_2)\!\times\!\Inj_\KK(A_2,A_r)$. The projection $\pr_1 : \rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r) \lra \rep(K_r;V_1,V_2)$ is open, so that $\pr_1(U)$ is open and
dense in $\rep(K_r;V_1,V_2)$. In view of Proposition <ref>(1), the subset $\pr_1(U)\cap\EKP(K_r)$ enjoys the same properties.

If $M \in O_{\gen}$, then there is $\alpha \in \Inj_\KK(A_2,A_r)$ such that

\[ \alpha^\ast(M) \cong (\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\!bP_{j+1}(2),\]

whence $(M,\alpha) \in U$ and $M \in \pr_1(U)\cap\EKP(K_r)$.

Conversely, let $M \in \pr_1(U)\cap\EKP(K_r)$. Then there is $\alpha \in \Inj_\KK(A_2,A_r)$ such that

\[ (\ast) \ \ \ \ \alpha^\ast(M) \cong (\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\!bP_{j+1}(2).\]

Let $\pi : \rep(K_2;V_1,V_2) \lra \Iso(K_2;V_1,V_2)$ be the canonical projection. Thanks to Lemma <ref>, $\Iso(K_2;V_1,V_2)_0\!:=\!\pi(\rep(K_2;V_1,V_2)_0)$ is a non-empty open subset of $\Iso(K_2;V_1,V_2)$.
Owing to ($\ast$), Proposition <ref> implies that

\[ U_M\!:=\! \res_M^{-1}(\Iso(K_2;V_1,V_2)_0) = \{\fv \in \Gr_2(A_r) \ ; \ [M|_\fv]\!=\![(\Delta_{(V_1,V_2)}\!-\!b)P_j(2)\!\oplus\!bP_{j+1}(2)]\}\]

is a dense open subset of $\Gr_2(A_r)$. In view of Corollary <ref>, $U_M$ therefore intersects the set

\[ O_M := \{ \fv \in \Gr_2(A_r) \ ; \  n_i(M,\fv) = n_i(M) \ \ \ \forall \ i \in \NN_0\}.\]

\[ n_i(M) = \left\{ \begin{array}{cc} 0 & i \not\in \{j,j\!+\!1\} \\ \Delta_{(V_1,V_1)}\!-\!b & i\!=\!j \\ b & i\!=\!j\!+\!1, \end{array} \right.\]

so that $M \in O_{\gen}$. It follows that $O_{\gen}\!=\!\pr_1(U)\cap\EKP(K_r)$ has the asserted properties.

Let $V_1,V_2$ be $\KK$-vector spaces such that $\Delta_{(V_1,V_2)}\!\ge\!\max\{\dim_\KK V_1,r\!-\!1\}$. Then we have

\[ O_{\gen} = \{ M \in \EKP(K_r)\cap\rep(K_r;V_1,V_2) \ ; \ M_{\gen}\! =\! (\Delta_{(V_1,V_2)}\!-\!\dim_\KK V_1)P_0(2)\!\oplus\!(\dim_\KK V_1)P_1(2)\}.\]

Since $\Delta_{(V_1,V_2)}\!\ge\!r\!-\!1$, Proposition <ref> shows that $\EKP(K_r)\cap\rep(K_r;V_1,V_2)\!\ne\!\emptyset$. If $\Delta_{(V_1,V_2)}\!\ge\!\dim_\KK V_1\!+\!1$, we obtain

\[ \dim_\KK V_1 = 0\Delta_{(V_1,V_2)}\!+\!\dim_\KK V_1,\]

where $\dim_\KK V_1\le\!\Delta_{(V_1,V_2)}\!-\!1$. The result thus follows from Corollary $\ref{GCD3}$. Alternatively, $\dim_\KK V_1\!=\!\Delta_{(V_1,V_2)}$ and Corollary <ref> implies
$M_{\gen}\!=\!(\dim_\KK V_1)P_1(2)$.

Let $V_1,V_2$ be $\KK$-vector spaces such that $\Delta_{(V_1,V_2)}\!\ge\!1$, $M \in \EKP(K_r)\cap\rep(K_r;V_1,V_2)$. If there exist $\fv_0 \in \Gr_2(A_r)$ and $c,d,j \in \NN_0$ such that

\[ [M|_{\fv_0}] = cP_j(2)\!\oplus\!dP_{j+1}(2),\]

then $M_{\gen}\!=\!cP_j(2)\!\oplus\!dP_{j+1}(2)$ and $M \in O_{\gen}$.

By assumption, we have $\Delta_{(V_1,V_2)}\!=\!\Delta_M\!=\!c\Delta_{P_j(2)}\!+\!d\Delta_{P_{j+1}(2)}\!=\!c\!+\!d$, while $\dim_\KK V_1\!=\!cj\!+\!d(j\!+\!1)\!=\!j\Delta_{(V_1,V_2)}\!+\!d$. If $c\!\ne\!0$,
then $0\!\le\!d\!\le\!\Delta_{(V_1,V_2)}\!-\!1$ and  Proposition <ref>(3) implies that

\[ U := \{(N,\alpha) \in \rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r) \ ; \  \alpha^\ast(N) \cong cP_j(2)\!\oplus\!dP_{j+1}(2)\}\]

is a dense open subset of $\rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r)$. Alternatively, $\dim_\KK V_1\!=\!(j\!+\!1)\Delta_{(V_1,V_2)}$ and we arrive at the same conclusion. In view of Proposition <ref>(1),
we conclude that

\[ \tilde{U} := U \cap ((\EKP(K_r)\cap\rep(K_r;V_1,V_2))\!\times\! \Inj_\KK(A_2,A_r))\]

is a dense open subset of $\rep(K_r;V_1,V_2)\!\times\! \Inj_\KK(A_2,A_r)$.

\[ \iota_M : \Inj_{\KK}(A_2,A_r) \lra \rep(K_r;V_1,V_2)\!\times\!\Inj_{\KK}(A_2,A_r) \ \ ; \ \ \alpha \mapsto (M,\alpha)\]

is continuous,  our current assumption ensures that $\tilde{O}\!:=\! \iota_M^{-1}(\tilde{U})$ is a dense open subset of $\Inj_{\KK}(A_2,A_r)$. As the morphism $\msim : \Inj_\KK(A_2,A_r) \lra \Gr_2(A_r)$ is open
(see the proof of Proposition <ref>),

\[ \msim(\tilde{O})\!=\!\{ \fv \in \Gr_2(A_r) \ , \ M|_\fv \cong  cP_j(2)\!\oplus\!dP_{j+1}(2)\}\]

is a dense open subset of $\Gr_2(A_r)$ and Corollary <ref> gives

\[ M_{\gen} =  cP_j(2)\!\oplus\!dP_{j+1}(2).\]

As a result, $M \in O_{\gen}$.

By way of example, we briefly discuss Schwarzenberger modules. Our result and its succeeding remark will provide the generic splitting type of the associated vector bundles and
show that these are $\GL(A_2)$-homogeneous, but not homogeneous (cf. Section <ref> below).

Given $\ell \in \NN$, we consider the Schwarzenberger module $M_\cS[\ell\!+\!r]$ of dimension vector $\udim M_\cS[\ell\!+\!r]\!=\!(\ell, \ell\!+\!r\!-\!1)$. Then we have $(M_\cS[\ell\!+\!r])_1\!=\!\KK[X_1,X_2]_{\ell-1}$ and
$(M_\cS[\ell\!+\!r])_2\!=\!\KK[X_1,X_2]_{\ell+r-2}$. Note that these spaces are $\ZZ^2$-graded. Without loss of generality, we may assume that $M_\cS[\ell\!+\!r](\gamma_i)(f)\!=\!X_i^{r-1}f$ for $i \in \{1,2\}$ and $f \in
\KK[X_1,X_2]_{\ell-1}$. In particular, we have $\deg(M_\cS[\ell\!+\!r](\gamma_1))\!=\!(r\!-\!1,0)$ and $\deg(M_\cS[\ell\!+\!r](\gamma_2))\!=\!(0,r\!-\!1)$.

If $\ell\!-\!1\!=\!a_\ell(r\!-\!1)\!+\!q_\ell$ for $q_\ell \in \{0,\ldots, r\!-\!2\}$, then

\[ M_\cS[\ell\!+\!r]_{\gen} = (r\!-\!2\!-\!q_\ell)P_{a_\ell}(2)\!\oplus\!(q_\ell\!+\!1)P_{a_\ell+1}(2).\]

In particular, $M_{\cS}[\ell\!+\!r] \in O_{\gen}$.

Let $\fv_0\!:=\!\KK\gamma_1\!\oplus\!\KK\gamma_2$. Writing $M\!:=\!M_\cS[\ell\!+\!r]$, we shall first find the decomposition of $M|_{\fv_0}$.

Given $(a,b) \in \NN_0^2$, we write $X^{(a,b)}\!:=\!X_1^aX_2^b$. For $q \in \{0,\ldots, r\!-\!2\}$ we consider the vector spaces

\[ M_{1,q} := \langle\{ X^{(a,b)} \in \KK[X_1,X_2]_{\ell-1} \ ; \ a\equiv q \ \modd(r\!-\!1)\} \rangle\]

as well as

\[M_{2,q} := \langle\{ X^{(a,b)} \in \KK[X_1,X_2]_{\ell+r-2} \ ; \ a\equiv q \ \modd(r\!-\!1)\}\rangle.\]

Then we have $M(\gamma_i)(M_{1,q}) \subseteq M_{2,q}$ for $i \in \{1,2\}$. Defining $M_q\!:=\!(M_{1,q},M_{2,q})$, we obtain a decomposition

\[ (\ast) \ \ \ \ \ \ \ \ M|_{\fv_0} = \bigoplus_{q=0}^{r-2} M_q\]

of $\KK.\fv_0$-modules. Moreover, setting $t_q\!:=\!|\{ n \in \NN_0 \ ; \ q\!+\!n(r\!-\!1)\!\le\!\ell\!-\!1\}|$, we have $\udim M_q\!=\!(t_q,t_q\!+\!1)$, so that $M_q \in \EKP(K_2)$ is of minimal type. Since $\EKP(K_2)$ is
closed under taking direct summands and $\Delta_X\!>\!0$ for all $X \in \EKP(K_2)\!\smallsetminus\!\{(0)\}$, this renders $M_q$ indecomposable. We conclude $M_q\!\cong\!P_{t_q}(2)$, so that ($\ast$) actually is the
decomposition of the $\KK K_2$-module $M|_{\fv_0}$ into indecomposables. For $0\!\le\!q\!\le\!q_{\ell}$, we have $t_q\!=\!a_\ell\!+\!1$, while $t_q\!:=\!a_\ell$ for $q_\ell\!+\!1\!\le\!q\!\le\!r\!-\!2$. As a result, ($\ast$) yields

\[  M|_{\fv_0} \cong (r\!-\!2\!-\!q_\ell)P_{a_\ell}(2)\!\oplus\!(q_\ell\!+\!1)P_{a_\ell+1}(2).\]

Corollary <ref> implies

\[ (M_{\cS}[\ell\!+\!r])_{\gen} = (r\!-\!2\!-\!q_\ell)P_{a_\ell}(2)\!\oplus\!(q_\ell\!+\!1)P_{a_\ell+1}(2),\]

and $M_{\cS}[\ell\!+\!r] \in O_{\gen}$.

Let $M\!:=\!M_\cS[\ell\!+\!r]$ and consider

\[ \fv_1 := \KK\gamma_1\!\oplus\!\KK\gamma_3,\]

\[ M(\gamma_1)(X_1^aX_2^b) = X_1^{a+r-1}X_2^b  \ \ ; \ \  M(\gamma_3)(X_1^aX_2^b) = X_1^{a+r-2}X_2^{b+1}  \ \ \ \ (a\!+\!b\!=\!\ell\!-\!1).\]

Then $(\id_{\KK [X_1,X_2]_{\ell-1}}, X_1^{r-2}\cdot)$ defines an embedding $P_\ell(2) \hookrightarrow M|_{\fv_1}$ such that

\[ M|_{\fv_1} \cong (r\!-\!2)P_0(2)\!\oplus\!P_{\ell}(2).\]

Thus, for $\ell\!\ge\!2$, the dense open subset $O_M$ defined in Corollary <ref> is properly contained in $\Gr_2(A_r)$.

§ COHERENT SHEAVES ON $\GR_D(A_R)$ AND REPRESENTATIONS OF $K_R$

This section is concerned with the equivalence of the categories $\repp(K_r,d)$ of relative projective representations and Steiner bundles on the Grassmannians $\Gr_d(A_r)$ along with some consequences thereof.
Our account builds on work by Jardim-Prata [40], where vector bundles over fields of characteristic $0$ were considered. For fields of characteristic $p\!>\!0$, Friedlander-Pevtsova [32] earlier employed
universal nilpotent operators to define functors between module categories of infinitesimal group schemes and vector bundles on their support spaces. In the context of elementary abelian $p$-groups (or equivalently
elementary abelian restricted Lie algebras), the two approaches are related via the functorial correspondence between Kronecker representations and modules of Loewy length $\le\!2$.

§.§ Conventions
Let $r\!\ge\!2$, $d \in \{1,\ldots, r\!-\!1\}$. We denote by $\Coh(\Gr_d(A_r))$ and $\Vect(\Gr_d(A_r))$ the categories of coherent sheaves and vector bundles (locally free coherent
sheaves) on $\Gr_d(A_r)$, respectively. Note that $\Coh(\Gr_d(A_r))$ is an abelian category, cf. <cit.>. Let $\cO_{\Gr_d(A_r)}$ be the structure sheaf of $\Gr_d(A_r)$, so that every $\cF \in \Coh(\Gr_d(A_r))$ is
an $\cO_{\Gr_d(A_r)}$-module. In view of [3], the category $\Vect(\Gr_d(A_r))$ is a Krull-Schmidt category. Thus, the indecomposable objects of any full subcategory $\cC \subseteq \Vect(\Gr_d(A_r))$ that is closed
under taking direct summands are just the indecomposable vector bundles belonging to $\cC$.

Given $\fv \in \Gr_d(A_r)$, we let $\cO_{\Gr_d(A_r),\fv}$ be the local ring of $\Gr_d(A_r)$ at $\fv$, whose maximal ideal we denote by $\fm_\fv$. Let $\cF \in \Coh(\Gr_d(A_r))$. For $\fv \in \Gr_d(A_r)$, the stalk $
\cF_\fv$ of $\cF$ at $\fv$ is an $\cO_{\Gr_d(A_r),\fv}$-module, and the finite-dimensional $\KK$-vector space $\cF(\fv)\!=\!\cF_\fv/\fm_\fv\cF_\fv$ is called the fiber of $\cF$ at $\fv$. If $\cF \in \Vect(\Gr_d(A_r))$, then $
\cF_\fv$ is a free $\cO_{\Gr_d(A_r),\fv}$-module of rank $\rk(\cF_\fv)\!=\!\dim_\KK\cF(\fv)$.

If $\cF,\cG \in \Coh(\Gr_d(A_r))$ are sheaves, we let $\msHom_{\Gr_d(A_r)}(\cF,\cG)$ be the sheaf of $\cO_{\Gr_d(A_r)}$-homomor- phisms from $\cF$ to $\cG$. By definition, we have $\msHom_{\Gr_d(A_r)}(\cF,\cG)
(\Gr_d(A_r))\!=\!\Hom_{\Gr_d(A_r)}(\cF,\cG)$, the space of homomorphisms from $\cF$ to $\cG$, cf. <cit.>. For $i \in \NN_0$, we let

\[ \Ext^i_{\Gr_d(A_r)}(\cF,-) : \Coh(\Gr_d(A_r)) \lra \modd \KK\]

be the $i$-th right derived functor of $\Hom_{\Gr_d(A_r)}(\cF,-) \cong \Ext^0_{\Gr_d(A_r)}(\cF,-)$. Setting

\[ \HH^i(\Gr_d(A_r),\cF) := \Ext^i_{\Gr_d(A_r)}(\cO_{\Gr_d(A_r)},\cF),\]

we recall that $\Ext^i_{\Gr_d(A_r)}(\cF,\cG) \cong \HH^i(\Gr_d(A_r),\msHom_{\Gr_d(A_r)}(\cF,\cG))$ for every $\cG \in \Coh(\Gr_d(A_r))$.

We denote by

\[ \chi(\cF) = \sum_{i=0}^{d(r-d)}(-1)^i \dim_\KK\HH^i(\Gr_d(A_r),\cF)\]

the Euler characteristic of $\cF$ (see <cit.>). We refer the reader to <cit.> for more details.

The following subsidiary result for “special exceptional pairs“ of coherent sheaves on an arbitrary variety $X$ is inspired by <cit.>. We begin by recalling the relevant terminology:

A vector bundle $\cF \in \Vect(X)$ is referred to as exceptional if $\dim_\KK \Ext^i_X(\cF,\cF)\!=\!\delta_{i,0}$ for all $i\!\ge\!0$. We say that a pair $(E_0,E_1)$ of coherent sheaves on $X$ is special
exceptional, provided

(a) $\dim_\KK\Ext^n_X(E_i,E_i)\!=\!\delta_{n,0}$ for all $n\!\ge\!0$, $i \in \{0,1\}$.

(b) $\dim_\KK\Ext^n_X(E_0,E_1)\!=\!0$ for all $n\!\ge\!1$.

(c) $\dim_\KK\Ext^n_X(E_1,E_0)\!=\!0$ for all $n\!\ge\!0$.[Our choice of terminology derives from Rudakov's more general notion of an exceptional pair, cf. [58]. Pairs satisfying the above conditions are
referred to as “strongly exceptional" in [1], which differs from the definition employed in [15].]
We put $r\!:=\!\dim_\KK\Hom_X(E_0,E_1)$. Suppose there are exact sequences

\[ (\ast) \ \ \ \ (0) \lra E_0^m \lra E_1^n \lra \cF \lra (0)\]

\[ (\ast\ast) \ \ \ (0) \lra E_0^s \lra E_1^t \lra \cG \lra (0).\]

Recall the from Section <ref> the definition of the Euler-Ringel bilinear form:

\[ \langle(x_1,y_1),(y_1,y_2)\rangle_r = x_1y_1\!+\!x_2y_2\!-\!rx_1y_2.\]

We have

\[\chi(\msHom_X(\cF,\cG))=\!\langle (m,n),(s,t)\rangle_r.\]

We proceed in several steps:

(i) We have an exact sequence

\[ (0) \lra \Hom_X(\cF,E_1^t) \lra \Hom_X(\cF,\cG) \lra \Ext^1_X(\cF,E_0^s) \lra \Ext^1_X(\cF,E_1^t) \lra \Ext^1_X(\cF,\cG) \lra (0).\]

The assertion follows by applying $\Hom_X(\cF, -)$ to ($\ast\ast$), once we know that $\Hom_X(\cF,E_0^s)\!=\!(0)\!=\!\Ext^2_X(\cF,E_0^s)$.

By applying $\Hom_X(-,E_0^s)$ to ($\ast$) while observing (c), we obtain a sequence $(0)\!\lra\!\Hom_X(\cF,E_0^s)$ $\lra\!\Hom_X(E_1^n,E_0^s)\!=\!(0)$, so that $\Hom_X(\cF,E_0^s)\!=\!(0)$. To show exactness at
$\Ext^1_X(\cF,\cG)$, we apply $\Ext^1_X(-,E_0^s)$ to $(\ast)$. Then (a) and (c) yield

\[ (0) = \Ext^1_X(E_0^m,E_0^s) \lra \Ext^2_X(\cF,E_0^s) \lra \Ext^2_X(E_1^n,E_0^s) = (0),\]

whence $\Ext^2_X(\cF,E_0^s)\!=\!(0)$.  $\diamond$

(ii) We have $\dim_\KK\Hom_X(\cF,E_1^t)\!-\!\dim_\KK\Ext^1_X(\cF,E_1^t)\!=\!nt\!-\!rmt$.

We apply $\Hom_X(-,E_1^t)$ to ($\ast$) and obtain

\[ (0) \lra \Hom_X(\cF,E_1^t) \lra \Hom_X(E_1^n,E_1^t) \lra \Hom_X(E_0^m,E_1^t) \lra \Ext^1_X(\cF,E_1^t) \lra \Ext^1_X(E_1^n,E_1^t)\!=\!(0).\]

As $\dim_\KK\Hom_X(E_1^n,E_1^t)\!=\!nt$, while $\dim_\KK\Hom_X(E_0^m,E_1^t)\!=\!rmt$, our assertion follows. $\diamond$

(iii) We have $\Ext^\ell_X(\cF,\cG)\!=\!(0)$ for $\ell\!\ge\!2$.

Let $\ell\!\ge\!2$. Application of $\Ext^{\ell-1}_X(-,E_0^s)$ to $(\ast)$ in conjunction with (a) and (c) yields

\[(0) = \Ext^{\ell-1}_X(E_0^m,E_0^s) \lra \Ext^\ell_X(\cF,E_0^s) \lra \Ext^\ell_X(E_1^n,E_0^s) = (0),\]

so that $\Ext^\ell_X(\cF,E_0^s)\!=\!(0)$. In view of (b) and (a), application of $\Ext^{\ell-1}_X(-,E_1^t)$ to ($\ast$) gives

\[(0) = \Ext^{\ell-1}_X(E_0^m,E_1^t) \lra \Ext^\ell_X(\cF,E_1^t) \lra \Ext^\ell_X(E_1^n,E_1^t) = (0),\]

so that $\Ext^\ell_X(\cF,E_1^t)\!=\!(0)$.

Applying $\Ext^\ell_X(\cF,-)$ to ($\ast\ast$), we finally arrive at

\[ (0) = \Ext^\ell_X(\cF,E_1^t) \lra \Ext^\ell_X(\cF,\cG) \lra \Ext^{\ell+1}_X(\cF,E_0^s)=(0),\]

so that $\Ext^\ell_X(\cF,\cG)\!=\!(0)$. $\diamond$

By applying (iii), (i) and (ii) consecutively, we obtain

\begin{eqnarray*}
\chi(\msHom_X(\cF,\cG)) & = & \dim_\KK\Hom_X(\cF,\cG)\!-\!\dim_\KK\Ext^1_X(\cF,\cG)\\
& = & \dim_\KK\Hom_X(\cF,E_1^t)\!-\!\dim_\KK\Ext^1_X(\cF,E_1^t)\!+\!\dim_\KK\Ext^1_X(\cF,E_0^s)\\
& = & nt\!-\!rmt\!+\!\!\dim_\KK\Ext^1_X(\cF,E_0^s).
\end{eqnarray*}

Application of $\Hom_X(-,E_0^s)$ to ($\ast$) gives

\[ (0)= \Hom_X(E_1^n,E_0^s) \lra \Hom_X(E_0^m,E_0^s) \lra \Ext^1_X(\cF,E_0^s) \lra \Ext^1_X(E_1^n,E_0^s)=(0),\]

\[ \chi(\msHom_X(\cF,\cG)) = nt\!-\!rmt\!+\!ms = \langle (m,n),(s,t)\rangle_r,\]

as desired.

§.§ The functor $\TilTheta_d$

For $d \in \{1,\ldots,r\!-\!1\}$ we consider the universal vector bundle $\cU_{(r,d)}$ of $\Gr_d(A_r)$. By definition, $\cU_{(r,d)}$ is the locally free sheaf corresponding to the locally trivial vector space fibration $(E,p)$
over $\Gr_d(A_r)$, where

\[ E\!:=\!\{(\fv,a) \in \Gr_d(A_r)\!\times\!A_r \ ; \ a \in \fv\} \ \ \text{and}  \ \  p : E \lra \Gr_d(A_r) \ ; \ (\fv,a) \mapsto \fv\]

denote the incidence variety and the canonical projection, respectively. Consequently,

\[ \cU_{(r,d)}(U) = \{s : U \lra E \ ; \ p\circ s\!=\!\id_U\} \cong \{ t : U \lra A_r \ ; \ t(\fv) \in \fv \ \ \ \forall \ \fv \in U\}\]

for every open subset $U \subseteq \Gr_d(A_r)$. The vector bundle $\cU_{(r,d)}$ is known to be simple and the canonical map $\cU_{(r,d)} \lra \cO^r_{\Gr_d(A_r)}$ is locally split injective. Hence its cokernel $\cQ_{(r,d)}$
is a vector bundle (cf. <cit.>), referred to as the universal quotient bundle, cf. <cit.>.

Given a pair $(V_1,V_2)$ of $\KK$-vector spaces, we put

\[ \widetilde{V_1} := V_1\!\otimes_\KK\!\cU_{(r,d)} \ \ \text{and} \ \ \widetilde{V_2} := V_2\!\otimes_\KK\!\cO_{\Gr_d(A_r)}.\]

Let $f_i : V_i \lra W_i$ ($i \in \{1,2\}$) be linear maps between $\KK$-vector spaces. These give rise to morphisms $\tilde{f_i} : \widetilde{V_i} \lra \widetilde{W_i}$, where

\[ \tilde{f_1} := f_1\!\otimes\id_{\cU_{(r,d)}} \ \ \ \text{and} \ \ \ \tilde{f_2} := f_2\otimes\id_{\cO_{\Gr_d(A_r)}}.\]

For $\fv \in \Gr_d(A_r)$, the standard isomorphisms $\ev_{1,\fv} : \cU_{(r,d)}(\fv) \lra \fv$ and $\ev_{2,\fv} : \cO_{\Gr_d(A_r)}(\fv) \lra \KK$ induce commutative diagrams

\begin{equation}\label{diagram1} \begin{tikzcd}   \widetilde{V_i}(\fv) \arrow[d, "\id_{V_i}\otimes \ev_{i,\fv}"] \arrow[r,"\tilde{f_i}(\fv)"] & \widetilde{W_i}(\fv) \arrow[d,"\id_{W_i}\otimes \ev_{i,\fv}"]\\
V_i\!\otimes_\KK\!\fu_i  \arrow[r,"f_i\otimes\id_{\fu_i}"] & W_i\!\otimes_\KK\!\fu_i,
\end{tikzcd} \end{equation}

\[ \fu_i := \left\{\begin{array}{cc} \fv & i\!=\!1 \\ \KK & i\!=\!2.\end{array} \right.\]

Similarly, the isomorphism $\cO_{\Gr_d(A_r)}(\Gr_d(A_r)) \cong \KK$ allows us to identify $(\tilde{f_2})_{\Gr_d(A_r)}$ and $f_2$.

Let $M \in \rep(K_r)$. In analogy with <cit.> (see also [32]), we consider for every open subset $U \subseteq \Gr_d(A_r)$, the morphism

\[ \TilTheta_{M,d}(U) : \widetilde{M_1}(U) \lra \widetilde{M_2}(U) \ \ ; \ \ m\otimes t \mapsto \sum_{i=1}^r \gamma_i\dact m\otimes \gamma_i^\ast\circ t\]

of $\cO_{\Gr_d(A_r)}(U)$-modules.[Here $\{\gamma_1^\ast,\ldots, \gamma_r^\ast\} \subseteq A_r^\ast$ denotes the dual basis of $\{\gamma_1,\ldots, \gamma_r\}.$] Then $\TilTheta_{M,d} : \widetilde{M_1} \lra
\widetilde{M_2}$ is a morphism of vector bundles, and we define

\[ \TilTheta_d(M) := \msCoker\TilTheta_{M,d} \in \Coh(\Gr_d(A_r)).\]

The definition of $\TilTheta_d(M)$ does not depend on the choice of the dual bases $\{\gamma_1,\ldots, \gamma_r\} \subseteq A_r$, $\{\gamma^\ast_1,\ldots, \gamma^\ast_r\} \subseteq A^\ast_r$. In fact, one can define
sheaves $\widetilde{M}'_1$ and $\widetilde{M}'_2$ via

\[ \widetilde{M}'_1(U) := \{ s : U \lra A_r\!\otimes_\KK\!M_1 \ ; \ s(\fv) \subseteq \fv\!\otimes_\KK\!M_1 \ \ \  \ \forall \ \fv \in U\} \subseteq \Mor(U,A_r\!\otimes_\KK\!M_1)\]

\[ \widetilde{M}'_2(U) := \Mor(U,M_2).\]

\[ \TilTheta'_{M,d}(U)(s) = \psi_M\circ s  \ \ \ \ \forall \ s \in \widetilde{M}'_1(U),\]

we obtain

\[ \TilTheta_d(M) \cong \coker\TilTheta'_{M,d}.\]

If $f : M \lra N$ is a morphism in $\rep(K_r)$ with components $f_i: M_i \lra N_i \ \ \ (i \in \{1,2\})$, then

\[ \tilde{f}_i : \widetilde{M_i} \! \lra \widetilde{N_i} \ \ ; \ \ m\otimes g \mapsto f_i(m)\otimes g \  \  \  \  \  \  \   (i \in \{1,2\})\]

are morphisms of vector bundles and there results a commutative diagram

\[ \begin{tikzcd}   \widetilde{M_1} \arrow[d, "\tilde{f_1}"] \arrow[r,"\TilTheta_{M,d}"] & \widetilde{M_2} \arrow[d,"\tilde{f_2}"] \arrow[r] & \TilTheta_d(M) \arrow[r] & (0)\\
\widetilde{N_1}  \arrow[r,"\TilTheta_{N,d}"] & \widetilde{N_2} \arrow[r] & \TilTheta_d(N) \arrow[r] & (0).
\end{tikzcd} \]

Consequently, there is a unique morphism

\[ \TilTheta_d(f) : \TilTheta_d(M) \lra \TilTheta_d(N),\]

which completes the diagram above. We thus obtain a functor

\[ \TilTheta _d: \rep(K_r) \lra \Coh(\Gr_d(A_r)).\]

We denote by $\reppi(K_r,d)$ the full subcategory of $\rep(K_r)$, whose objects $M$ have the property that $\psi_{M,\fv}\!=\!\psi_M|_{\fv\otimes_\KK M_1}$ is surjective for every $\fv \in \Gr_d(A_r)$. (This category
coincides with the essential image of $\repp(K_r,d)$ under the standard duality.)

Let $M \in \rep(K_r)$. We say that $M$ has constant $d$-rank, provided there is $\rk_d(M) \in \NN_0$ such that $\rk(\psi_{M,\fv})\!=\!\rk_d(M)$ for all $\fv \in \Gr_d(A_r)$.
We let $\CR(K_r,d)$ be the full subcategory of $\rep(K_r)$ whose objects have constant $d$-rank.

The category $\CR(K_r,d)$ is the analog of the category of modules of constant $(d,1)$-radical rank that was considered in <cit.>.

We record a few basic properties:

The following statements hold:

* The functor $\TilTheta_d$ is right exact.

* Let $M \in \rep(K_r)$. Then $M \in \CR(K_r,d)$ if and only if $\TilTheta_d(M) \in \Vect(\Gr_d(A_r))$. In that case, we have $\rk(\TilTheta_d(M))\!=\!\dim_\KK M_2\!-\!\rk_d(M)$.

* We have $\ker\TilTheta_d\cap\CR(K_r,d)\!=\!\reppi(K_r,d)$.

(1) The right exactness of $\TilTheta_d$ is a direct consequence of (the proof of) the Snake Lemma.

(2) Suppose that $M \in \CR(K_r,d)$. Given $\fv \in \Gr_d(A_r)$, the commutative diagram

\[ \begin{tikzcd}   \widetilde{M_1}(\fv) \arrow[d, "\id_{M_1}\otimes \ev_{1,\fv}"] \arrow[r,"\TilTheta_{M,d}(\fv)"] & \widetilde{M_2}(\fv) \arrow[d,"\id_{M_2}\otimes \ev_{2,\fv}"]\\
M_1\!\otimes_\KK\!\fv \arrow[r,"\psi_{M,\fv}\circ \omega"] & M_2,
\end{tikzcd} \]

where $\omega$ flips the tensor factors, yields

\[\rk(\TilTheta_{M,d}(\fv)) = \rk(\psi_{M,\fv}) = \rk_d(M).\]

It now follows from <cit.> that $\TilTheta_d(M)$ is a vector bundle of rank $\rk(\TilTheta_d(M))\!=\!\dim_\KK M_2\!-\!\rk_d(M)$.

Conversely, assume that $\TilTheta_d(M)$ is a vector bundle. Then <cit.> implies that there is $n \in \NN_0$ such that

\[ \rk(\TilTheta_{M,d}(\fv)) = n  \ \ \ \ \ \forall \ \fv \in \Gr_d(A_r).\]

The observation above thus implies that $\rk(\psi_{M,\fv})\!=\!n$ for all $\fv \in \Gr_d(A_r)$, so that $M \in \CR(K_r,d)$.

(3) Let $M \in \CR(K_r,d)$. By definition, we have $M \in \reppi(K_r,d)$ if and only if $\rk_d(M)\!=\!\dim_\KK M_2$. In view of (2), this is equivalent to $\TilTheta_d(M)\!=\!(0)$.

Steiner bundles on projective space were first systematically studied by Dolgachev-Kapranov [23]. The following definition for Grassmannians is taken from [1].

Let $(s,t) \in \NN_0^2$. A vector bundle $\cF \in \Vect(\Gr_d(A_r))$ is referred to as an $(s,t)$-Steiner bundle if there exists an exact sequence

\[ (0) \lra \cU^s_{(r,d)} \lra \cO^t_{\Gr_d(A_r)} \lra \cF \lra (0).\]

We denote by $\StVect(\Gr_d(A_r))$ the full subcategory of $\Vect(\Gr_d(A_r))$, whose objects are Steiner bundles (for some $(s,t) \in \NN_0^2$).

In view of $\cU_{(r,1)}\cong \cO_{\PP(A_r)}(-1)$ (cf. <cit.>), we retrieve the original definition of [23].

The relationship between $\repp(K_r)$ and $\StVect(\Gr_d(A_r))$, which for $d\!=\!1$ is implicit in Brambilla's work [13, 14] on Steiner bundles (for $\KK\!=\!\CC$), was investigated by Jardim and
Prata [40] for $\Char(\KK)\!=\!0$ in the more general context of cokernel bundles. In our context the category of “globally injective representations“ defined in [40] coincides with $\repp(K_r,d)$.

Since we will employ the functors $\TilTheta_d$ extensively, we recall the basic arguments of the proof of <cit.>, thereby hopefully convincing the reader that the result does
not necessitate any assumption on the characteristic of the base field. To that end we require the following subsidiary result.

The pair $(\cU_{(r,d)},\cO_{\Gr_d(A_r)})$ is special exceptional.

Given a partition $\alpha$, we let $|\alpha|\!=\!\sum_i\alpha_i$ be its degree, and $\alpha'$ be its transpose partition. Let $\cB_{r-d,d}$ be the set of those partitions, whose corresponding Young tableaux
have at most $r\!-\!d$ rows and at most $d$ columns. We pick a total ordering $\prec$ on $\cB_{r-d,d}$ such that $|\alpha| < |\beta|$ implies $\alpha \prec \beta$.

According to <cit.>, the vector bundle

\[ \cT := \bigoplus_{\alpha \in \cB_{r-d,d}}\bigwedge^{\alpha'}(\cU_{(r,d)})\]

is a tilting object in the bounded derived category $\msD^b(\Gr_d(A_r))$ of $\Coh(\Gr_d(A_r))$. In particular,  $\Ext^i_{\Gr_d(A_r)}(\cT,\cT)\!=\!(0)$ for all $i\!>\!0$. We consider the partitions $\alpha\!:=\!0$ and
$\beta\!:=\!1$. Then $\alpha\!\prec\!\beta \in \cB_{r-d,d}$, so that  $\cO_{\Gr_d(A_r)}\!=\!\bigwedge^\alpha(\cU_{(r,d)})$ and $\cU_{(r,d)}\!=\!\bigwedge^\beta(\cU_{(r,d)})$ are direct summands of $\cT$. Consequently,
$\Ext^i_{\Gr_d(A_r)}(\cX,\cY)\!=\!(0)$ for $\cX,\cY \in \{\cU_{(r,d)},\cO_{\Gr_d(A_r)}\}$ and $i\!>\!0$. Since both bundles are simple, our assertion follows from the fact that $\Hom_{\Gr_d(A_r)}(\cO_{\Gr_d(A_r)}, \cU_{(r,d)})\!
=\!\cU_{(r,d)}(\Gr_d(A_r))\!=\!(0)$, cf. <cit.>.

The following statements hold:

* $\cF \in \Vect(\Gr_d(A_r))$ is a Steiner bundle if and only if there is $M \in \repp(K_r,d)$ such that $\cF \cong \TilTheta_d(M)$.

* The functor $\TilTheta_d : \repp(K_r,d) \lra \StVect(\Gr_d(A_r))$ is an equivalence.

* Let $M \in \repp(K_r,d)$. Then we have $\rk(\TilTheta_d(M))\!=\!\Delta_M(d)$.

Let $M \in \CR(K_r,d)$. The proof of Lemma <ref> yields  $\ker \TilTheta_{M,d}(\fv) \cong \ker \psi_{M,\fv}$ for every $\fv \in \Gr_d(A_r)$.

(1) Suppose that $M \in \repp(K_r,d)$. By virtue of Theorem <ref> and the above we have

\[ \ker \TilTheta_{M,d}(\fv) \cong \ker \psi_{M,\fv}\!=\!(0)\]

for all $\fv \in \Gr_d(A_r)$. According to <cit.>, the morphism $\TilTheta_{M,d}$ is locally split injective, whence $\msKer\TilTheta_{M,d}\!=\!(0)$. Consequently, $\TilTheta_d(M)$ is a Steiner bundle.

Conversely, suppose that $\cF$ is a Steiner bundle. Then there exists a pair $(M_1,M_2)$ of vector spaces and an exact sequence

\[ (0) \lra \widetilde{M_1} \stackrel{\Psi}{\lra} \widetilde{M_2} \lra \cF \lra (0)\]

of vector bundles. It follows from <cit.>, which holds for fields of arbitrary characteristic, that

\[ \dim_\KK\Hom_{\Gr_d(A_r)}(\cU_{(r,d)},\cO_{\Gr_d(A_r)}) = \dim_\KK\HH^0(\cU_{(r,d)}^\vee) = r.\]

Hence the injective canonical map

\[ \Hom_\KK(M_1,M_2)\!\otimes_\KK\!A_r^\ast \lra \Hom_{\Gr_d(A_r)}(\widetilde{M_1},\widetilde{M_2})\]

sending $\zeta\otimes f$ to the map $m\otimes t \mapsto \zeta(m)\otimes f\circ t$ is also surjective and thus an isomorphism. There thus exist $\KK$-linear maps $M(\gamma_i) : M_1 \lra M_2$ such that

\[ \Psi(m\otimes t) = \sum_{i=1}^rM(\gamma_i)(m)\otimes \gamma_i^\ast\circ t \ \ \ \ \ \ \forall \ m \in M_1, t \in \cU_{r,d}.\]

Letting $M\!:=\!(M_1,M_2,(M(\gamma_i))_{1\le i\le r}) \in \rep(K_r)$, we see that $\cF\!\cong\!\TilTheta_d(M)$, while $\msKer\TilTheta_{M,d}\!=\!(0)$. Consequently, $M \in \repp(K_r,d)$, as $\ker \psi_{M,\fv} \cong
(\msKer\TilTheta_{M,d})(\fv)\! =\! (0)$ for every $\fv \in \Gr_d(A_r)$.

(2) Lemma <ref> ensures that $\Ext^1_{\Gr_d(A_r)}(\cO_{\Gr_d(A_r)},\cU_{(r,d)})\!=\!(0)$ and we may now proceed as in <cit.>, observing that the assumption $\Char(\KK)\!=\!0$ is not needed.

(3) This follows directly from Lemma <ref>.

For $d\!=\!1$, the bundles $\TilTheta_1(M_{\cS}[m]), \ (m\!\ge\!r\!+\!1)$ are those introduced by Schwarzenberger, cf. <cit.>. Using Corollary <ref> one obtains an explicit formula for
the generic splitting types of these bundles.

Let $(0) \lra A \stackrel{f}{\lra} B \stackrel{g}{\lra} C \lra (0)$ be an exact sequence in $\rep(K_r)$ such that $C \in \repp(K_r,d)$. Then the sequence $(0) \lra \TilTheta_d(A) \lra \TilTheta_d(B) \lra
\TilTheta_d(C) \lra (0)$ is exact.

We have exact sequences $(0) \lra A_i \stackrel{f_i}{\lra} B_i \stackrel{g_i}{\lra} C_i \lra (0)$ for $i \in \{1,2\}$. Tensoring with $\cU_{(r,d)}$ and $\cO_{\Gr_d(A_r)}$  yields a commutative diagram

\[  \begin{tikzcd}
(0) \arrow[r] & \widetilde{A_1} \arrow[r] \arrow[d,"\TilTheta_{A,d}"] & \widetilde{B_1} \arrow[r] \arrow[d,"\TilTheta_{B,d}"] & \widetilde{C_1} \arrow[d,"\TilTheta_{C,d}"] \arrow[r] & (0)\\
(0) \arrow[r] & \widetilde{A_2} \arrow[r] & \widetilde{B_2} \arrow[r] & \widetilde{C_2} \arrow[r] & (0)
\end{tikzcd} \]

with exact rows. The arguments of the proof of Theorem <ref> show that

\[ 0 = \dim_\KK \ker \psi_{C,\fv}  = \dim_\KK\ker(\TilTheta_{C,d}(\fv))\]

for all $\fv \in \Gr_d(A_r)$. By virtue of <cit.>, we obtain $\msKer \TilTheta_{C,d}\!=\!(0)$. The assertion now follows by applying the Snake Lemma.

For $M,N \in \rep(K_r,d)$ the following statements hold:

* We have

\[ \chi(\msHom_{\Gr_d(A_r)}(\TilTheta_d(M),\TilTheta_d(N))) = \langle\udim M, \udim N\rangle_r.\]

* We have

\[ \dim_\KK\Ext^1_{K_r}(M,N) = \dim_\KK\Ext^1_{\Gr_d(A_r)}(\TilTheta_d(M),\TilTheta_d(N)).\]

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

(2) Proposition <ref> and its proof yield

\begin{eqnarray*}
\dim_\KK\Ext^1_{\Gr_d(A_r)}(\TilTheta_d(M),\TilTheta_d(N)) &= & \dim_\KK \Hom_{\Gr_d(A_r)}(\TilTheta_d(M),\TilTheta_d(N))\\
&- & \chi(\msHom_{\Gr_d(A_r)}(\TilTheta_d(M),\TilTheta_d(N)))\\
& = & \dim_\KK \Hom_{K_r}(M,N)\!-\!\langle\udim M,\udim N\rangle_r\\
& = &\dim_\KK\Ext^1_{K_r}(M,N). \ \ \  \ \ \  \qedhere
\end{eqnarray*}

§.§ Direct consequences

Following [26], we begin by collecting a few facts concerning Chow rings of Grassmannians. We emphasize that our statements are valid for the field $\KK$, which has arbitrary characteristic.

For $\ell \in \{0,\ldots, d(r\!-\!d)\}$, we let $Z_\ell(\Gr_d(A_r))$ be the free abelian group with basis given by the $\ell$-dimensional irreducible closed subsets of $\Gr_d(A_r)$. The elements
of $Z_\ell(\Gr_d(A_r))$ are called $\ell$-cycles, those of

\[ Z(\Gr_d(A_r)) := \bigoplus_{\ell=0}^{d(r\!-\!d)} Z_\ell(\Gr_d(A_r))\]

are referred to as cycles.

The Chow group of $\Gr_d(A_r)$ is the factor group

\[ A(\Gr_d(A_r)) := Z(\Gr_d(A_r))/\Rat(\Gr_d(A_r))\]

given by the subgroup $\Rat(\Gr_d(A_r))$ defined in <cit.>. If $C$ is a cycle, we write $[C]$ for its residue class. The class $[\Gr_d(A_r)] \in A(\Gr_d(A_r))$ is called the fundamental class of $\Gr_d(A_r)$.

Given $\ell \in \{0,\ldots, d(r\!-\!d)\}$, we put

\[ A^\ell(\Gr_d(A_r)) := \{ [C] \ ; \ C \in Z_{d(r-d)-\ell}(\Gr_d(A_r))\}.\]

We have

\[ A(\Gr_d(A_r)) = \bigoplus_{\ell=0}^{d(r-d)} A^\ell(\Gr_d(A_r)).\]

The group $A(\Gr_d(A_r))$ has the structure of a $\ZZ$-graded, commutative ring, see <cit.>. One can show that $A^0(\Gr_d(A_r))\!=\!\ZZ\cdot [\Gr_d(A_r)] \cong \ZZ$.

The ring $A(\Gr_d(A_r))$ is called the Chow ring of $\Gr_d(A_r)$.

\[ \cV : \ \ \ \ \ \ (0) \subseteq V_1 \subseteq V_2 \subseteq \cdots \subseteq V_{r-1} \subseteq V_r\!=\!A_r\]

be a complete flag in $A_r$, so that $\dim_\KK V_i\!=\!i$.

We let $\NN_0^d(r)\!:=\!\{ a \in \NN_0^d \ ; \  r\!-\!d\!\ge\! a_1\! \ge\! \cdots\! \ge\! a_d\! \ge\! 0\}$. For $a \in \NN_0^d(r)$ we define the Schubert variety $\Sigma_a(\cV)$ via

\[ \Sigma_a(\cV) := \{\fv \in \Gr_d(A_r) \ ; \ \dim_\KK(\fv\cap V_{r-d+i-a_i})\!\ge\!i \ \ \ \forall \ i \in \{1,\ldots, d\}\}.\]

Then $\Sigma_a(\cV)$ is smooth, closed, irreducible and of codimension $|a|\!:=\!\sum_{i=1}^da_i$ in $\Gr_d(A_r)$, cf. <cit.>. Its class

\[ \sigma_a := [\Sigma_a(\cV)]\]

is the Schubert class of $a$. It does not depend on the choice of $\cV$.

Given $\ell \in \{0,\ldots, r\!-\!d\}$, we write

\[ \Sigma_\ell := \Sigma_{(\ell,0,\ldots,0)},\]

so that

\[\Sigma_\ell(\cV) =  \{ \fv \in \Gr_d(A_r) \ ; \ \fv\cap V_{r-d+1-\ell} \!\ne\!(0)\}.\]

The classes $\sigma_\ell\!:=[\Sigma_\ell(\cV)]$ are referred to as special Schubert classes.

We have

\[ \sigma_\ell \in A^\ell(\Gr_d(A_r))\]

for all $\ell \in \{0,\ldots,r\!-\!d\}$.

(cf. <cit.>) The Schubert classes $\{\sigma_a \ ; \ a \in \NN_0^d(r)\}$ form a basis of the $\ZZ$-module $A(\Gr_d(A_r))$.

It follows that

\[\{\sigma_a \ ; \ a \in \NN_0^d(r), \ |a|\!=\!i\}\]

is a basis of $A^i(\Gr_d(A_r))$ for every $i \in \{0,\ldots,d(r\!-\!d)\}$.

Let $\cF$ be a vector bundle on $\Gr_d(A_r)$. We denote by

\[ c(\cF) = 1\!+\!\sum_{i=1}^{\rk(\cF)} c_i(\cF)t^i \in A(\Gr_d(A_r))[t]\]

the Chern polynomial of $\cF$. The coefficient $c_i(\cF) \in A^i(\Gr_d(A_r))$ is called the $i$-th Chern class of $\cF$, cf. <cit.>.

The Chern polynomial of the universal quotient bundle $\cQ_{(r,d)}$ on $\Gr_d(A_r)$ is given by

\[ c(\cQ_{(r,d)}) = 1\!+\!\sum_{\ell=1}^{r-d} \sigma_\ell t^{\ell},\]

see <cit.>.

Let $M \in \repp(K_r,d)$. Then we have

\[ c_1(\TilTheta_d(M)) = (\dim_\KK M_1)\sigma_1.\]

Hence the dimension vector of $M$ can be recovered from $c_1(\TilTheta_d(M))$ and $\rk(\TilTheta_d(M))$ (cf. Theorem <ref>(3)).

By definition, there is an exact sequence

\[ (0) \lra \cU_{(r,d)} \lra A_r\!\otimes_\KK\! \cO_{\Gr_d(A_r)} \lra \cQ_{(r,d)} \lra (0), \]

so that the Whitney sum formula yields $c_1(\cU_{(r,d)})\!=\!-c_1(\cQ_{(r,d)})\!=\!-\sigma_1$. By the same token, the defining sequence

\[ (0) \lra \widetilde{M}_1 \lra \widetilde{M}_2 \lra \TilTheta_d(M) \lra (0)\]

implies $c_1(\TilTheta_d(M))\!=\!-c_1(\widetilde{M}_1)\!=\!-(\dim_\KK M_1)c_1(\cU_{(r,d)})\!=\!(\dim_\KK M_1)\sigma_1$.

The foregoing result allows us to view the first Chern class $c_1(\cF)$ of a Steiner bundle $\cF$ as a number $c_1(\cF) \in \NN_0$. In the sequel, we will freely use this interpretation without further notice.

We next employ the functor $\TilTheta_d$ to translate some results of Section <ref> to the context of Steiner bundles.

Let $\cF \in \StVect(\Gr_d(A_r))$ be a Steiner bundle.

* We have $\rk(\cF)\!\ge\!\min\{c_1(\cF),d\}(r\!-\!d)$.

* If $c_1(\cF)\!\le\!d$ or $\rk(\cF)\!<\!d(r\!-\!d)$, then $\cF \cong \cO_{\Gr_d(A_r)}^{\rk(\cF)-(r-d)c_1(\cF)}\!\oplus\!\cQ_{(r,d)}^{c_1(\cF)}$.

* If $\rk(\cF)\!=\!d(r\!-\!d)$, then $\cF$ is either as in (2), or $\cF$ is simple.

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

* Suppose that $\rk(\cF)\!>\!d(r\!-\!d)$. Then there exists a Steiner bundle $\cG$ such that $\rk(\cG)\!=\!d(r\!-\!d)$ and a short exact sequence

\[ (0) \lra \cO_{\Gr_d(A_r)}^{\rk(\cF)-d(r-d)} \lra \cF \lra \cG \lra (0).\]

Theorem <ref> provides $M \in \repp(K_r,d)$ such that $\TilTheta_d(M) \cong \cF$ and $\rk(\cF) \cong \Delta_M(d)$. In addition, Corollary <ref> gives $\dim_\KK M_1\!=\!c_1(\cF)$.

(1) This is a direct consequence of Theorem <ref>.

(2) In view of Corollary <ref> and the remark at the beginning of Section <ref>, the representation $M$ is projective, whence $M \cong \Delta_M(r)P_0(r)\!\oplus\!(\dim_\KK M_1)P_1(r)$.
We have $\Delta_M(r)\!=\!\Delta_M(d)\!-\!(r\!-\!d)\dim_\KK M_1 = \rk(\cF)\!-\!(r\!-\!d)c_1(\cF)$. Moreover, $\TilTheta_d(P_0(r))\cong \cO_{\Gr_d(A_r)}$ and since $\TilTheta_{P_1(r),d}(t)\!=\!\sum_{i=1}^r \gamma_i\otimes
\gamma_i^\ast\circ t$ for all $t \in \cU_{(r,d)}(U)$ and $U \subseteq \Gr_d(A_r)$ open, we obtain $\TilTheta_d(P_1(r))\cong \cQ_{(r,d)}$.

(3) In view of Corollary <ref>, $M$ is projective or a brick. In the former case, $\cF$ is as in (2), in the latter case, Theorem <ref> gives $\End_{\Gr_d(A_r)}(\cF) \cong \End_{K_r}(M) \cong \KK$, so that $\cF$
is simple.

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

(5) Let $n\!:=\!\Delta_M(d)\!-\!(r\!-\!d)$. Proposition <ref> provides a short exact sequence

\[ (0) \lra nP_0(r) \lra M \lra N \lra (0),\]

where $N \in \repp(K_r,d)$ has minimal type. Observing Corollary <ref>, we may apply $\TilTheta_d$ to obtain the asserted sequence.

Suppose that $\Char(\KK)\!=\!0$.

(1) In this case, part (1) was proved in <cit.>, using Porteous' formula.

(2) For $d\!=\!1$, part (5) was obtained in <cit.>.

§ INDECOMPOSABLE STEINER BUNDLES: KAC'S THEOREM AND THE AUSLANDER-REITEN QUIVER

In this section, we introduce our two main representation-theoretic tools: Kac's Theorem concerning dimension vectors of indecomposable representations of $K_r$ and the Auslander-Reiten quiver $\Gamma(K_r)$ of
$\rep(K_r)$. In combination with the functors $\TilTheta_d$ the latter will be used to endow the set of isoclasses of indecomposable Steiner bundles with the structure of a quiver, whose connected components can be
studied in terms of those of $\Gamma(K_r)$.

§.§ Kac's Theorem
Recall that an indecomposable module $M \in \rep(K_r)$ is called regular, if it is neither preinjective nor preprojective. For future reference it will be convenient to have the following version of  <cit.>, which also builds on <cit.>, at our disposal:

[Kac's Theorem for $K_r$]  Let $r\!\ge\!2$ and $\delta \in \NN^2_0\!\smallsetminus\!\{0\}$.

* If $\delta\!=\!\udim M$ for some indecomposable $M \in \rep(K_r)$, then $q_r(\delta)\!\le\!1$.

* If $q_r(\delta)\!=\!1$, then there is a, up to isomorphism, unique indecomposable module $M \in \rep(K_r)$ such that $\udim M\!=\!\delta$. The module $M$ is preprojective or preinjective.

* If $q_r(\delta)\!\le\!0$, then every indecomposable module $M \in \rep(K_r)$ such that $\udim M\!=\!\delta$ is regular. Moreover, there are infinitely many isoclasses of these modules.

§.§ Exceptional Steiner bundles
In this section we shall classify exceptional Steiner bundles. Throughout, we assume that $r\!\ge\!2$ and $d \in \{1,\ldots, r\!-\!1\}$. Our first result characterizes the
modules corresponding to exceptional Steiner bundles.

Suppose that $r\!\ge\!3$, and let $\cF \in \StVect(\Gr_d(A_r))$ be a Steiner bundle. Then the following
statements are equivalent:

* $\cF$ is exceptional.

* There is $M \in \rep(K_r)$ indecomposable preprojective such that $\cF \cong \TilTheta_d(M)$.

(1) $\Rightarrow$ (2). By definition, we have $\End_{\Gr_d(A_r)}(\cF)\cong \KK$, while $\Ext^i_{\Gr_d(A_r)}(\cF,\cF)\!=\!(0)$ for all $i\!\ge\!1$. Theorem <ref> provides $M \in
\repp(K_r,d) \subseteq \EKP(K_r)$ such that $\TilTheta_d(M)\cong \cF$. In particular, $M$ is a brick and hence indecomposable. In view of <cit.>, the module $M$ is not preinjective. Corollary
<ref> yields

\[ \Ext^1_{K_r}(M,M) \cong \Ext^1_{\Gr_d(A_r)}(\TilTheta_d(M),\TilTheta_d(M)) = (0),\]

so that Theorem <ref> in conjunction with the Euler-Ringel form (cf. Section <ref>) ensures that $M$ is preprojective.

(2) $\Rightarrow$ (1). Since $M$ is preprojective, an application of <cit.> in conjunction with Corollary <ref> implies

\[\Ext^1_{\Gr_d(A_r)}(\cF,\cF) \cong  \Ext^1_{K_r}(M,M)\!=\!(0).\]

Now Theorem <ref> ensures that $M$ is a brick, so that the $\KK$-space $\End_{\Gr_d(A_r)}(\cF,\cF) \cong \End_{K_r}(M)$ is one-dimensional. In view of Lemma <ref>, part (iii) of the proof of Proposition <ref>
yields $\Ext^i_{\Gr_d(A_r)}(\cF,\cF)\!=\!(0)$ for all $i\!\ge\!2$. As a result, the Steiner bundle $\cF$ is exceptional.

Part (3) of our next result extends <cit.>, where the bundles involved were assumed to be generic. Part (5) shows that <cit.> holds for Grassmannians over fields of arbitrary characteristic.

Recall from Section <ref> that $(P_n(r))_{n\ge 0}$ denotes the family of preprojective $K_r$-representations (of dimension vectors $\udim P_n(r)\!=\!(a_n(r),a_{n+1}(r))$).

Suppose that $r\!\ge\!3$. Then the following statements hold:

* For each $n \in \NN_0$ there exists an exceptional Steiner bundle $\cE_{n,d} \in \StVect(\Gr_d(A_r))$ such that $\rk(\cE_{n,d})\!=\! a_{n+1}(r)\! -\!da_n(r)$ and $c_1(\cE_{n,d})\! =\! a_n(r)\sigma_1$.

* If $\cE \in \StVect(\Gr_d(A_r))$ is an exceptional Steiner bundle, then $\cE \cong \cE_{n,d}$ for some $n \in \NN_0$.

* An indecomposable Steiner bundle $\cF \in \StVect(\Gr_d(A_r))$ is exceptional if and only if $\rk(\cF)\!=\!a_{n+1}(r)\!-\!da_n(r)$ and $c_1(\cF)\! =\! a_n(r)\sigma_1$ for some $n \in \NN_0$.

* A non-zero Steiner bundle $\cF \in \StVect(\Gr_d(A_r))$ satisfies $\Ext^1_{\Gr_d(A_r)}(\cF,\cF)\!=\!(0)$ if and only if there exist $(a,b) \in \NN\!\times\!\NN_0$ and $n \in \NN_0$ such that $\cF \cong \cE_{n,d}^a\!