Framed moduli and Grassmannians of submodules

In this work we study a realization of moduli spaces of framed quiver representations as Grassmannians of submodules devised by Marcus Reineke. Obtained is a generalization of this construction for finite dimensional associative algebras and for quivers with oriented cycles. As an application we get an explicit realization of fibers for the moduli space bundle over the categorical quotient for the quiver $A_{n-1}^{(1)}$.


Introduction
A quiver Q is a diagram of arrows, determined by two finite sets Q 0 (the set of "vertices") and Q 1 (the set of "arrows") with two maps h, t : Q 1 → Q 0 which indicate the vertices at the head and tail of each arrow. A representation (W, ϕ) of Q consists of a collection of finite dimensional k-vector spaces W i , for each i ∈ Q 0 , together with linear maps ϕ a : W ta → W ha , for each a ∈ Q 1 . The dimension vector α ∈ Z Q0 of such a representation is given by α v = dim k W i . A morphism f : (W i , ϕ a ) → (U i , ψ a ) of representations consists of linear maps f i : W v → U v , for each i ∈ Q 0 , such that f ha ϕ a = ψ a f ta , for each a ∈ Q 1 . Evidently, it is an isomorphism if and only if each f i is. Thus, isomorphism classes of representations of Q with dimension vector α coincide with orbits of the action of GL(α) = i∈Q0 GL i (k) on the representation space Rep(Q, α).
In studying quiver representations standard approaches of Invariant Theory often fail because the algebra of invariants is poor or even trivial, as in case of the quiver without oriented cycles, and so the categorical quotient Rep(Q, α)/ /GL(α) := Spec k[Rep(Q, α)] GL(α) is a point. Constructions of Geometric Invariant Theory may help to compensate this defect. Indeed, one can consider the trivial linearization twisted by a character χ of GL(α), which restricts our attention to an open subset of Rep(Q, α), consisting of χ-semistable representations. Within the open set there are more closed orbits and the corresponding algebraic quotient is more interesting. In his paper [7] A.D. King showed that the notions of semistability and stability, that arise from Geometric Invariant Theory, coincide with more algebraic notions, expressed in the language of abelian categories. Namely, he devised a link between this concept and the so-called θ-stability. All the characters of GL(α) are given by χ θ (g) = v∈Q0 det (g v ) θv for θ ∈ Z Q0 , and a representation W is a χ θ -(semi-)stable point of the variety Rep(Q, α) if and only if it is θ-(semi-)stable as an element of the abelian category Rep(Q) (see [7,Section 2]). King proved the existence of coarse moduli space for semistable representations of quivers and guaranteed that for stable representations there is a fine moduli space. This technique allowed a straightforward and convenient adaptation to the study of representations of finite dimensional algebras [7,Section 4].
An alternative approach to the problem considered was introduced by B. Huisgen-Zimmermann [5]. Let A be a finite dimensional associative algebra with unity. Fix a semisimple A-module T with projective cover P and a positive integer m. Denote by Grass T m the Grassmannian of all (dim P − m)-dimensional submodules of the radical radP . There is then a bijection between Aut A P -orbits in Grass T m and isomorphism classes of m-dimensional modules with top isomorphic to T , sending Aut A P ·C to the isomorphism class of P/C. In [5] investigated are such triples (A, T , m), that Grass T m itself provides a moduli space classifying the d-dimensional A-modules with top T , up to isomorphism. It is also proved there that Grass T m admits an open covering by representation-theoretically defined affine charts. For further generalization and systematical treatment of the whole hierarchy of moduli-parametrizing Grassmannian varieties see [6].
Another possible way of applying Invariant Theory to the study of quiver representations is to consider framed representations thus achieving better precision at the expense of extending the representation space. They first appeared in [10] as one of the steps in the construction of Nakajima Varieties.
Let Q be a quiver and α be a dimension vector. Fix an additional dimension vector ζ and consider the space Rep(Q, α, ζ) := Rep(Q, α) ⊕ i∈Q0 Hom k (k αi , k ζi ). Its elements are said to be framed representations of Q. Define a GL(α)-action on Rep(Q, α, ζ) by g · (M, (f i ) n i=1 ) = (g · M, (f i g −1 i ) n i=1 ). A framed representation (M, f ) is called stable if there is no nonzero subrepresentation N of M which is contained in ker f . Denote by Rep s (Q, α, ζ) the space of stable framed representations. One can show that the quotient of Rep s (Q, α, ζ) is more efficient in orbits discriminating than the standard categorical quotient. Furthermore, it can be shown that it is in a sense reducible to King's construction and thus enjoys all its properties.
Grassmanians of submodules of injective modules arise independently in the course of this approach. In [11] M. Reineke obtained for acyclic quivers a realization of framed moduli space as a Grassmannian of subrepresentations of an injective representation depending only on dimension vectors α and ζ. He further investigated its cohomology (see also [2]) and applications to quantum groups.
The aim of this work is to explore possible generalizations of Reineke's construction. So far, there are two of them. First, we can adapt it for quivers with relations, i.e. at least for finite dimensional algebras. We may also try to eliminate the condition of having no oriented cycles. Both possibilities are discussed below. Section 2 is devoted to giving some basic results on the connection between finite dimensional algebras and quivers. In Subsections 3.1 and 3.2 we remind the concept of stability as it was formulated by A. D. King and A. N. Rudakov for abelian categories. We also introduce the notion of framed representation space for finite dimensional algebras.
A straightforward generalization of Reineke's ideas for finite dimensional algebras is presented in Subsection 3.3. We prove that for a finite dimension algebra and two dimensional vectors α and ζ the quotient space Rep s (A, α, ζ)/ /GL(α) is isomorphic to the Grassmannian of submodules of a certain injective A-module J.
In Sections 4 to 6 we investigate a possible way to generalize this result to quivers with oriented cycles. It is crucial for Reineke's construction that A is finite dimensional, since without it we are no longer able to embed all stable framed representations in a finite dimensional module. In general situation we thus come to considering Grassmannians of submodules in infinite dimensional spaces, that makes impossible to apply the usual techniques. So, having established a bijection between such a Grassmannian and the moduli space we provide the former with a structure of algebraic variety, but this gives us no information about the moduli space itself. Hence we are forced to restrict our attention to fibers of the moduli space bundle over the categorical quotient.
From that point we work over the ground field C. In sections 4 and 5 we study the quiver Q = A (1) n−1 with cyclic orientation. We obtain an embedding of Rep s (Q, α, ζ) into an infinite dimensional representation J !! , with each J !! i a space of holomorphic vector functions on A 1 . Recall that by Procesi-Razmyslov's Theorem [9, Theorem 1] the algebra of invariant polynomial functions on Rep(Q, α, ζ) is generated by traces of oriented cycles in Q. Since in Q there is only one cycle τ i of minimal length starting at each vertex i ∈ Q 0 , traces of τ k i are polynomials in coefficients of characteristic polynomials χ i of τ i . Hence, the categorical quotient Rep(Q, α, ζ)/ /GL(α) may be identified with a subspace in the Cartesian product C[t] n , each point presented by a tuple of characteristic polynomials. Let χ = (χ 1 , . . . , χ n ) be in Rep(Q, α, ζ)/ /GL(α), λ 1 , . . . , λ N be all different roots of χ 1 , . . . , χ n , and r ij be the multiplicity of λ j as a root of χ i . For each j = 1, . . . , N consider a new dimension vector r j with (r j ) i = r ij . Define sub- These representations are finite dimensional and it may be shown that the fiber of the framed moduli space over (χ 1 , . . . , χ n ) ∈ C[t] n is isomorphic to the product of the Grassmannians of their subrepresentations. The details are discussed in Section 4. In Section 5 obtained is an explicit presentation by equations in projective space of fibers of moduli space bundle over the categorical quotient. Section 6 is devoted to applying of this technique to any quiver Q where all oriented cycles pairwise commute. We consider the natural projection π s : Rep s (Q, α, ζ)/ /GL(α) → Rep(Q, α, ζ)/ /GL(α) and describe the fibers π −1 s (x), for x ∈ Rep(Q, α, ζ)/ /GL(α). The main result here is that for every x and two dimension vectors α and ζ there is a quiver Q ♠ , a dimension vector α ∈ (Z 0 ) Q ♠ 0 , and a finite dimensional representation W ♠ of Q ♠ such that the fiber π −1 s (x) is isomorphic to the Grassmannian of α-dimensional submodules of W ♠ .
After having written this paper the author found that fibers of the projection π s were described by J. Engel and M. Reineke for arbitrary quivers using Luna's stratification as nilpotent parts of the framed moduli space for some new quiver Q and dimension vectors α and ζ (see [4,Theorem 4.1]). Theorem 3 together with Proposition 4 thus imply that each fiber of π s is isomorphic to the Grassmannian of submodules of a module over some finite dimensional algebra with a certain dimension vector. Our description is less general than the one given in [4], but more explicit and convenient when applicable.
The author thanks Ivan V. Arzhantsev and Markus Reineke, who also brought to his attention the paper [4] and the universal description of layers of π s , for useful discussions.

Background information
First, recall some general facts concerning finite dimensional algebras and their connection with quivers.
Let A be an associative finite dimensional algebra over an arbitrary field k. An element e ∈ A is called idempotent if e 2 = e. We say that two idempotents e 1 and e 2 are orthogonal if e 1 e 2 = e 2 e 1 = 0. An idempotent e is primitive if it is not a sum of two nonzero orthogonal idempotents. It is well known that for an algebra A with unity there always exists a decomposition 1 = e 1 + . . . + e n , where e i are primitive orthogonal idempotents. Note that this decomposition induces a decomposition A = Ae 1 ⊕ . . . ⊕ Ae n of the regular A-module called Peirce decomposition.
An algebra is said to be splitting if the quotient A/r, where r stands for the radical of A, is isomorphic to a direct product of matrix algebras over the ground field. Note that all algebras over an algebraically closed field are splitting. An algebra is called basic if A/r is isomorphic to a direct product of division rings. This condition is equivalent to the following: in the decomposition A = P 1 ⊕. . .⊕ P k of the regular module, where all P i are indecomposable projective modules, all the summands are pairwise non-isomorphic, [3,Theorem 3.5.4].
Fix a decomposition 1 = e 1 + . . . + e n of the unity 1 ∈ A, where all e i are primitive orthogonal idempotents. It is not hard to see that every A-module M as a k-vector space may be decomposed as The dimension vector of M is the vector α = dimM with α i = dim M i . Decomposing in this way the ideals Ae i of A (which are submodules of the regular module), we obtain the two-sided Peirce decomposition of A: A = i,j e j Ae i . The components e j Ae i are neither left, nor right ideals, but they provide a convenient matrix interpretation of the elements of A (see [3,Chapter 1,§7]).
The set of all A-modules with dimension vector α will be denoted by Rep(A, α). The group GL(α) = n i=1 GL αi (k) acts naturally on this set, each factor acting by base change in M i . Namely, for an element a ∈ e j Ae i the corresponding operator ϕ(a) of the representation A → L(M ) maps M i to M j and all the rest components to zero; thus we may define the action g = (g t ) n t=1 ∈ n i=1 GL(M i ) = GL(α) as follows: (g · ϕ(a))(m) = (g j ϕ(a)g −1 i )(m), for all g ∈ G, m ∈ M i . Since A admits the two-sided Peirce decomposition A = i,j e j Ae i , the actions is well defined. Now we remind the connection between algebras and quivers. Let k be a field. For a quiver Q one defines a path algebra kQ. As a linear space it is the span of all paths in Q, including those of length 0, which we identify with vertices of Q. Multiplication in kQ is defined by for two paths σ and τ in Q.
A relation in Q is a k-linear combination of paths in Q of length not less than 2 with the same source and target. For a set of relations ρ denote by ρ the ideal of the algebra kQ generated by these relations. Theorem 1. [1, Prop. II.2.5] For a finite dimensional algebra A the category of finitely generated A-modules is equivalent to the category of finitely generated Γ-modules for some basic algebra Γ.
Thus the problem of classifying the representations of arbitrary finite dimensional algebras can be in a sense reduced to the case of basic algebras. So we will be considering only basic algebras.
We now remind briefly the procedure of assigning a quiver Q(A) to a splitting basic algebra A. Let r be the radical of A, 1 = e 1 + . . . + e n be a decomposition of unity in A/r, 1 = e 1 + . . . + e n the corresponding decomposition of unity in A; further, let W = (r/r 2 ). Denote t ij = dim k e j W e i . Now set Q(A) = (Q 0 , Q 1 ) with Q 0 = {1, . . . , n} and t ij arrows from the i-th vertex to the j-th one.
Let (kQ) 1 be the ideal generated by the arrows in Q. An ideal I kQ is said to be regular if (kQ) 2 1 ⊇ I ⊇ (kQ) t 1 , for some t 2.
Theorem 2. [1, Theorem III.1.9] Every splitting basic finite dimensional algebra with quiver Q is isomorphic to a factor algebra kQ/I, where I is a regular ideal. On the level of representation spaces this correspondence looks as follows: the set Rep(A, α) is a Zariski closed subvariety of Rep(Q(A), α), since it is the subset where X p ≡ 0 for all p ∈ ρ (for a representation X ∈ Rep(Q(A), α) and an element λ 1 a i11 . . . a i1k(1) + . . . + λ s a is1 . . . a isk(s) we denote by X p the linear transformation λ 1 X ai 11 . . . X ai 1k(1) + . . . + λ s X ai s1 . . . X ai sk(s) ). Sometimes we will denote this subvariety by Rep(Q(A), ρ, α).

Framed representations of finite dimensional algebras
3.1. Semistabile representations. Consider a quiver Q. A character of the category Rep(Q) of representations of Q is a linear function θ : ZQ 0 → Z (in other words, to each vertex of the quiver this function assigns an integer). For . This approach devised by A. D. King, was generalized and reformulated in a more flexible form by A. N. Rudakov [12, §3].
From now on, since for the subsets Rep ss ξ (Q, α) and Rep s ξ (Q, α) the existence of the categorical quotient was proved in [7], we can use the notion of moduli spaces of µ-(semi-)stable points, where µ is a slope. Namely, denote by M ss µ (Q, α) (respectively by M s µ (Q, α)) the categorical quotient Rep ss µ (Q, α)/ /GL(α) (respectively Rep s µ (Q, α)/ /GL(α)) Now let A be a finite dimensional algebra, Q = Q(A) be its quiver, ρ be a set of relations such that A ∼ = kQ/ ρ , and µ : ZQ 0 → Q be a slope. The fact that the set Rep(A, α) is embedded in Rep(Q, α) as a Zariski closed GL(α)-invariant subvariety allows us to define µ-semistable and µ-stable A-modules and, consequently, the subsets . Elements of Rep(A, α, ζ) will be called framed representations of the algebra A.
The set consisting of such pairs will be denoted by Rep s (A, α, ζ).
Let ζ be a dimension vector. We introduce a new quiver Q with Q 0 = Q 0 ∪{∞}, the arrow of Q being those of Q together with ζ i arrows from i (i ∈ Q 0 ) to ∞. We also extend the dimension vector α to α, setting α i = α i for i = 1, . . . , n and α ∞ = 1.
Observe that the elements of ρ are relations in Q; consider the ideal I = ρ in k Q. Then (k Q) 2 1 ⊇ I ⊇ (k Q) t+1 1 for t 2 such that ρ kQ ⊇ (kQ) t 1 (here ρ kQ stands for the ideal of kQ generated by ρ). The last statement is not that obvious; but recall that all the new arrows terminate in ∞, which means that no path starts in this vertex. Therefore, if p is a path of length not less than t + 1 in Q, then either it is entirely contained in Q and so p ∈ I, since ρ ⊇ (kQ) t 1 = (kQ) t , or p = bq, where q is a path entirely contained in Q and b is an arrow ending in ∞. But in the latter case the length of q is not less than t yielding that q ∈ I. Thus, the algebra A = k Q/I is finite dimensional.
Further, for β ∈ Z Q 0 set θ(β) = −β ∞ , κ(β) = i β i and consider the corresponding slope µ = θ κ . We are now going to use Corollary 1 to get an interpretation of our notion of stability in the spirit of King's construction. Proof. The isomorphism at the level of quivers, i. e. for A = kQ, is proved in [11,Prop. 3.3]. We just point out that crucial here is the existence of the In order to pass to the general case we need to show that the im- But this follows from the fact that the relations ρ only affect the summand a:i→j Hom(k αi , k αj ), which is common for both sets Rep(Q, α, ζ) and Rep( Q, α), while the restriction of (1) to this summand is the identity map.
Proof. We will prove these properties for M s µ ( A, α). First of all, note that orbits of points from Rep s µ ( A, α) are closed in Rep ss µ ( A, α) = Rep s µ ( A, α) (see the geometric definition of stability in [7]) and so the quotient is geometric. Furthermore, standard results of algebraic geometry imply that From now on for σ ∈ kQ we will denote by σ its image σ + ρ in kQ/ ρ ∼ = A. It is easy to see that in kQ/ ρ there is a (finite) k-basis Ξ consisting of images of paths in kQ. Its elements will be referred to as paths in the algebra A. Denote by I i the injective A-module associated with the i-th vertex of the quiver. Recall that the corresponding representation from Rep(Q, ρ) may be described as follows: where "τ : j i" means that τ is the image of a path τ starting in the j-th vertex and ending in the i-th one; in this case M → J by the following rule: (2) Here we view τ as an element of A; i. e. τ (m) = τ · m.
Proof. First of all, we recall how A acts on J. Write J as For a path σ ∈ Ξ denote by τ * the linear function defined by σ * (σ ) = δ σσ , for each σ ∈ Ξ, where δ σσ stands for the Kronecker delta. As a k-linear space I i has basis τ * i1 , . . . , τ * ir(i) with τ ij being all the paths ending at i. To the basis elements of the p-th copy of I i attach the index (p); thus, For the image of an arrow a of Q we have a · τ a * (p) = τ * (p) ; and a · λ * (p) = 0 in case if λ = e i or λ = λ b, where b is an arrow different from a. Now pass to the isomorphism (2). If a : i → k, then a acts on the summand of the right hand side of (2) corresponding to a path τ : i j (note that the element τ * is in (I j ) i ) as follows In the first case λ : k → j, so that the image lies in J k . We now check the Now use the above alternative: if τ = λa for a path λ : k → j, then the corresponding component ϕ l (m) is mapped to the summand V (λ) j ⊆ J k without being changed, and otherwise vanishes. On the other hand, which coincides with the above description of a · ϕ l (m). Proof. Denote by IHom α (V ) the set of all injective graded vector space homomorphisms from a space with dimension vector α to V . It is easy to see that Gr α (J) is a quotient of i∈Q0 IHom k (M i , J i ) by the natural action of GL(α) (GL(α i ) acts on M i by base change). Denote the inverse image of where the last map is a projection on a summand associated to τ = e i . As for the module M , the following lemma gives the possibility to recover it.
Proof. To recover M ∈ Rep s (A, α, K) means to define the action of A on the vector space i∈Q0 M i , i.e. of the elements A a : i → j for a ∈ Q 1 . Furthermore, for each of them we have the following commutative diagram: may be considered as a matrix equation. Analogously, f j may be regarded as a matrix of dimension dim K j × dim M j . Its rank is maximal and equals dim M j , since f j are injections. Therefore, IHom A α (K) may be covered by open subsets, where various minors of the matrix of f j do not vanish, and M a are recovered from matrix elements of f i , f j and f a using Kramer's Theorem.
Together with the above described way of recovering f the morphism s : IHom A α (J) → Rep s (A, α) gives a morphism that is inverse to Φ. Consequently, Φ is an isomorphism and, being GL(α)-invariant, it descends to . Theorem 3 is proved. Theorem 3 describes a variety that may serve as a substitude of a moduli space of A-modules with dimension vector α whenever Rep s (A, α, ζ) is nonempty. So, it is important to have a criterion of existence of a stable pair. For quivers M. Reineke proved that Rep s (A, α, ζ) = ∅ if and only if ζ i (i, α i ) Q , for all i ∈ Q 0 , where i stands for the vector with all coordinates zero except 1 on the i-th place, and (·, ·) Q is the Euler form, i.e. (i, j) Q = δ ij − (number of arrows from i to j) , see [11,Prop. 4.3]. For arbitrary finite dimensional algebras we do not have such a result. However, we can state a weaker proposition. Recall that a socle of an A-module M is the sum of all its simple submodules.
Taking the dimensions of (soc J) i and (soc M ) i his condition may be reformulated as: For quivers this obviously coincides with [11,Lemma 4.1]. When using this criterion it is also convenient to have in mind that Now, we need a way of determining, for a point of the classical Grassmannian Gr α (J) = i∈Q0 Gr αi (J i ), whether it lies in the Grassmanian of submodules. To formulate the next proposition, we should recall that the summands V s of J i are indexed by paths in A. Furthermore, since an arrow a : i → s induces, for each ⊆ J s such that σa = 0. This injection acts as a simple index change, and moreover, a † a = id J (a) s . Note that for hereditary algebras, i.e. for quivers with no relations, J coincides with J, so J s is embedded in J i whenever there is an arrow i → s.
Proof. As it was shown above, It is straightforward to check that these conditions are equivalent to (3).
Denote by kQ (1) an ideal in the path algebra generated by all oriented cycles in Q. We also use the notation kQ (t) := (kQ (1) ) t .
Let G be a reductive group acting on a vector space X. Recall, that the null cone of this action is the set N = {x ∈ X | f (x) = 0, for all nonconstant homogeneous G-invariant functions on X}. By Hilbert-Mumford's Criterion [8,Theorem III.2.4] this is equivalent, for x ∈ X, to the existence of a one-parameter subgroup λ : k\{0} → G with lim t→0 λ(t)x = 0.
One of the possible applications of the above construction is the study of the null cone of Rep(Q, α, ζ) in case when Q is a quiver with oriented cycles. This is made possible by the following proposition. where M are such representations on which all the oriented cycles in Q act as nilpotent operators. Since A k = 0 for a nilpotent operator A in a k-dimensional space, all the oriented cycles as operators in M vanish in the (max i α i )-th power. But we state that the stronger property holds: that the product of any max i α i oriented cycles is zero. If σ 1 and σ 2 are two such cycles, then by our conditions σ 1 σ 2 and σ 2 σ 1 are also nilpotent, as well as any their product. Moreover, all of them vanish in the same power. This implies that the commutator [σ 1 , σ 2 ] is also nilpotent. Indeed, any its power equals a sum of products of σ i . Since these products are nilpotent, their traces are all zero, providing that the trace of any power of the operator [σ 1 , σ 2 ] is zero. Therefore, the commutator [σ 1 , σ 2 ] is nilpotent. Now, using Engel's Theorem, we can conclude that there is a basis in the representation space, in which the matrices of both σ 1 and σ 2 are uppertriagonal with zeroes on the diagonal. Applying induction, we get a basis, in which the matrices of all oriented cycles starting in a given vertex are upperniltriagonal. As a product of oriented cycles starting in different vertices is zero by definition, the product of any max i α i such operators is zero.
. Note that in this example we do not need to write the index (e i ) over V i , since there is only one summand V i in J i . Example 2. Consider the algebra A = k[x]/(x) n . It is also a Nakayama algebra with admissible sequence (n), but now its quiver contains a loop. In fact it is a Jordan quiver with a single vertex and a single loop a, and the only relation is a n . Let α = (m), ζ = (k), V be a k-dimensional vector space. Then J = J 1 = V (e) ⊕ V (a) ⊕ . . . ⊕ V (a n−1 ) . Observe that a acts on J as follows: .
It is easy to check that J can be viewed as and the moduli space is isomorphic to the Grassmannian of m-dimensional d dt -invariant subspaces in J. This interpretation is in fact rather fruitful and will be further explored in next sections.
Example 3. Let A be the algebra with quiver 2 a2 a a a a a a a a Q : 1

Framed moduli spaces for quiver
In three remaining sections we work over the ground field k = C.
In this section our aim is to describe fibers of the projection π s for a cyclic quiver Q of type A (4) Here we identify τ 0 i with e i . Furthermore, we use the symbol of direct product instead of direct sum, since we allow our tuples to contain infinitely many nonzero terms. It should be explained how a i ∈ kQ acts on J. In order to do this we use the alternative formulated in the proof of Lemma 2. Considering the restrictions of a i to the summands of (4) we have takes place and, in particular, a i : , hence T is really an isomorphism of kQ-modules. We may further Inspired by this example, we'll try to generalize this interpretation for arbitrary n and ζ. Namely, consider the space this map is componentwise and its components Lemma 5. The map T is a kQ-isomorphism.
Proof. We need to show that for every i, k there is a commutative diagram of restrictions: Observe that for k = i the restriction of a i to the subspaces considered is the identity operator in both rows, hence the diagram is commutative.
Now it will be shown that some of the useful properties of its finite dimensional analog hold for the above constructed map. In the finite dimensional case to each element (M, f ) ∈ Rep s (Q, α, ζ) associated is a submodule of J ! with dimension vector α, that is a point in the Grassmannian of submodules Gr kQ α (J ! ). However, to obtain a one-to-one correspondence, we need the following simplification.

Denote by A[t] the subspace in k[[t]]
consisting of power series converging for all t. Since k = C, this conditions means that the series is a Taylor series of a holomorphic function, and hence the uniqueness theorem implies that A[t] is the ring of entire functions O(C). As ϕ ! i are expressed in terms of exponents, they all It is not hard to see that J !! is a kQ-submodule in J ! , and so our task is now to describe all kQ-submodules in J !! with dimension vector α, i.e. the set Gr kQ α (J !! ) and to prove that such submodules are in one-to-one correspondence with stable pairs (M, f ) ∈ Rep s (Q, α, ζ).
Let U ⊆ J !! be a kQ-submodule with dimension vector α. Then for each i = 1, . . . , n we have an inclusion a i (U i ) ⊆ U i+1 . This in particular implies that Hence if U is a submodule, then it is preserved by the componentwise differentiation of elements G ∈ U i . This means that and so its elements may be considered as n j=1 ζ j -tuples of functions). Further, let D (i) = (d (i) pq ) be the matrix of the restricted operator d dt | Ui and set k = α i , m = n j=1 ζ j . Fix a base g 1 , . . . , g k in U i (following the idea expressed in a recent remark we here consider g j as a m-tuple of functions: g j = (g jl ) n l=1 ). Then, for all j, we get: It is an easy calculation to check that (g 1 (t), . . . , g k (t)) = (g 1 (0), . . . , g k (0)) exp(D (i) t).
Now Cayley-Hamilton's Theorem implies that χ i (D (i) ) = 0 in U i , where χ i is a characteristic polynomial of D (i) . Thus each vector function g ∈ U i satisfies the differential equation χ i ( d dt )g = 0. We can finally show that the above constructed correspondence between stable pairs and points in Grassmannian is one-to-one.
Proof. We need to show that having an inclusion Φ ! ∈ IHom kQ α (J !! ) we can uniquely recover a pair (M, f ). First of all note that since, for each j,r, we have τ r j a j−1 . . . a i = a j−1 . . . a i τ r i , the map ϕ ! i may be written as: . . , Let now U = (U i ) n i=1 ⊆ J !! be a submodule with dimension vector α. Let also U i = span g  Recall the equality (5), setting g r (t) = g (i) The matrices g  αi (0) may be divided into horizontal blocks of size ζ j × α i , those blocks corresponding to the natural projections Thus, for each j, we obtain where E αi is the identity matrix of size α i × α i . These equalities can be interpreted as follows: there is a map Ψ = (Ψ ij ) n i,j=1 with Ψ ij = P j g (i) where M is a graded vector space with dimension vector α; one may easily establish that Ψ is bijective. We need to show that Ψ = Φ ! (M,f ) for a map f : M → V and a certain kQ-module structure on M .
As for the module structure, it is quite clear: viewing (6) as formulas defining the natural inclusion . Hence, αi (0) m. Now it is left to prove that (7) defines a map Ψ coinciding with Φ (M,f ) for the above M and f . From (6) it follows, that it is sufficient to show, that for j = i. But D (i) is just a notation for the matrix of τ i , so exp(D (i) t) ≡ exp(τ i t). Canceling this exponent, we turn (8) into 1 (0), . . . , P j g (j) αj (0) · a j−1 . . . a i . We are going to prove this as a matrix equality, and so we can immediately write where a j−1 . . . a i g 1 (t), . . . , P j g (j) αj (t) · a j−1 . . . a i , but a r acts on J !! i as (id, . . . , d dt , . . . , id), where the differentiation takes place only at the r-th position. Consequently, the product a j−1 . . . a i does not change the j-th projection of g (i) , implying that P j a j−1 . . . a i g   This, however, does not give us a desired isomorphism. The reason is that instead of describing the quotient we rather have proved that Gr kQ α (J !! ) is an algebraic variety. Now consider the standard categorical quotient π GL(α) : Rep(Q, α, ζ) → M(Q, α) = Rep(Q, α, ζ)/ /GL(α). As it was pointed out before, this quotient parameterizes all possible characteristic polynomials of the cycles τ 1 , . . . , τ n in Q; following this observation, we will consider M(Q, α, ζ) as embedded in the product r is a space of polynomials of degree nor higher than r + 1. It is clear that not each tuple of polynomials can be obtained from a representation. Although we are not going to give an explicit description of M(Q, α, ζ) as a subvariety in k[x] α1 × . . . × k[x] αn , we prove the following useful lemma.
Proof. As was shown before, each vector function in (Φ (M,f ) (M )) i satisfies the equation N , r N ) i . This yields the required inclusion.
Let now W = N j=1 J(λ j , r j ) and Φ j (M,f ) be Φ ! (M,f ) followed by the projection on J(λ j , r j ) along p =j J(λ p , r p ). Proof. The "only if" part is straightforward.
Therefore the multiplicity of λ j as an eigenvalue of τ i as an operator on M i equals r ij . This implies that the characteristic polynomial of τ i acting on This lemma ensures that Φ ! : IHom kQ r j (J(λ j , r j )). As J(λ j , r j ) are finite dimensional, we can use Lemma 4 to prove that π −1 s (χ) is isomorphic (this time as an algebraic variety) to N j=1 Gr kQ r j (J(λ j , r j )). Collecting the results obtained we can state the following: (1) Points of the quotient M s (Q, α, ζ) are in one-toone correspondence with points of the Grassmannian of submodules Gr kQ α (J !! ). (2) Let χ = (χ 1 , . . . , χ n ) be an admissible tuple of polynomials, λ 1 , . . . , λ N be all different roots of χ 1 , . . . , χ n , and r ij be the multiplicity of λ j as a root of χ i . Then π −1 s (χ) ∼ = N j=1 Gr kQ r j (J(λ j , r j )), where r j = (r 1j , . . . , r nj ).

An explicit realization of fibers
Let Q be a Jordan quiver consisting of a single vertex and a single loop (both this loop and the corresponding operator in a representation will be denoted by a). Set also α = (m), ζ = (q). This is the case when our construction becomes as clear as possible.
It is evident that the standard categorical quotient for the action GL(m) : Rep(Q, m) is isomorphic to A m : points of the quotient are tuples of characteristic polynomial coefficients of the operator corresponding to the arrow a; having this in mind we will further assume that We have J !! = J !! 1 = A ⊗ V 1 , and the map Φ ! (M,f ) becomes ϕ(m) = f 1 (exp(at)m). Further, for a subspace U ∈ Gr m (J !! ) the equivalence U ∈ Im(Φ ! ) ⇔ d dt (U ) ⊆ U holds. Thus, the fiber over χ ∈ k[x] is precisely Gr  , 1) there is at most one point. Such a result is rather upsetting, though the situation will be more favorable for q > 1. We may even guarantee that for q = m each kQ-module M arises as a member of a stable pair (M, f ) ∈ Rep s (Q, m, q) (it is, for instance, (M, id)).
The next theorem describes the fibre structure of π s for arbitrary q and m.
Theorem 5. Let m, q be positive integers. , where λ ∈ k. But each onedimensional d dt -invariant subspace in J(λ, 1) is generated by a vector function g satisfying d dt g = λg that is by (α 1 e λt , . . . , α q e λt ), where α i ∈ k. Hence, Gr d dt 1 (J(λ, 1)) is a projectivization of the linear span of such functions, i. e. it is isomorphic to P q−1 .
(c) Before starting to prove this, we must confess that the relations of group (ii) are in fact unnecessary, for they follow from (i) and (iii). Although, in practice they may help to simplify much of the group (iii) relations and to shorten their list, so we couldn't help mentioning them. Because of this, we first prove that the equations (ii) are satisfied in our variety, and then we show that it is in fact given by (i) and (iii).
ij e ij | k = 1, . . . , m , where base elements will be chosen in the following way. Decompose U into a direct sum of subspaces U t satisfying the property that each characteristic polynomial χ t (x) = x n + c n−1 x n−1 + . . . + c 1 x + c 0 of the restriction of d dt to U t is minimal. But in this case in a certain base the matrix of d dt | Ut will be written as which means that the first basic vector is a derivative of the second, the second one is the derivative of the third and so on until the last one which satisfies the differential equation χ t ( d dt )g = 0. Collecting the bases of all U t , we obtain a convenient spanning set.
In P( m J(λ, m)) to the subspace U associated is a line spanned by ij e ij ). It is easy to see that coefficients in the decomposition ω U = l1m+ν1<...<lmm+νm p l1m+ν1,...,lmm+νm e ν1l1 ∧ . . . ∧ e νmlm satisfy the relations (ii). Let im (λe im +e i,m−1 ))) = λω U + U . The last term denoted by U is to be investigated. Its summands are obtained when e i,j−1 are taken instead of e ij in the above wedge product. Therefore, the coefficient of e ν1l1 ∧ . . . ∧ e νmlm in this term equals But let l * 1 , . . . , l * m be such a tuple that p mr1+η1,...,mrm+ηm = 0 or not defined for all vectors (mr 1 + η 1 , . . . , mr m + η m ) with r i > l * i ∀i, and, moreover, p mr1+µ1,...,mrm+µm = 0 for some µ 1 , . . . , µ m . Then the coefficient e µ1r1 ∧ e µmrm of d dt · ω U equals λ m p mr1+µ1,...,mrm+µm , and thus if ω U is a relative and only if U is zero. But we have already shown that, rewritten in terms of Plucker coordinates of ω λ , this condition becomes (iii). This completes the proof of the theorem.
Example 5. For m = q = 2 these relations are very simple. It is a straightforward computation to check that in this case, for µ = λ which is a non-degenerate quadric. When the eigenvalues coincide, the fiber is i.e. a degenerate quadric. Now let Q be an arbitrary quiver of type A (1) n (we use the notation from the previous section for its vertices and arrows ). As it was shown before, the layer of M s (Q, α, ζ) over a point χ = (χ 1 , . . . , χ n ) of the standard categorical quotient is isomorphic to N i=1 Gr kQ r j (J(λ j , r j )), where the notation is as in Theorem 4. On the other hand, ⊗ V i and trivially on all other components.
Convenient is to fix the following basis in all J(λ, r j ) i : e (i,λ) qrs = (0, . . . , t s e λt , . . . , 0) ∈ A[t] ⊗ V q , where the only nonzero component is the r-th one. Furthermore, in Gr rij (J(λ j , r j ) i ) we will be considering Plucker's coordinates corresponding to this basis; they will be denoted by p (j) k1,...,kα j , where k h are in fact triples of indices (q h , r h , s h ). Recall that by Theorem 4 it is sufficient to describe the quotient for the case, when all λ i coincide.
Proof. At first, all M i are d dt -invariant, so (i) -(iii) of Theorem 5 hold. In addition, there is a condition a i (M i ) ⊆ M i+1 . If α i = α i+1 , this is equivalent to the corresponding skew symmetric tensors being proportional, i. e. to (b).
Theorem 6 is proved.

Quivers with successive cycles
A quiver Q will be called a quiver with successive cycles, if whenever two oriented cycles in Q have a common vertex, they are both powers of a certain cycle. It is easy to see that all such quivers may be constructed through the following procedure (that justifies our choice of terminology). Take a quiver without oriented cycles and replace some of its vertices by oriented cycles so that the arrows that used to start from the replaced vertex may now start from any chosen vertex of the pasted cycle. Here is an example of a quiver obtained through such a transformation: Using our description of the quotient for the case of one oriented cycle, we can generalize the technique we possess to such quivers. As before, we denote by π s the natural projection M s (Q, α, ζ) → M(Q, α, ζ).
Theorem 7. Let Q be a quiver with successive cycles. Let also α and ζ be two dimension vectors and y be a point in M(Q, α, ζ). There exists a quiver Q ♠ , a dimension vector α ∈ (Z 0 ) Q ♠ 0 , and a finite dimensional representation W ♠ of Q ♠ such that π −1 s (y) ∼ = Gr kQ ♠ α (W ♠ ). Proof. We begin the proof by giving a construction of a module J !! such that points in Gr kQ α (N ) are in one-to-one correspondence with points in the quotient space M s (Q, α, ζ).
Let Q be the quiver without oriented cycles, from which Q may be obtained through the above procedure, and Ξ i be the set of all paths in Q starting at i. For the vertices of the oriented cycle that we place instead of the i-th vertex of Q we set the following notation: v v l l l l l l l l l l l l l l l Let also τ i,j be the cycle of minimal nonzero length starting at i (j) , if there is any, or e i = e i (0) , otherwise. As before, we consider the kQ-module J with J i = σ:i j V σ j . Every path σ in Q is of the form σ = B l τ kl il,jl a l . . . B 2 τ k2 i2,j2 a 2 B 1 τ k1 i1,j1 a 1 B 0 τ k0 i0,j0 , where l = l(σ) is the length of σ, a j are arrrows of Q, and B t are segments of τ it,jt , that are not oriented cycles. Let a 0 = i and u t = ha t . We then have that, for r = 0, . . . , n ul , k, otherwise. Thus we may set Combining the proofs of Lemma 2 and Lemma 6 we see that thus defined ϕ ! (M,f ) enjoys its usual properties, i.e. that the statement of Lemma 6 holds in this situation.
Proof. Let U ⊆ J !! be a subrepresentation with dimension vector α. We need a pair (M, f ) such that Im(Φ ! (M,f ) ) = U . Take M = U . The maps f i (j) are then reconstructed as compositions of the projections The proof is by "downward induction over Q". If i ∈ Q is a sink, we apply Proposition 5. For an arbitrary vertex i, fix a basis G 1 , . . . , G dj in each U i (j) and observe that, as in (5), where G p (x ei , 0) are in fact functions in x ρ,t , for ρ = e i . We need to prove that the right hand side of this equality coincides with (9). Recall that a:i→p aB 0,a , where B 0,a is the shortest segment of τ i,j linking i j with ta, acts as evaluation at x ei,0 = 0. So, G 1 (0), . . . G dj (0) consists of horizontal blocks representing the bases of aB 0,a (U i j ), where a are arrows starting at vertices of τ i,j . By the induction hypothesis all these blocks are of the form (9). Thus the claim follows. This, however, rather characterizes Gr kQ α (J !! ), than the quotient space. So, as we did in Section 4, we restrict our attention to the fibers of π s .
Recall that the algebra k[Rep(Q, α, ζ)] is generated by coefficients of characteristic polynomials of all oriented cycles in Q. So we can consider y as a tuple χ i,j | i ∈ Q 0 , j = 0, . . . , n i , where χ i,j := the characteristic polynomial of τ i,j , if l(τ (i, j) > 0, 0, otherwise.
We are now ready to prove the theorem. Introduce the module W with Observe that W may be considered as a representation W ♠ of Q ♠ with . The dimension vector of W ♠ is α with α i (j,l) = r (i,j) l . It is now straightforward to check that IH(W )/ /GL(α) is isomorphic to the Grassmannian Gr kQ ♠ α (W ♠ ). This finishes the proof.
Remark. For Q = A n−1 the quiver Q ♠ is non-connected. In fact, it is a union of N copies of Q, where N is as in Theorem 4. So, a kQ ♠ -module is a N -tuple of representations of Q. In particular, W ♠ splits as j J(λ j , r j ). So, Gr kQ ♠ α (W ♠ ) ∼ = j Gr kQ r j (J(λ j , r j )). To illustrate both the theorem and the proof, we give an example.
Our purpose is now to show that fibers of M s (Q, α, ζ) over points of categorical quotient are realized as a kQ ♠ -module Grassmannians Gr kQ ♠ α (W ♠ ) for some quiver Q ♠ , dimension vector α, and finite dimensional module W ♠ . The categorical quotient Rep(Q, α)/ /GL(α) is isomorphic to A α2 × A α4 , and its points may be viewed as pairs χ = (χ b , χ c ) of characteristic polynomials of b and c. Having fixed such a pair, we construct the required modules W by "downward induction".