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On degenerations and extensions of symplectic and orthogonal quiver representations

We discuss degenerations of symplectic and orthogonal representations of symmetric quivers and algebras with self-dualities. As in the non-symmetric case, we define a partial ordering, that we call symmetric Ext-order which gives a sufficient criterion for a symmetric degeneration. Then a detailed discussion of type A quivers and their (symmetric) representation theory via Auslander-Reiten theory leads to our main theorem which states that the symmetric degeneration order of a symmetric quiver of finite type is induced by the usual degeneration order between representations of the underlying quiver.


Introduction
Let X be a complex algebraic variety acted upon by a group G. Suppose that the pair (G, X) satisfies the following: (1) both G and X are equipped with involutions ρ : G → G, g → g ρ and ∆ : X → X, x → ∆x such that ∆(g · ∆x) = g ρ · x. Denote by G ρ ⊂ G and X ∆ ⊂ X the sets of fixed points; (2) the group G is a subgroup of the group of invertible elements E × of a finite-dimensional associative algebra E over C; (3) the anti-involution of G given by g → g * := (g ρ ) −1 extends to a C-linear anti-involution f → f * on the algebra E; (4) for every fixed point x ∈ X ∆ , its stabilizer H = Stab G (x) is the group of invertible elements of its linear span Span C (H) ⊂ E. Then Magyar, Weyman and Zelevinsky [20, Section 2.1] (generalizing a result of Richardson [21] and one of Derksen and Weyman [15,Theorem 2.6]) prove that for every fixed point x ∈ X ∆ (1.1) G x ∩ X ∆ = G ρ x.
Equation (1.1) means that the orbit of a fixed point by the "small" group G ρ is the intersection of its orbit by the "big" group G with the set of fixed points. It is then natural to ask if the same happens for the Zariski orbit closures: Question 1.1. Is it true that G x ∩ X ∆ = G ρ x, for every x ∈ X ∆ ? Example 1.1. Let X ⊂ Mat n (C) be the variety of n × n nilpotent complex matrices and G = GL n acting on X by conjugation. Let ∆ and ρ be the involutions on X and G such that G ρ = SP n is the symplectic group (or G ρ = O n is the orthogonal group) and X ∆ ⊂ sp n = Lie(SP n ) (or X ∆ ⊂ o n = Lie(O n )) is the nilpotent cone. It is straightforward to check that the hypotheses (1)-(4) are satisfied for the pair (X, G). By classical work of Freudenthal, Gerstenhaber and Hesselink (compare with [19,Proposition 2.1]) it is known that the answer to Question 1.1 is positive in this case.
A pair (X, G) as above can be found in the context of representation theory of finite dimensional algebras endowed with a self-duality (see Section 2.4) and we give another proof of Magyar, Weyman and Zelevinsky's result in this specific context in Section 2.6. We are aware of an example arising in this context where the answer to Question 1.1 is negative [9]. In this paper we provide a positive answer to Question 1.1 in case of symmetric quiver algebras of finite representation type.
Let us explain the context, the results and our motivations to seek Question 1.1.
Quiver algebras. Let k = C be the field of complex numbers. A quiver Q is a finite oriented graph given by a quadruple Q = (Q 0 , Q 1 , s, t) where Q 0 denotes the finite set of vertices of Q, Q 1 denotes the finite set of edges and s, t : Q 1 → Q 0 are two functions that provide the orientation α : s(α) → t(α) of the edges. For simplicity of notation we consider the elements of Q 1 as oriented edges or arrows. A path in Q is defined to be a sequence of arrows ω = α s · · · α 1 , such that t(α k ) = s(α k+1 ) for all k; formally we include a path ε i of length zero for each i ∈ Q 0 starting and ending in i. The path algebra kQ of Q is defined to be the k-vector space with a basis given by the set of all paths in Q. The multiplication of two paths is defined by concatenation of paths. Let R be the arrow ideal of kQ which is the (two-sided) ideal generated by the arrows of Q. Let I ⊆ kQ be an ideal such that there is an integer s with R s ⊆ I ⊆ R 2 (i.e. it is admissible). The quotient algebra A := kQ/I is finite-dimensional and associative, and we refer to it as a quiver algebra (standard references are [4,6,12,23]).
Symmetric quiver algebras. A symmetric quiver as defined in [15] is a pair (Q, σ) where Q is a finite quiver and σ : Q → Q op is an involutive isomorphism of Q with its opposite Q op , i.e. it is an involution of Q 0 and of Q 1 , reversing the orientation of arrows. Notice that σ is defined on the underlying graph of Q and once this is fixed we say that Q and σ are compatible. For example, let us consider the Dynkin diagram of type A 3 , 1 α 2 β 3 , and let σ be its non-trivial involution given by σ(1) = 3, σ(2) = 2, and σ(α) = β. Let Q(1) = 1 α G G 2 β G G 3 and Q(2) = 1 α G G 2 3 β o o be two quivers of type A 3 . Then Q(1) is compatible with σ, i.e. (Q(1), σ) is a symmetric quiver, but Q(2) is not compatible with σ. Actually, Q(2) cannot be endowed with the structure of a symmetric quiver, since σ is the only non-trivial involution of its underlying graph. Let (Q, σ) be a symmetric quiver. For every admissible ideal I ⊂ kQ such that σ(I) = I, the quiver algebra A = kQ/I is isomorphic (via σ) to its opposite A op = kQ op /σ(I). The pair (A, σ) is called a symmetric quiver algebra.
(Symmetric) representation theory. The representation theory of symmetric quivers or generalized quivers with dimension vectors was developed by Derksen and Weyman in [15] and it has strong connections with the theory of symmetric spaces and θ-groups of Vinberg [26] and the results of Kac [17,18], as explained in [15] and [25]. We recall here a few facts about it. Let (A, σ) be a symmetric quiver algebra. An A-module or an A-representation is a representation of the quiver Q satisfying the relations from I. Thus, a representation is a pair M = (V, f ) where V = ⊕ i∈Q0 V i is a finite dimensional Q 0 -graded vector space and f = (f α : V s(α) → V t(α) ) α∈Q1 is a collection of linear maps such that f π = 0 for every π ∈ I. The vector space V is called the underlying vector space of the representation M and its graded dimension d = dim V = (dim V i ) i∈Q0 is called the dimension vector of M or V . Notice that V is itself an A-module, by choosing all the linear maps to be zero; it is called the semisimple module of dimension vector d. A morphism of A-modules h : (V, f ) → (W, g) is a collection of linear maps (h i : V i → W i ) i∈Q0 such that h t(α) • f α = g α • h s(α) . We denote by Rep(A) the category of finite dimensional representations of A. Let V = ⊕ i∈Q0 V i be a Q 0 -graded complex vector space of dimension vector d. We denote by R(A, V ) the variety of A-modules having V as underlying vector space. Thus Hom k (V i , V j ).
We denote by GL • (V ) := i∈Q0 GL(V i ) the group of graded automorphisms of V . We sometimes think of GL • (V ) as embedded into GL(V ) as diagonal block matrices (see [15,Section 1]). The group GL • (V ) acts on R(A, V ) by change of basis, i.e. given g = (g i ) i∈Q0 ∈ GL • (V ) and M = (M α ) α∈Q1 ∈ R(A, V ) the representation g · M is defined by (g · M ) α = g t(α) • M α • g −1 s(α) . The GL • (V )-orbits are the isomorphism classes of A-representations with underlying vector space V .
Let ε be +1 or −1. We say that a bilinear form −, − : V × V → C on V is a σ-compatible ε-form on V if: (i) the form −, − is non-degenerate; (ii) the form −, − is compatible with σ, i.e. −, − | Vi×Vj = 0 if j = σ(i); (iii) the form −, − is an ε-form: i.e. v, w = ε w, v for every v, w ∈ V . The pair (V, −, − ) is called an ε-quadratic space for (Q, σ) or for (A, σ). Notice that the dimension vector of V is σ-symmetric in this case, i.e. d σ(i) = d i for every i ∈ Q 0 . Indeed, by (ii), the non-degenerate bilinear form −, − induces an isomorphism between V i and the linear dual of V σ(i) .
Every endomorphism f of V has a unique adjoint f with respect to the nondegenerate bilinear form −, − defined by the condition f (v), w = v, f (w) , for all v, w, ∈ V . We denote by G(V, −, − ) = {g ∈ GL(V )|g = (g ) −1 } the group of isometries of (V, −, − ). A point M ∈ R(A, V ) can be considered as an endomorphism of V . Following [15], we say that M ∈ R(A, V ) is an ε-representation of (A, σ) with respect to (V, −, − ) if (iv) M + M = 0.
Condition (iv) means that M lies in the Lie algebra of the group G(V, −, − ) and condition (ii) implies that M α = −M σ(α) , i.e. M α (v), w + v, M σ(α) (w) = 0 for every arrow α : i → j, v ∈ V i and w ∈ V σ(j) . For ε = +1, M is hence called an orthogonal representation of (A, σ) and for ε = −1, it is called a symplectic representation of (A, σ).  . Let R = Mat n×n (C) be the vector space of complex n × n matrices. For ε = −1 let R ε = Sym n denote the subspace of symmetric matrices and for ε = 1 let R ε = ASym n denote the space of anti-symmetric matrices. Let V = C n ⊕ C n . We view R ⊂ End(V ) as 0 0 R 0 so that the elements of R are considered as linear maps from the first copy of C n to the second one. Let Q = 1 G G 2 be a quiver of type A 2 and let σ be the anti-involution on Q.
Then R can be seen as the representation variety R (n,n) (CQ). On V we define the . Then, a short calculation shows that the variety of ε-representations of Q with respect to (V, −, − ) is R ε = {A ∈ R|A t + εA = 0} and thus we recover the two subspaces defined above. The group GL • (V ) = GL n × GL n acts on R by base change ((g 1 , g 2 ), A) → g 2 Ag −1 1 , and the group G • (V, −, − ) = {(g 1 , g 2 )| g 1 = (g t 2 ) −1 } GL n acts on R ε by congruence (g 2 , A) → g 2 Ag t 2 . Two n × n matrices are in the same GL • (V )-orbit if and only if they have the same rank. The theorem of Magyar, Weyman and Zelevinky applies here (see Section 2.4) and states the well-known fact that two symmetric (or anti-symmetric) matrices are congruent if and only if they have the same rank.
Let U be a finite dimensional complex vector space and let U * denote its linear dual. Let ε be +1 or −1 and let (W, −, − ) be an ε-quadratic vector space. The isometry group G(W, −, − ) equals SP(W ) if ε = −1 and O(W ) if ε = 1. Let R = Hom(U, W ) ⊕ Hom(W, U * ). Then R is a representation variety for the quiver Q = 1 α G G 2 β G G 3 of type A 3 and we denote its elements as (f α , f β ). Given f α ∈ Hom(U, W ) we denote by f α ∈ Hom(W, U * ) the linear map defined by f α (w)(u) = f α (u), w for all w ∈ W and u ∈ U . Let R ε = {(f α , f α )} ⊂ R be the subspace of fixed points for the involution (f α , f β ) → (f β , f α ). It is easy to see that the fixed point set R ε can be realized as a variety of ε-representations for the symmetric quiver (Q, σ).
Let us consider the group G = GL(U ) × GL(W ) × GL(U * ). As usual, for an endomorphism h of U we denote by h * the endomorphism of U * given by h * (g)(u) = g(h(u)) for all g ∈ U * and u ∈ U . Similarly, for an endomorphism h of W we denote by h its adjoint with respect to −, − which is the unique endomorphism of W such that h(w 1 ), w 2 = w 1 , h (w 2 ) for every w 1 , w 2 ∈ W . Let us consider the involution on G given by (g 1 , g 2 , g 3 ) → ((g * 3 ) −1 , (g 2 ) −1 , (g * 1 ) −1 ) and let G ε be the subgroup of fixed points. We notice that G ε GL(U ) × G(W, −, − ).
The group G acts on R by change of basis, i.e. given g = (g 1 , g 2 , ). It is well-known that the only invariants for this action are rk The action of G on R restricts to an action of G ε on R ε . We notice that for a point f = (f α , f α ) in R ε , the rank of f α is equal to the rank of f α and the rank of f α • f α is the rank of the form on W restricted to the image of f α , and thus these two ranks are preserved by the action of G ε on R ε .
The theorem of Magyar, Weyman and Zelevinsky applies to this example (see Section 2.4) and states that two points f = (f α , f α ) and f = (f α , (f α ) ) in R ε are in the same G ε -orbit if and only if rk(f α ) = rk(f α ) and the rank of the form restricted to the image of f α is equal to the rank of the form restricted to the image of f α .
and say that M degenerates symmetrically to N or that N is a symmetric degeneration of M . In Section 2.4 we show that the theorem of Magyar, Weyman and Zelevinsky applies to this situation and a positive answer to Question 1.1 means that ≤ deg and ≤ ε deg coincide on R(A, V ) −,− ,ε . In Section 3 we define a third order that we call symmetric Ext-order, denoted by ≤ ε Ext (see Definition 3.2). We obtain Corollary 3.3, which states that ≤ ε Algebras of finite representation type. Question 1.1 is particularly interesting when the orbit closures can be described by a finite number of invariants. In the context of symmetric representation varieties, this requirement is fullfilled by an algebra A which is of finite representation type. Indeed, in this case, Zwara [27] proved that the degeneration order ≤ deg is equivalent to the so-called Hom-order (see Section 4.1). This implies that the orbit closures in X are described by checking a finite number of inequalities which is independent on X and G. Even in this special case, the answer to Question 1.1 is negative [9]. It is then an open problem to find conditions on (X, G) or on the algebra A such that the answer to Question 1.1 is positive. In this paper we investigate this problem in the case of a symmetric hereditary quiver algebras of finite representation type. These algebras are the path algebras of symmetric quivers of finite type. A symmetric quiver is called of finite type if it admits only a finite number of indecomposable ε-representations (see Section 2.8 for a description of indecomposable ε-representations). In [15,Proposition 3.3] it is shown that a (connected) symmetric quiver is of finite type if and only if it is a symmetric orientation of a Dynkin diagram of type A. We hence restrict ourselves to this situation. For n ≥ 2, we denote by A n the Dynkin diagram of type A n : with the following labelling of vertices and edges: Let σ be the non-trivial automorphism of A n (i.e. σ(i) = n + 1 − i and σ(α i ) = α n−i ). In case n is odd there is a unique σ-fixed vertex and no σ-fixed arrow; if n is even there is a unique σ-fixed edge and no σ-fixed vertex. We distinguish the two cases by writing A odd and A even . Let (Q, σ) be a symmetric quiver of type A n . A typical example is the equioriented quiver that we denote by Ò Ò 1 3 n − 1 n + 1 The following is the main result of the paper: Theorem 7.1. Let (Q, σ) be a symmetric quiver of type A. Let ε be +1 or −1.
Then on the ε-representations we have Ext is the most surprising and it is proved in Section 7 (see Theorem 7.1). The proof is constructive in the sense that given two ε-representations M, N ∈ R(Q, V ) −,− ,ε such that M ≤ deg N it shows how to inductively find a sequence of ε-representations M = M (0), M (1), · · · , M (k) = N such that there is a one-parameter subgroup λ i (t) ∈ G • (V, −, − ) which fullfills lim t→0 λ i (t) · M (i) = M (i + 1) for every i (see Section  Our goal is to make the paper self-contained and spare the reader a long search through the literature. In Section 5 we recall with proofs many special features of the representation theory and Auslander-Reiten theory of quivers of type A, used in the proof of Theorem 7.1. We address the reader to the references only for those aspects which hold in greater generality (e.g. for Dynkin quivers, or for gentle algebras) and whose proofs are not simpler when restricted to our context. Subsection 5.3 contains an explicit description of generic quotients by indecomposables. Section 6 is dedicated to highlight essential facts about ε-representations of quivers of type A, including the key concepts of generic isotropic subrepresentations and generic ε-subquotients.
We conjecture that Theorem 7.1 holds for every symmetric algebra which is representation-directed. It is an open problem to investigate if Theorem 7.1 holds true for every symmetric quiver algebra of finite representation type whose indecomposable modules are rigid (compare with [27,Theorem 2] Motivations. One of our motivations to seek Theorem 7.1 comes from the study of linear degenerations of the symplectic (complete) flag variety, in analogy with [13,14]. Indeed, let A = kQ be the path algebra of the equioriented quiver of type A 2n−1 and let (V, −, − ) be a (−1)-quadratic space for (Q, σ) of dimension vector d = (2n, 2n, · · · , 2n). Then one can construct a GL • (V ) −,− ,(−1) -equivariant family X → R(A, V ) −,− ,(−1) whose fiber over a symplectic representation consists of its isotropic (actually Lagrangian) subrepresentations of dimension vector e = (1, 2, · · · , 2n − 1). One sees immediately that the fiber over the generic point is the complete symplectic flag variety. It is then an interesting problem to understand the geometric properties of the special fibers, which can be considered as "linear" degenerations of the symplectic flag variety. The first step to study this family, is to understand the orbit closures in the base. This is now clear thanks to Theorem 7.1.
The general answer of Question 1.1 is furthermore interesting from an algebraic Lie-theoretic point of view: It helps to answer questions about orbit closures in the nilpotent cone. In more detail, the above described classical example of reductive actions on the nilpotent cone can be generalized in many ways (e.g. restrict to Borel-actions on certain subvarieties of nilpotent matrices as in [10]). This setup can be translated via an associated fibre bundle to the (symmetric) representation theory of a particular quiver with relations. and on morphisms Here and always throughout the paper, given a vector space V , we denote by V * = Hom(V, k) its linear dual and given a linear map f : U → V between two vector spaces, its dual is the linear map f * : V * → U * defined by f * (h)(u) = h(f (u)) for every h ∈ V * and u ∈ U .
2.2. The functor ∇. It is convenient to introduce the following twist of the selfduality * . For a Q 0 -graded vector space V = ⊕ i∈Q0 V i , we define its "twisted dual" ∇V as the Q 0 -graded vector space whose i-th component is (∇V ) i = V * σ(i) . We define a functor ∇ : Rep(A) → Rep(A) as follows: given a representation M ∈ R(A, V ) its twisted dual is a representation ∇M ∈ R(A, ∇V ) defined by ∇(M ) α = −M * σ(α) for every arrow α (notice the minus sign); given a morphism h : M → N its twisted dual is defined by (∇h) i = h * σ(i) , for every vertex i ∈ Q 0 .
The functor ∇ is defined on objects by . G G and on morphisms as the self-duality * (see Example 2.1).
2.3. The isomorphism Ψ. Let (V, −, − ) be an ε-quadratic space for the symmetric quiver (Q, σ). The form −, − induces the isomorphism Ψ : V → ∇V given by Ψ(v) = v, − for every v ∈ V . Moreover, this isomorphim satisfies the relation: Indeed, by canonically identifying ∇∇V V via the evaluation map, we get . Viceversa, if Ψ : V → ∇V is an isomorphism of Q 0 -graded vector spaces satisfying (2.1), then the bilinear form on V given by v 1 , v 2 := Ψ(v 1 )(v 2 ) is a σ-compatible ε-form on V . Thus the datum of a σ-compatible ε-form on V is equivalent to the datum of an isomorphism Ψ : V → ∇V satisfying (2.1). With abuse of terminology we sometimes say that Ψ is an ε-form on V .
With this ingredient, the definition of an ε-representation is reformulated as follows: a representation M with underlying vector space V is an ε-representation with respect to −, − if and only if Ψ is an A-isomorphism from M to ∇M . Thus we sometimes use the following notation for the variety R(A, V ) −,− ,ε : In terms of Ψ, an element g ∈ GL • (V ) belongs to the group of graded isometries G • (V, −, − ) if and only if ∇g • Ψ • g = Ψ and we use the following notation: Two ε-representations M, N ∈ R(A, V ) Ψ,ε are isomorphic as ε-representations if there exists an isomorphism θ : M → N of A-representations such that ∇θ • Ψ • θ = Ψ. We sometimes write a point of R(A, V ) Ψ,ε as (M, Ψ) to highlight the dependency on Ψ and we say that M is an ε-representation with respect to the ε-form Ψ.
3. An ε-representation M is isomorphic to its " twisted dual" ∇M via the isomorphism Ψ. We sometimes omit to mention this isomorphism and freely identify M and ∇M . Thus, for example, if f : L → M is an homomorphism then ∇f : ∇M → ∇L and we sometimes write ∇f • f : L → ∇L instead of ∇f • Ψ • f .

2.4.
Embedding into the general context. In this section we show that the theorem of Magyar, Weyman and Zelevinsky mentioned in the introduction applies to the symmetric representation varieties. Let (A, σ) be a symmetric quiver algebra, let (V, −, − ) be an ε-quadratic space for (A, σ), let X = R(A, V ) and Notice that denotes the adjoint with respect to the bilinear form −, − and thus the functor ∆ is slightly different from the functor ∇ defined in Section 2.2. On the pair (X, G) we consider the two involutions ∆ : X → X : M → ∆M and ρ : G → G : g → g ρ = ∆(g) −1 . The variety R(A, V ) −,− ,ε of ε-representations of (A, σ) with respect to (V, −, − ) is hence the variety X ∆ of ∆-fixed points and the group G • (V, −, − ) of graded isometries of (V, −, − ) is the group G ρ of ρ-fixed points. We now have to check that the four hypotheses mentioned in the introduction are satisfied by the pair (X, G) endowed with the two involutions ∆ and ρ.
(2) The group G is the group of invertible elements of the algebra E.
(3) The anti-involution of G given by g → (g ρ ) −1 = ∆g is the restriction to G of the linear anti-involution of E given by f → ∆f . (4) For every ε-representation M , let H = Aut(M ) denote its G-stabilizer.
Then an element h ∈ H is a collection h = (h i : The map ζ is an isomorphism of vector spaces: indeed it is clearly linear and the inverse is given by ζ : 2.6. Isomorphism classes of symmetric representations. The following result was proved by Derksen and Weyman in [15, Theorem 2.6] for symmetric quivers without relations and it is a special case of the result proved by Magyar, Weyman and Zelevinsky mentioned in the introduction (see Section 2.4). We provide a different proof for symmetric quiver algebras.
Then they are isomorphic as ε-representations if and only if they are isomorphic as Arepresentations.
Proof. If M and N are isomorphic as ε-representations, then they are isomorphic as A-representations since the group G • (V, Ψ) is a subgroup of GL • (V ). To prove the converse, we choose an isomorphism θ : M → N and we prove that there exists The group Aut(M ) acts from the right on Hom(M, ∇M ) ε∇ as follows We hence see that every point in Hom 0 (M, ∇M ) ε∇ has a dense Aut(M )-orbit. Since Hom(M, ∇M ) ε∇ is irreducible, being a vector space, two such orbits meet. Both Ψ and ∇θ • Ψ • θ lie in Hom 0 (M, ∇M ) ε∇ . It follows that there exists ρ ∈ Aut(M ) such that (∇θ • Ψ • θ) · ρ = Ψ, which is what we wanted to prove.
The following fact is very well-known. It basically says that two complex symmetric or anti-symmetric non-degenerate forms are conjugate under GL. In terms of symmetric representation theory, it says that if M is an ε-representation with respect to some ε-quadratic space (V, −, − ), then for any other choice −, − of a σ-compatible ε-form on V there is M M such that M is an ε-representation with respect to (V, −, − ). This fact is very useful when one makes computations with ε-representations, since it allows to change the form at the best convenience.
In particular, if M is an ε-representation with respect to Ψ, then M = g −1 · M is an ε-representation with respect to Ψ .
Proof. In the proof of Theorem 2.5 replace M with the semisimple representation V . Since Aut(V ) = GL • (V ) we get the first part. To get the second part, we notice Remark 2.7. In view of Corollary 2.6 we can omit to mention the form with respect to which a representation is an ε-representation and we just say that a representation is an ε-representation, without specifying the form. We hence sometimes use to denote the variety of ε-representations of a fixed dimension vector d. This is sloppy, since its points are defined only up to isomorphisms, but we will be very careful and use it only when there is no possibility of confusion. Similarly, we use the notation G ε d instead of In terms of the isomorphism Ψ, ι(N ) ⊥ is described in the following lemma.
By definition, ι(N ) ∩ ι(N ) ⊥ is the kernel of the form restricted to ι(N ) ⊥ . Thus,  Proof. The proof of Derksen and Weyman [Proposition 2.7] [15] uses only the fact that the endomorphism ring of an indecomposable representation is local; since this fact is true for every algebra by Fitting's lemma [6,Corollary 4.8], the same proof applies to the case of symmetric quiver algebras.

Isotropic degenerations
In order to approach the degeneration order for (non-symmetric) A-representations there is a famous second ordering, namely the Ext-order. It is defined on the isoclasses of A-representations of dimension vector d as follows ( [1,7,22,21]): Given M, N ∈ R(A, V ), , M k such that for every i there exists a short exact sequences Indeed, by [7, Lemma 2.1]: ≤ Ext C Q ≤ deg . In this section we provide an analogue of this result for ε-representations.
be an ε-representation and let ι : L → M be an isotropic subrepresentation. Then where Y i denotes the span of the basis elements of ι(L) ⊥ i which do not belong to such that for every i = 2, · · · , k there exists a short Theorem 3.1 implies at once: The opposite implication is not true in general, since an ε-representation could degenerate to an indecomposable ε-representation of type (I) (see [9]). This is not the case for symmetric quivers of type A, though, as we show in Section 7.

Degenerations and Bongartz's cancellation theorem
In this section we recall the definition of the degeneration, the Hom, and the Ext order on the variety of A-representations of a fixed dimension vector. Then we recall Bongartz's cancellation theorem and how to use it to show that these three orders coincide for Dynkin quivers. This argument is analogous to the argument that we will use to prove the main theorem in Section 7.
4.1. Degeneration, Hom and Ext order. Let A be a quiver algebra. Given two A-representations X and Y we use the notation [X, Y ] := dim Hom A (X, Y ) and [X, Y ] 1 := dim Ext 1 A (X, Y ). We denote by ind(A) the set of isoclasses of indecomposable A-modules. Let d ∈ Z Q0 ≥0 be a dimension vector and let V be a Q 0 -graded complex vector space of dimension vector d.
These orders were first considered by Abeasis-Del Fra for quivers of type A [1,2], and then extended to general quiver algebras by Riedtmann [22], Bongartz [7] and Zwara [27]. It is well-known that    A connected quiver Q is called a Dynkin quiver if it is an orientation of a simplylaced Dynkin diagram of type A, D or E.
representation-directed algebras and it does not seem to simplify when restricted to Dynkin quivers.

Quivers of type A
In this section we recall with proofs some useful features of the representation theory and the Auslander-Reiten theory of quivers of type A. In the last subsection, Proposition 5.18 contains a description of the generic quotients by indecomposables which turns out to be very useful in the proof of Theorem 7.1. Throughout the section, Q denotes a quiver of type A n and A denotes its complex path algebra.

Representation Theory.
We denote the category of finite-dimensional complex Q-representations by Rep(Q) . The category Rep(Q) is abelian and Krull-Schmidt. The set ind(Q) of isoclasses of indecomposable Q-representations is finite by Gabriel's Theorem and the dimension vector induces a bijection between ind(Q) and the positive roots of the root system of type A n . We denote by α i , i = 1, · · · , n the simple roots. A positive root α i,j = j k=i α k , 1 ≤ i ≤ j ≤ n, is the dimension vector of the indecomposable Q-representation that we denote by U i,j , here (U i,j ) k = C for i ≤ k ≤ j and every linear map between two non-zero vector spaces is an isomorphism. Given a vertex j ∈ Q 0 , we denote by S j = U j,j the corresponding simple representation, by P j its projective cover, and by I j its injective envelope. More concretely, P j = U t1,t2 and I j = U s1,s2 where s 1 , s 2 , t 1 , t 2 are the vertices (possibly equal to j) such that there are maximal directed paths j G G t k and s k G G j in Q (i.e. t k is either equal to j or it is a sink of Q and s k is either equal to j or it is a source of Q), for k = 1, 2. The radical of P i , i.e. the kernel of the map P i G G G G S i , is the direct sum of at most two indecomposable projectives rad(P i ) = P j1 ⊕ P j2 where j 1 and j 2 are the terminal vertices of the arrows of Q starting from i (there can be at most two).
Every M ∈ Rep(Q) admits a canonical projective resolution, or standard resolution (see e.g. [12, page 7 Here, for e, d ∈ Z Q0 the Ringel-Euler form −, − Q : Since ind(Q) is finite, for every dimension vector d, the representation variety R d contains a dense G  [12,Section 6]). In particular, τ P and τ − I are zero for every projective P and every injective I. For L ∈ ind(Q) not projective, τ L is indecomposable and τ − τ L L. For E ∈ ind(Q) not injective, τ − E is indecomposable and τ τ − E E. In [24, Section 3.1.2] several methods to compute τ L and τ − L are collected. The following Auslander-Reiten formulas hold (see e.g [12, Lemma 6.1]): For every L ∈ ind(Q) there exist two vertices i, j ∈ Q 0 and two non-negative integers s, of non-zero non-isomorphisms between the indecomposable Q-representations L 1 , L 2 , · · · , L k+1 (notice that the composition of these maps can be zero). A path ( be a non-split short exact sequence. Then the following properties hold for F : (i) F is rigid and it has at most two indecomposable direct summands.  6) ξ : 0 This sequence has the remarkable property that it is almost split, or an ARsequence; this means that 1) it is not split, 2) for every E ∈ ind(Q), E L, the induced map The Auslander-Reiten quiver or AR-quiver of Rep(Q) is a translation quiver [23] denoted by Γ Q and defined as follows: It has ind(Q) as set of vertices; thus a vertex of Γ Q denotes an isoclass of an indecomposable Q-representation; with abuse of notation, we denote a vertex of Γ Q and an element of the corresponding isoclass with the same symbol. The arrows of Γ Q are defined as follows: given X i , L ∈ ind(Q) there is an arrow X i → L in Γ Q if X i is a direct summand of the middle term of the almost sequence ending in L in case L is non-projective, or, if L is projective, X i is a direct summand of the radical of L. Thus, an arrow X i → L corresponds to a homomorphism from X i to L which by Lemma 5.1 is unique, up to nonzero-scalar multiples. Finally, the translation is given by the functor τ ; it sends a non-projective vertex L to the non-injective vertex τ L and an arrow f : E → L between non-projectives to the arrow τ f : τ E → τ L between non-injectives. We always use the standard convention to depict Γ Q so that the translation is represented by a horizontal dashed arrow which points from right to left, and the arrows of Γ Q either point towards the north-east or the south-east.
There are several methods to compute Γ Q (see e.g. [24, Section 3.1]). One is the so-called knitting algorithm which works as follows: the subquiver of Γ Q supported on the indecomposable projectives is a copy of Q op , with P i put at vertex i; starting from this one inductively knits by using the formula dim τ − L = dim X − dim L to get the vertex τ − L knowing X and L; the algorithm stops when dim X − dim L is not a dimension vector. We choose to draw P 1 at the top in order to fix a unique way to depict Γ Q . Let us sketch three examples.
. Its AR-quiver Γ Q is given by . Its AR-quiver Γ Q is given by Paths and homomorphisms. A path of length k in Γ Q is a directed path of the quiver Γ Q . For k = 0 this is the trivial path from L 1 to itself. We identify this path with the path of Rep(Q) obtained by choosing a representative of each arrow f i . If L 1 is non-projective, then the translation τ induces the translated path τ p : τ L 1 (τ f1) G G τ L 2 (τ f2) G G · · · (τ f k ) G G τ L k+1 . We say that a path p in Γ Q as (5.8) factors through an almost split sequence if k ≥ 2 and there exists an index 1 ≤ i ≤ k − 1 such that L i = τ L i+2 .
A zero-relation is a path in Γ Q that factors through an almost split sequence whose middle term is indecomposable. By Lemma 5.4, a zero-relation is zero.
Lemma 5.8. Let E, L ∈ ind(Q) and let p be a path from E to L in Γ Q which factors through an almost split sequence. Then L is not projective and there exists a path from E to τ L. In particular, there exists a path from E to every direct summand of the middle term of the almost split sequence ending in L.
Proof. To fix notation, we assume that p is given by (5.8) with E = L 1 and L = L k+1 and let i be an index such that L i = τ L i+2 . Then L i+2 is not projective and there is an induced path from L i+2 to L. Thus there is the translated path from L i = τ L i+2 to τ L and hence from E to τ L. The statement is now obvious.
We say that two vertices E and L of Γ Q have distance dist(E, L) := d if there exists a path in Γ Q from E to L of length d. Since every path from τ L to L has length two, we see that this notion is well-defined. Proof. We fix E ∈ ind(Q) and let L vary in ind(Q). If L E then [E, L] = 1 and there is a unique (trivial) path from E to L. Let us assume that L E. For L ∈ ind(Q) let d L = max{dist(P i , L)| i ∈ Q 0 } be the maximal distance from an indecomposable projective to L. We proceed by induction on d L ≥ 0. For simplicity, in this proof, by a path we mean a path in Γ Q .
Suppose, first, that L = P i is projective (this case includes the base of induction). Then the only paths ending in L start from a projective. Thus if E is not projective, [E, L] = 0 and there are no paths from E to L. If E = P j is projective, then [E, L] = 1 if and only if the vertices i and j of Q are joined by a directed path i G G j of Q. In this case, there is a unique path from P j to P i and by Lemma 5.8 it is not a zero-relation. This proves the lemma in the case when L is projective. Let us assume that L ∈ ind(Q) is not projective, in particular d L ≥ 1. Let us consider the almost split sequence ξ (5.6) ending in L. In view of Lemma 5.4 either its middle term X is indecomposable or it has two non-isomorphic direct summands X = X 1 ⊕ X 2 . For simplicity, by a path from E to X we mean a path from E to an indecomposable direct summand of X. We notice that d τ L < d L and d Xi < d L .
Since E L, the almost split properties give Moreover, again because E L, every path from E to L factors through X. Clearly, every path from E to X gives rise to a path from E to L. Thus, there are no paths from E to L if and only if there are no paths from E to X. In the first case, ξ is a zero-relation and by composing with p, there exists a zero-relation from E to L. In the second case, by composing with p, we see that there exists a path from E to X 2 , and since [E, X 2 ] = 0 there exists a zero-relation from E to X 2 and hence there exists a zero-relation from E to L, too.
If [E, L] = 1 then by (5.9) either [E, X] = 2 and [E, τ L] = 1 or [E, X] = 1 and [E, τ L] = 0. In the first case every path from E to X is not a zero-relation and hence the same holds for every path from E to L, since ξ is not a zero-relation. In the second case, either X = X 1 is indecomposable or [E, X 1 ] = 1 and [E, X 2 ] = 0. Then there are no paths from E to τ L; indeed, a zero-relation from E to τ L induces a zero-relation from E to X 1 , contradicting [E, X 1 ] = 1; in particular, by Lemma 5.8, there are no paths from E to X 2 . Thus, every path from E to L factors through X 1 , does not factor through τ L and therefore it is not a zero-relation.
Proof. Let p 21 be a path from L 1 to L 2 and let p 32 be a path from L 2 to L 3 in Γ Q . By Proposition 5.9, p 21 is a basis of Hom(L 1 , L 2 ) and p 32 is a basis of Hom(L 2 , L 3 ). The composite map p = p 32 • p 21 is a path from L 1 to L 3 and again by Proposition 5.9 is a basis of Hom(L 1 , L 3 ).  For L ∈ ind(Q) the function dim Hom(L, −) : ind(Q) → Z ≥0 is called a hammock function in the terminology of [8]. The support of the hammock function dim Hom(L, −) (or for brevity of Hom(L, −)) is the set of all E ∈ ind(Q) such that [L, E] = 0; it is called a hammock. By using Proposition 5.9, Lemma 5.10 and Corollary 5.11 one can show that these hammocks are "convex". They have a typical rectangular shape, which explains the terminology. Dually, the same holds for the function dim Hom(−, L) and for its support. In The relation . Given E, L ∈ ind(Q), we say that E L if there exists a path in Γ Q which starts in E and ends in L. Since Rep(Q) is representation-directed the relation is a partial order. If [E, L] = 0, then E L by Proposition 5.9. The opposite is not necessarily true, since a path from E to L could be a zero-relation.
Sectional paths. A sectional path in Γ Q is a path in Γ Q which does not factor through an almost split sequence. Then there exists a zero-relation from L to E, too, against the hypothesis.
A sectional path in Γ Q can be directed either towards the north-east or the southeast. We call the former an N E sectional path and the latter an SE sectional path. Two consecutive sectional paths of Γ Q either form a sectional path (if they have the same direction) or they form a broken line (if they have different directions).
In view of Lemma 5.13 (iii), if there is a sectional path from L to E then [E, L] 1 = 0. It then follows from the Happel-Ringel Lemma [16,Lem. 4.1] that a sectional path is either mono or epi. In particular if an arrow of a sectional path (5.8) f i : L i → L i+1 is epi, then all the successive maps f i+1 , · · · , f k must be epi, too.
Lemma 5.14. The cokernel of a sectional mono is a sectional epi and the kernel of a sectional epi is a sectional mono. They form a broken line. Proof. We prove (i). The proof of (ii) is dual. If p is mono then by Lemma 5.14 there is a sectional path from E to its cokernel C, and Rectangles and extensions. Let us consider two subquivers of Γ Q of the form: (5.10) where p 1 , p 2 , q 1 , q 2 , p and q are sectional paths, q = coker(p) and p = ker(q) (compare with Lemma 5.16). The subquiver on the left is called a non-degenerate rectangle of Γ Q and the one on the right is called a degenerate rectangle of Γ Q . By a rectangle from L to E of Γ Q we mean either a non-degenerate or a degenerate rectangle shown in (5.10). It is clear that there exists at most one such rectangle. Proof. If [E, L] 1 = 1, let ξ ∈ Ext 1 (E, L) be a basis and let F be its middle term. Let F i be an indecomposable direct summand of F . By Lemmas 5.4 and 5.13, the non-zero morphisms L → F i and F i → E are sectional paths. Viceversa, we show that a rectangle gives rise to a short exact sequence. If the rectangle is degenerate, this is by definition. If there is a non-degenerate rectangle (5.10) from L to E in Γ Q , then it does not contain zero-relations but it contains every path from L to E and it contains τ E. By Proposition 5.9, [L, τ E] = [L, E] = 1. Thus [E, L] 1 = 1. Let ξ ∈ Ext 1 (E, L) be non-zero. By Lemma 5.4, the middle term of ξ has two nonisomorphic direct summands F 1 and F 2 . By Lemmas 5.4 and 5.13, the morphisms L → F i and F i → E are sectional. It follows that F 1 = F 1 and F 2 = F 2 .
Proposition 5.17 states that the rectangles of Γ Q are cartesian. As a consequence we get the following useful properties of a non-degenerate rectangle (5.10): opposite sides are either both mono or both epi; (5.11) ker(p i ) ker(q i ) and coker(p i ) coker(q i ). (5.12) Join and Meet. Recall the partial order . Given E, L ∈ ind(Q) we denote Of course, E ∨L and E ∧L might not exist. If E ∨L exists then it is the intersection of two sectional paths, one starting from E and one starting from L; dually, if E ∧ L exists then it is the intersection of two sectional paths, one ending in E and one ending in L. They both exist if and only if there is a rectangle

Generic quotients.
Let Q be a quiver of type A. Let L be an indecomposable Q-representation and M be a Q-representation. In this section we give a necessary and sufficient condition for L to embed into M and we specify the position of the indecomposable direct summands of the generic quotient of M by L in Γ Q . Suppose that [L, M ] = 0. Let T 0 1 , · · · , T 0 r be the -minimal indecomposable direct summands (one for each isoclass) of M which belong to the support of Hom(L, −). We order them from top to bottom in Γ Q as depicted in Table 1 for We also define T 1 0 and T 1 r as follows: if there is an NE sectional path from L to T 0 1 we define T 1 0 to be zero, otherwise T 1 0 is the -minimal vertex of the NE sectional path starting from T 0 1 not belonging to the support of Hom(L, −); similarly, T 1 r is zero if there is an SE sectional path from L to T 0 r and otherwise it is the -minimal vertex of the SE sectional path starting from T 0 r not belonging to the support of Hom(L, −). (Compare with Table 1.) Clearly, T 1 i might not exist. In case they all exist, we put T 1 = ⊕ r i=0 T 1 i . By construction, using Proposition 5.17, one immediately sees that T 0 and T 1 are rigid (see Remark 5.19). For i = 1, · · · , r we denote by L (i) and L (i) the two indecomposables such that L (i) ∧ L (i) = L and L (i) ∨ L (i) = T 0 i . We place L (1) , · · · , L (r) in the same NE sectional path starting from L.
. This picture represents a part of Γ Q containing the support of Hom(L, −). Every side of every square represents an arrow of Γ Q which points towards the north-east or the south-east. For every i = 1, · · · , r, let ι i := f i : L → T 0 i and set ι = (ι 1 , · · · , ι r ) t : L → T 0 . We claim that there exists an automorphism of M such that • f = (ι, 0) t . Indeed, given two indices i = j such that i ≤ r, f j = 0 and [M i , M j ] = 1, i.e. either M i M j or M j is not -minimal in the support of Hom(L, −), then there is a homomorphism ji : M i → M j and by Lemma 5.1, up to a non-zero scalar factor, f j = ji • f i . We then define the automorphism in the obvious way to get the claimed equality ( Since f and • f have the same quotient, f is a generic embedding of L into M if and only if ι is a generic embedding of L into T 0 and in this case the generic quotient is the direct sum of M and the generic quotient of T 0 by L.
Since T 1 is rigid, it remains to prove that ι : L → T 0 is mono if and only if T 1 is defined, and in this case there exists a short exact sequence We notice that, for 1 ≤ i ≤ r − 1, with the convention that L (0) = L, T 0 0 and L (r+1) are zero. We show that T 1 is defined if and only if dim T 0 − dim L is a dimension vector. In this case, it equals dim T 1 by the discussion above. We put t i := dim T 0 i + dim L (i+1) − dim L (i) . Let us prove that if there exists an index j such that T 1 j does not exist then dim T 0 −dim L is not a dimension vector. The argument we use is slightly different in the case when j = 0, and we include this case in the general discussion by highlighting the differences in parentheses. The SE sectional path starting from T 0 j (or the N E sectional path starting from L (1) , if j = 0) terminates with an injective I. The almost split sequence ending in I is not a zero-relation, because, otherwise, I would not be in the support of Hom(L (j) , −) and T 1 j would exist by Lemma 5.16. Thus, there exists an indecomposable injective I k and an NE arrow (or SE arrow for j = 0) I k → I in Γ Q . Then, [L (i) , I k ] = 1 for every 0 ≤ i ≤ j, [L (i) , I k ] = 0 for every i > j and [T 0 , I k ] = 0. By (5.2) we get that the k-th component of t i is We conclude that dim T 0 − dim L is not a dimension vector, and hence L cannot embed into T 0 and a fortiori into M .
Suppose that T 1 is defined. Then dim T 1 = dim T 0 − dim L. Obviously, L embeds into L ⊕ T 1 and [L, L ⊕ T 1 ] 1 = 0. Since T 0 is rigid, it is generic and hence T 0 ≤ deg L ⊕ T 1 . By Bongartz's Theorem 4.2 L embeds into T 0 and since T 1 is rigid, T 1 is the generic quotient of T 0 by L, finishing the proof.
Remark 5.19. By construction both T 0 and T 1 are Hom-and Ext-orthogonal and T 0 ⊕ T 1 can be completed to a separating tilting object T of Rep(Q) whose indecomposable direct summands form a section of Γ Q (see [6] for the definitions). Then L is a torsion-free module for T and the short exact sequence (5.13) is the add T resolution of L.

Symmetric quivers of type A
Let (Q, σ) be a symmetric quiver of type A n . Let ε be +1 or −1. Let (V, Ψ) be an ε-quadratic space for (Q, σ) of dimension vector d. We denote the variety of Q-representations of dimension vector d either by R(A, V ) or by R d and the structure group by G d . We denote by R(A, V ) Ψ,ε ⊂ R(A, V ) the subvariety of ε-representations. Sometimes, for simplicity of notation and in view of Remark 2.7, we use the shorter notation R ε d ⊂ R d for the subvariety R(A, V ) Ψ,ε . By G ε d we denote the group of graded isometries. In this section we describe useful facts about ε-representations of (Q, σ). The main results are Propositions 6.10, 6.12 and 6.17. They are crucial in our proof of Theorem 7.1. In the other types (A odd , −1) and (A even , 1) the indecomposable ε-representations are all of the form L ⊕ ∇L, for L ∈ ind(Q). This is due to the fact that the indecomposable Q-representations are thin and a symplectic vector space must be even dimensional, compare with [15, Proposition 3.6 and 3.8]. For this reason we write that (A odd , −1) and (A even , 1) are the split types while (A odd , 1) and (A even , −1) are the non-split types. Accordingly, we say that (Q, ε) is of split or non-split type.
In the next lemmas we use the notation introduced in (2.3). Proof. On Im(f ) we define the bilinear form f (x), f (y) = f (x)(y). This is a non-degenerate, σ-compatible ε-form on the Q 0 -graded vector space Im(f ).  Γ Q from L to ∇L whose middle vertices G 1 , G 2 ∈ ind(Q) are ∇-invariants: This is called a ∇-invariant rectangle of Γ Q . It has the following property. Lemma 6.3. Either q 1 or q 2 is mono.
Proof. Since G 1 and G 2 are ∇-invariants, q 1 and q 2 are non-zero scalar multiples of ∇p 1 and ∇p 2 , respectively. Thus, if q 2 is epi, then p 1 is epi by (5.11) and hence ∇p 1 is mono by duality. It follows that q 1 is mono. . This is a well-known additive function with many useful properties (cf. [27]). One of them is that if E ∈ ind(Q) is nonprojective and F is the middle term of the almost split sequence ending in E, then where µ(N, E) (resp. µ(M, E)) denotes the multiplicity of E as a direct summand of N (resp. M ). This follows directly from the almost split properties. for every E ∈ ind(Q).
Proof. We compute Definition 6.6. We say that Clearly, F is δ-fixed if and only if all its indecomposable direct summands are.
Example 6.7. For the quiver Q considered in Example 5.5, P 2 is the only δ-fixed indecomposable Q-representation. In Example 5.6 the δ-fixed indecomposables are U 4,3 and U 1,3 and in Example 5.7 they are U 3,5 and U 1,4 .
Proof. Suppose that F is δ-fixed. If F is injective, then ∇F is projective and hence F is, thus, not injective. Let 0 → F → G → τ − F → 0 be the almost split sequence starting from F . Then τ − F ∇F and hence G is ∇-invariant. Since G is an indecomposable direct summand of G , it is ∇-invariant, too. Viceversa, suppose that G is ∇-invariant. Then G is not projective. Let 0 → τ G → F → G → 0 be the almost split sequence ending in G. Then 0 → ∇G → ∇F → ∇τ G → 0 is the almost split sequence starting in ∇G = G. Since τ ∇G = τ G and τ ∇τ G = G, necessarily, τ ∇F = F and F is δ-fixed. Since F is a direct summand of F , F is δ-fixed, too.
Lemma 6.9. There exists a ∇-invariant representation G ∈ ind(Q), such that there is only one arrow in Γ Q ending in G.
Proof. Let ω be the unique source of Q such that the path ω G G σ(ω) is equioriented. We claim that G = U ω,σ(ω) has the required property. Indeed, if G is projective then its radical is indecomposable, since G has a simple socle. If G is not projective there exists an arrow ω → j of Q which does not belong to the path ω G G σ(ω) . Let k = ω be the source of Q such that there exists an equioriented path k G G j (if there is no such path, i.e. if j is not a sink, put k = j). Then τ G = U k,j . Since [U k,j , G] = 0 it follows from Lemma 5.4 that the middle term of the almost split sequence ending in G is indecomposable (namely, it is U k,σ(ω) ). Since M and N do not have ε-indecomposable direct summands of type (I) and G is ∇-invariant, its multiplicity both in M and in N is even. We hence see that δ M,N (F ) is even. Let F 1 , · · · , F k ∈ ind(Q) be the δ-invariant isoclasses, ordered from top to bottom in Γ Q . Then F i ⊕ F i+1 is the middle term of an almost split sequence ending in a ∇-invariant indecomposable representation, and by (6.6), δ M,N (F i ) and δ M,N (F i+1 ) have the same parity, for all i = 1, · · · , k − 1. Thus, δ M,N (F 1 ), · · · , δ M,N (F k ) have all the same parity. To conclude the proof it is enough to show that one of them is even. By Lemma 6.9, either F 1 or F k is the middle term of an almost split sequence ending in an indecomposable ∇-invariant representation, and hence, by (6.6), either δ M,N (F 1 ) or δ M,N (F k ) is even.
6.3. Generic isotropic embeddings. We state and prove a surprising result that says that under certain mild hypotheses, it is always possible to embed an indecomposable inside an ε-representation so that the embedding is isotropic and generic.
In the split types this is obvious since every embedding is isotropic: Lemma 6.11. Let (Q, ε) be of split type. Let M ∈ R(A, V ) Ψ,ε and L ∈ ind(Q). Then every embedding of L into M is isotropic.
In the non-split types the statement of Lemma 6.11 is not true (see Examples 6.13 and 6.14) and one needs to add extra hypotheses on L and M . Let us assume that [L, ∇L] = 1. By Lemma 6.2 Hom(L, ∇L) = Hom(L, ∇L) ε∇ = kϕ for a non-zero homomorphism ϕ = ε∇ϕ : L → ∇L. We need to prove that there exists an automorphism g of M such that j := g • ι is isotropic, i.e. ∇j • Ψ • j = 0.
Let Z be an indecomposable direct summand of M which is -minimal in the support of Hom(L, −). We say that Z is a split (resp. non-split) -minimal direct summand of M relatively to L if Z ⊕ ∇Z is (resp. is not) a direct summand of M . Thus if Z is non-split then Z ∇Z, but the converse is not necessarily true: if Z ∇Z and there is more than one copy of Z in M , then Z is split even if it is ∇-invariant (compare with example 6.16 below). for k = s + 1, · · · , N . In view of Theorem 2.5 we can assume that Ψ is: is a non-zero homomorphism for k = 1, · · · , s and We denote by ι k : L → M (k) the component of ι along the direct summand M (k). The generic embedding ι has hence the form Let us discuss ∇ι • Ψ • ι. With the choice we made for the form, we have ∇ι For k ≤ s, M (k) is indecomposable and ∇-invariant, and thus ∇ι k • Ψ k • ι k is non-zero by Lemma 6.2. and the fact that [L, ∇L] = 0.
For k > s we choose two homomorphisms j k : L → T 0 k and k : T 0 k → ∇T 0 k such that j k is non-zero and k is non-zero whenever possible or stated differently. (Notice that k can be chosen to be non-zero in case [L, ∇T 0 k ] = 0 by Corollary 5.11.) We can choose ι k = (j k , k • j k ) t , since the generic quotient Q does not depend on k . By Lemma 6.2 and our assumption on ε, we get In particular, if [L, ∇T 0 k ] = 0, or k is chosen to be zero, then ∇ι k • Ψ k • ι k is zero.
Let us reorder the direct summands M (k)'s of M so that ∇ι k • Ψ k • ι k is non-zero for 1 ≤ k ≤ h and it is zero for k > h.
Assume that h ≥ 1. For every k = 1, · · · , h there exists a non-zero complex number x k such that ∇ι k • Ψ k • ι k = x k ϕ. In particular we get ∇ι • ι = (x 1 + · · · + x h )ϕ. Example 6.14. Let Q be of type → A 5 . We consider L ∈ ind(Q) and the orthogonal Q-representation M = S ⊕ T 0 ⊕ ∇T 0 given by In this case it is not true that a generic embedding of L into M is isotropic. To see this, let us fix an orthogonal basis of M as follows Then for some x = 0 and thus ι(L), ι(L) = xu, xu = x 2 = 0, proving that ι(L) is not isotropic. (If x = 0, the map is still an embedding, it is isotropic but it is not generic, since the quotient is ∇H ⊕ S ⊕ ∇T 0 which is a degeneration of Q.) Let us illustrate the proof of Proposition 6.12 in an example.
Example 6.15. Let us consider the following indecomposable representation L and the following orthogonal representation M = S ⊕ T 0 ⊕ ∇T 0 of a quiver of type → A 5 : We choose non-zero homomorphisms ι S , ι T 0 and as depicted. Then the homo- There exists a non-zero complex number x such that ∇ι S • ι S = x(2∇ι T 0 • • ι T 0 ). We define j T 0 := √ −x √ Example 6.16. Let Q : 1 → 2 → 3 be of type − → A 3 . Its AR-quiver is shown in Example 5.5. Let M = P n 1 and let L = P 2 . Then M is an orthogonal representation for every n ≥ 1 and it is also symplectic for n even. If n = 1, then L does not embed isotropically into M (compare with Example 6.13). For n ≥ 2, L embeds isotropically into M because we can change the form so that one copy of P 1 is isotropic, i.e. we can write M as T 0 ⊕ ∇T 0 ⊕ M where T 0 ∇T 0 P 1 ; then the embedding P 2 → P 1 is isotropic as well and it is generic. This example motivates our choice for the (unexpected) decomposition M = T 0 ⊕ ∇T 0 ⊕ M which is not the natural decomposition into indecomposable ε-representations (for ε = 1).
6.4. Generic ε-subquotients. Let M ∈ R(A, V ) Ψ,ε be an ε-representation and let L ∈ ind(Q). If there exists an isotropic embedding ι : L → M then the quotient ι(L) ⊥ /ι(L) is called the ε-subquotient of M by ι(L). In this section we study εsubquotients of M induced by a generic isotropic embedding of L into M . It turns out that there are substantial differences between the split and the non-split types.
Let ι : L → M be a monomorphism with generic quotient π : M G G G G Q and assume further that ι(L) is isotropic. Let Y = ι(L) ⊥ /ι(L) be the ε-subquotient.
We use the notation introduced in Section 5.3 and in the proof of Proposition 6.12 Proposition 6.17.
(i) In the split types, Y T 1 ⊕ ∇T 1 ⊕ M . (ii) In the non-split types, there exists a generic embedding ι so that the corresponding ε-subquotient Y is the generic quotient of ∇Q by L.
Proof. The proof of Proposition 6.17 needs a little preparation.
We decompose M = S ⊕T 0 ⊕∇T 0 ⊕M as in the proof of Proposition 6.12, with the convention that S is zero in the split types. The direct summand M does not play any role, since it splits off in M , Q and Y for every choice of a generic embedding. Thus we assume, for simplicity of notation, that it is zero.
By Theorem 2.5, or its Corollary 2.6, we can choose the ε-form at our convenience. We thus assume that Ψ : M → ∇M is given in the following block form For every index k, such that [L, ∇T 0 k ] = 1, we choose arbitrarily a morphism k : T 0 k → ∇T 0 k and let = diag( k ) : T 0 → ∇T 0 be the induced diagonal morphism. As in the proof of Proposition 6.12, and by Lemma 6.11, we assume that the generic embedding ι depends on the choice of and it is given by ι = (ι S , ι T 0 , • ι T 0 ) t . Thus, there is a short exact sequence of the form A short computation shows that condition ∇ι • Ψ • ι = 0 is equivalent to In the split types this condition is redundant, by Lemma 6.2. On the other hand, in the non-split types, since ε∇ = , this becomes ∇ι S • Ψ S • ι S + 2∇ι T 0 • • ι T 0 = 0. By Lemma 2.8, since ι(L) is isotropic, it is contained in the kernel of the map ∇ι • Ψ : M → ∇L. Thus, there exists j : T 1 → ∇L and a surjective map that we denote by f : T 1 ⊕ ∇T 0 → ∇L which makes the following diagram commutative: We notice that j satisfies the following equalities In the split types, equation (6.9) is not present since S is zero; moreover, since by Lemma 6.2 ε∇ = − , equation (6.10) becomes j • p T 0 = 0; since p T 0 is surjective (since (p S , p T 0 ) = p T 0 is surjective) we arrive at the following crucial observation: (6.11) j = 0 in the split types.
In the non-split types, since by Lemma 6.2 ε = ∇ , equation (6.10) becomes We have constructed the following commutative diagrams whose rows and columns are short exact sequences (6.13) We remark that all the maps involved depend on the choice of . We need to show that in the split types Y T 1 ⊕ ∇T 1 , for every choice of , and in the non-split types, there is a choice of such that ∇f is a generic embedding of L into ∇Q.
In the split types, by (6.11), we have ∇f = (0, ει) t and thus Y T 1 ⊕ ∇T 1 for every choice of , proving the first part of the proposition.
In the non-split types, we choose k = 0 for every possible index k (see the proof of Proposition 6.12). Then ∇f = (∇j, ει T 0 ). We claim that we can choose j such that this map is generic. In view of Proposition 5.18, it is enough to show that ∇f hits all the -minimal indecomposable direct summands of ∇T 1 ⊕ T 0 contained in the support of Hom(L, −). By definition of ι T 0 , every direct summand of T 0 is hit by the map. Thus, we only need to discuss the direct summands of ∇T 1 . Let ∇T 1 i be a -minimal direct summand of ∇T 1 ⊕ T 0 contained in the support of Hom(L, −). We claim that we can choose j such that the component ∇j i : ∇T 1 i and hence we are in one of the following where the dotted arrows G G denote sectional paths, the wavy arrows G G denote a composition of sectional paths and S j denotes an indecomposable direct summand of S. In the situation on the right, ∇j i is non-zero by (6.9). In the left-hand situation, if either i or i+1 is non-zero then ∇j i is non zero by (6.12).
If both i and i+1 are zero, then any choice of ∇j i makes the equation 6.12 true; in particular we can choose (∇j) i to be non-zero. This concludes the proof.
Definition 6.18. The representation Y constructed in Proposition 6.17 is called the generic ε-subquotient of M by L.
Remark 6.19. We called Y generic since it comes from a generic embedding. In the proof of Theorem 7.1 we will see that Y has "generic properties" (see remark 7.2).
Example 6.20. Let Q be the quiver of Example 5.6 and let us consider the following indecomposable Q-representations:

Main result
In this section we state and prove the main result of the paper. For convenience of the reader, we recall the notation used throughout the paper. Let (Q, σ) be a symmetric quiver of type A n , for some fixed n ≥ 2. Let A = kQ be its complex path algebra. Let ε be +1 or −1. Let (V, Ψ) be an ε-quadratic space for (Q, σ) (see Section 2.3) and let d = dim V . For simplicity of notation, we denote by R d the variety R(A, V ) of Q-representations with underlying vector space V and by R ε d the subvariety R(A, V ) Ψ,ε of ε-representations with respect to (V, Ψ) (see (2.2) for the definition). We denote by G d = i∈Q0 GL(V i ) the structure group and by G ε d := G • (V, Ψ) the subgroup of graded isometries of (V, Ψ). Given M, N ∈ R ε d we define the ε-degeneration order by M ≤ ε deg N if N ∈ G ε M . Theorem 7.1. For every M, N ∈ R ε d the following equivalences hold: Proof. The implication ≤ deg ≤ ε deg k s is obvious since G ε d is a subgroup of GL • (V ), and the implication ≤ ε deg ≤ ε Ext k s holds in general by Corollary 3.3. We hence prove ≤ deg C Q ≤ ε Ext . Let M, N ∈ R ε d such that M ≤ deg N . If M and N are isomorphic as Q-representations then they are isomorphic as ε-representations by Theorem 2.5. In the rest of the proof we use the symbol to denote that two ε-representations or two Q-representations are isomorphic. Let us discuss, first, the case when L ∇L. Then N E ⊕ X where E = L (in the non-split types) or E = L ⊕ ∇L (in the split types) is an indecomposable ε-representation and X is an ε-representation. We have [E, N ] 1 = 0 and also [N, E] 1 = 0 by duality. By Theorem 4.2, E embeds into M and the generic quotient M/E degenerates to X; it then follows that [M/E, E] 1 ≤ [X, E] 1 = 0 and thus M E ⊕ M/E. Then M/E is an ε-representation and it degenerates to X. By induction, we get that M/E ≤ ε Ext X and hence M E ⊕ M/E ≤ ε Ext N . We assume that L ∇L. Then N L ⊕ ∇L ⊕ X where X is an ε-representation.
In the split types the generic embedding ι is isotropic by Lemma 6.11.
In the non-split types, we claim that we can apply Proposition 6.12. Indeed, by We hence see that Proposition 6.12 applies and we can thus assume that a generic embedding ι : L → M is isotropic.
We denote by Y the generic ε-subquotient of M by L. We claim that If the claim holds then the proof finishes by induction as follows: M ≤ ε Ext L ⊕ Y ⊕ ∇L ≤ ε Ext L ⊕ X ⊕ ∇L = N (the first inequality holds by Theorem 3.1, and the second is given by induction since Y and X are ε-representations of smaller dimension.) It remains to prove (7.2).
In the non-split types the claim is true since, by Proposition 6.17, we can choose ι so that Y is the generic quotient of ∇Q by L; thus, since ∇Q ≤ deg L ⊕ X and [L, ∇Q] = [L, L ⊕ X], by Theorem 4.2 we get Y ≤ deg X.
We prove (7.2) in the split types. We write M = T 0 ⊕ ∇T 0 ⊕ M , where T 0 is the multiplicity-free direct summand of M consisting of -minimal direct summands which lie in the support of Hom(L, −). By Proposition 5.18, Q T 1 ⊕ ∇T 0 ⊕ M and ∇Q T 0 ⊕ ∇T 1 ⊕ M . By Proposition 6.17, Y = T 1 ⊕ ∇T 1 ⊕ M . Thus there is a commutative diagram with exact rows and columns In view of Theorem 4.3 it is enough to prove that  We claim that under the assumptions (7.6) there exists an index i such that E is contained in the ∇-invariant square (7.7) Suppose that τ ∇E = E. Without loss of generality we can assume that τ ∇E ≺ E, since δ Y,E (E) = δ Y,X (τ ∇E) by Lemma 6.5. The rectangle from τ ∇E to E is a square and its middle vertices F 1 and F 2 are δ-invariants. This square is all contained in the rectangle (7.7) and gives rise to a short exact sequence The only indecomposable direct summand of Y which could belong to this square is and ε = −1 (non-split type).
Let Q be of type → A 5 and ε = −1 (split type): Example 8.4. Let Q be of type → A 5 and ε = 1 (non-split type):