LOCALLY SEMI-SIMPLE REPRESENTATIONS OF QUIVERS

Transcription

1 LOCALLY SEMI-SIMPLE REPRESENTATIONS OF QUIVERS A Dissertation presented to the Faculty of the Graduate School at the University of Missouri In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy by DANIEL KLINE Dr. Calin Chindris, Dissertation Supervisor MAY 2016

2 The undersigned, appointed by the Dean of the Graduate School, have examined the dissertation entitled: LOCALLY SEMI-SIMPLE REPRESENTATIONS OF QUIVERS presented by Daniel Kline, a candidate for the degree of Doctor of Philosophy and hereby certify that, in their opinion, it is worthy of acceptance. Dr. Calin Chindris Dr. Ian Aberbach Dr. David Meyer Dr. Randy Prather Dr. Zhenbo Qin

3 For Melissa, Caleb, Adelaide, and Gabrielle. Soli Deo Gloria

4 ACKNOWLEDGMENTS I would like to thank my advisor, Professor Calin Chindris, for his skillful guidance and support throughout this project. Working and learning under Dr. Chindris has without a doubt been the highlight of my time at the University of Missouri. I would also like to thank my lovely wife Melissa for her tireless dedication, love, and support throughout my time in graduate school. I could not have been successful without her. ii

5 TABLE OF CONTENTS ACKNOWLEDGMENTS ii 1 Introduction Representation Theory of Quivers Quivers Krull-Schmidt property Exact sequences of quiver representations Path algebras Gabriel s Theorem Geometric Invariant Theory Background on Invariant Theory Affine varieties Linear Algebraic Groups Hilbert s Fourteenth Problem & Affine Quotient Varieties Semi-invariants for quivers The Weight Space Decomposition iii

6 3.2.2 The Space of β-dimensional Quiver Representations Stability Weights for Quiver Representations A Conjecture of Victor Kac Locally Semi-simple Representations Locally Semi-simple Representations and Stability Conditions Semi-Simple Endomorphism Rings Orthogonal Exceptional Sequences The Cone of Effective Weights Dynkin Quivers Euclidean Quivers Non-regular case Mixed case The Regular Category Regular Simples Tubes Common Stability Weights iv

7 7 Proof of the Main Theorem References VITA v

8 CHAPTER 1 Introduction A quiver is a directed graph and a representation of a quiver is an assignment of vector spaces to each vertex and (appropriately sized) matrices to each arrow. Quivers and their representations were initially devised to resolve classification problems in linear algebra, such as the classification of tuples of subspaces of a given vector space. Soon thereafter, through the work of P. Gabriel, the representation theory of quivers became an integral part of the representation theory of finite-dimensional algebras. Since then, quiver representations have found interesting connections to many other areas of mathematics, such as invariant theory, Lie theory, and algebraic combinatorics. More recently, they play a key role in persistent homology, a powerful tool in topological data analysis. One of the fundamental goals of the representation theory of quivers is to classify the indecomposable representations (see Section 2.5). Based on the complexity of their indecomposables, quivers come in three flavors: representation-finite, tame, and wild. The classification of all indecomposable representations of wild quivers is regarded as extremely difficult (if not impossible). Therefore, in order to make progress, especially in the presence of wild quivers which occur frequently in representation theory and other areas, we are naturally led to impose conditions on either the class of representations to be characterized or the quiver in question. In this thesis, we focus on the large class of locally semi-simple quiver representations. These representations arise most naturally when one studies quiver representations within the general framework of Geometric Invariant Theory. As ex- 1

9 plained by Kac in [12], every quiver representation can be expressed as the sum of a locally semi-simple representation and a nilpotent representation. Thus locally semi-simple representations play a fundamental role in the general classification problem for quiver representations. In [12], published in 1982, Victor Kac conjectured that for tame quivers (see Section 2.5), a representation is locally semi-simple if and only if its endomorphism ring is a semi-simple algebra. The main goal of this work is to prove this conjecture of Kac. To deal with locally semi-simple representations in general, we use ideas and techniques from quiver invariant theory to reduce the considerations to a problem about the existence of a common stability weight for, what we call, an orthogonal Schur sequence of quiver representations. In particular, this approach allows us to prove that for any acyclic quiver a locally semi-simple representation has a semi-simple ring of endomorphisms. To establish the converse, we need to assume that the quiver in question is representation-finite or tame. Specifically, for such a quiver, we first show that for any orthogonal Schur sequence of representations there exists a common weight θ such that each representation of the given sequence is θ-stable. It turns out that this is equivalent to saying that in the tame case, any representation with a semi-simple ring of endomorphisms is locally semi-simple. Going beyond the tame case, we show that any wild quiver posses a representation which is not locally semi-simple but whose algebra of endomorphisms is semisimple. In particular, this shows that the validity of Kac s conjecture precisely captures the tameness of a quiver. The main result of this thesis is: Theorem 1: (see Theorem 16) Let Q be an acyclic quiver. Then the following statements are equivalent: 2

10 (i) Q is tame; (ii) for a representation V of Q, V is locally semi-simple if and only if its ring of endomorphisms is semi-simple. The layout of this thesis is as follows. In Chapters 2-3, we present the background material leading up to Kac s conjecture. In Chapter 4 we present the conjecture and develop some important tools. In Chapters 5-6 we show that there exists a common stability weight for any orthogonal Schur sequence of representations over a tame quiver. This essentially proves one implication of our main result. In Chapter 7 we deal with wild quivers and prove our main result. 3

11 CHAPTER 2 Representation Theory of Quivers 2.1 Quivers A quiver is a directed graph. Throughout we will use the notation Q = (Q 0, Q 1, t, h) where Q 0 is the set of vertices, Q 1 is the set of arrows, and t, h : Q 1 Q 0 are functions assigning the tail ta and head ha to each arrow a. Note that loops, oriented cycles, and multiple arrows between vertices can occur, as Example 1 illustrates. Throughout, we assume Q is a finite, connected quiver. Definition 1: A path is a collection of arrows p = a 1... a m such that ta i = ha i+1 for i < m. In this case we say tp = ta m, hp = ha 1, and p has length m. An oriented cycle is a path p (of length at least one) such that tp = hp. We call a quiver acyclic if it has no oriented cycles. Example 1: Q = a 1 b 2 c d 3 i e 4 6 f h 5 7 g Using the notation above, we have Q 0 = {1, 2, 3, 4, 5, 6, 7} and Q 1 = {a, b, c, d, e, f, g, h, i}. Also, tb = 1, hb = 2, etc. Q has a path p = ghi from vertex 3 to vertex 5. Also, Q has two oriented cycles, namely p 1 = a and p 2 = cd. 4

12 Lemma 1: Let Q be an acyclic quiver. Then there exists a bijection f : Q 0 {1,..., n} such that f(ta) < f(ha) for each a Q 1. The central objects of study in our thesis are quiver representations. Let us fix a quiver Q. Definition 2: A representation of Q (or Q - representation) is a collection V = (V (i), V (a)) i Q0, a Q 1 of: K-vector spaces V (i), i Q 0 ; K-linear maps V (a) : V (ta) V (ha), a Q 1. The dimension vector of a representation V of Q is the function dim V : Q 0 Z defined by (dim V )(i) = dim K V (i) for i Q 0. Example 2: Let Q be the quiver D 4 : 1 4 α β 5 γ ɛ 2 3 For any λ K, we construct the representation V λ as follows: V λ = K K [ 1 0 ] [ 1 1] [ 0 1] K 2 [ 1 λ ] K K In this example, V λ (5) = K 2, V λ (β) = [ 1 1 ], and dim V λ = (1, 1, 1, 1, 2). 5

13 Definition 3: Let V and W be Q-representations. We say that W is a subrepresentation of V if: a) W (i) is a subspace of V (i) for each i Q 0 ; b) For each a Q 1, V (a)(w (ta)) W (ha) and W (a) = V (a) W (ta). Definition 4: Let V and W be Q-representations. A morphism of representations φ : V W is a collection of K-linear maps φ = (φ(i)) i Q0 with φ(i) : V (i) W (i), i Q 0, and such that W (a)φ(ta) = φ(ha)v (a). Remark 1: (i) Condition b) simply says that for each a Q 1, the following diagram commutes: V (ta) V (a) V (ha) φ(ta) φ(ha) W (ta) W (a) W (ha) (ii) A Q-representation W, with W (i) V (i), i Q 0, is a subrepresentation of V if (τ(i): W (i) V (i)) i Q0 is a morphism of Q-representations. We denote by Hom Q (V, W ) the vector space of morphisms from V to W. A morphism φ is injective, surjective, or an isomorphism if and only if for each i Q 0, φ(i) is an injection, surjection, or isomorphism of K-vector spaces (respectively). Let Rep(Q) denote the category of representations of Q. We use rep(q) to denote the (additive) sub-category of finite-dimensional representations, and we will 6

14 largely restrict our attention to this smaller category. Note that rep(q) satisfies the Krull-Scmidt property (see Section 2.2) but Rep(Q) does not. Example 3: Let Q = A 3 := Consider the representations V = K 1 K 1 K, S 2 = 0 K 0, and S 3 := 0 0 K Then Hom Q (V, S 2 ) = Hom Q (S 2, V ) = 0, and S 3 V. However, S 2 V. Definition 5: A Q-representation V is called simple if it is non-zero and the only subrepresentations of V are 0 and V, i.e. V has no proper subrepresentations. Next we define an important family of simple representations. For each vertex i Q 0, let S i rep(q) be defined as follows: K i = j S i (j) = 0 i j j Q 0, and S(a) = 0, a Q 1. In particular, each representation S i is simple. The next proposition shows that when Q is acyclic, these are the only simple representations. Proposition 1: Let Q be an acyclic quiver. Then {S i } i Q0 forms a complete set of pairwise non-isomorphic simple representations of Q. Remark 2: There are other simple representations when Q is not acyclic, as the next example shows. Example 4: Let Q: = 1 b a 2. For each λ K, consider the Q- 7

15 representation V λ : = K λid Id K. Then each V λ is simple and V λ = Vλ whenever λ λ. Thus there exists an infinite one-parameter family of pairwise non-isomorphic simple representations. 2.2 Krull-Schmidt property Let V, W Rep(Q). Then the direct sum V W is defined as follows: (V W )(i) = V (i) W (i), i Q 0 ; (V W )(a) = V (a) W (a), a Q 1. Note that when V and W are both finite-dimensional, after fixing bases, we can view each (V W )(a) as a block diagonal matrix: (V W )(a) = V(a) 0 0 W(a). Definition 6: A nonzero representation V Rep(Q) is said to be indecomposable if V = X Y implies either X = 0 or Y = 0. Otherwise, V is called decomposable. Definition 7: A category A is additive if: a) A admits finite direct sums; b) Hom A (A, B) forms an abelian group for every pair of objects A, B and the composition of maps is Z-bilinear; c) A has a zero object 0 (i.e. Hom A (X, 0) and Hom A (0, X) are trivial groups). 8

16 Theorem 2: [1] Let A be an additive category. Assume that: 1. Every object of A decomposes into a finite direct sum of indecomposable objects; 2. For any indecomposable object V, End A (V ) is local. Then every object decomposes uniquely (up to isomorphisms and permutation of summands) into a finite direct sum of indecomposable objects. Definition 8: An additive category satisfying the conditions of Theorem 2 is called a Krull-Schmidt category. Recall that a ring R (not necessarily commutative) is called local if R has a unique maximal left (equivalently right, or two-sided) ideal. Lemma 2: [2] Let R be a ring. Then R is local if and only if the set of non-unit elements of R forms a two-sided ideal. Theorem 3: The category rep(q) is a Krull-Schmidt Property category Proof. Given any W rep(q), an induction argument on the dimension shows that W must decompose into a finite direct sum of indecomposable representations. Now assume W is indecomposable. We need to show that End Q (W ) is local. Let φ be an arbitrary endomorphism of W. For each i Q 0, we have φ(i): W (i) W (i). For λ K, consider the generalized λ-eigenspace of W (i) with respect to φ(i): W λ (i) := {w W (i) (φ(i) λid) N (w) = 0 for N >> 0} 9

17 Since K is algebraically closed, we have W (i) = λ K W λ (i) (2.2.1) where all but finitely many W λ (i) are zero. Claim: For each λ K, the collection W λ : = (W λ (i)) i Q0 forms a subrepresentation of W. It is clear that W λ (i) W (i) for each i. To show that W (a)(w λ (ta)) W λ (ha), observe that: (φ(ha) λid) N W (a) = = N ( ) N φ(ha) l ( λid) N l W (a) l l=0 N l=0 ( N l ) W (a) φ(ta) l ( λid) N l (2.2.2) = W (a) (φ(ta) λid) N. Suppose w W λ (ta). Then we have W (a) (φ(ta) λid) N (w) = W (a)(0) = 0. On the other hand, by (2.2.2), W (a) (φ(ta) λid) N (w) = (φ(ha) λid) N W (a)(w). Thus, W (a)(w) ker(φ(ha) λid) N = W λ (ha) and in particular W λ is a subrepresentation of W. Thus, we have: W = λ W λ rep(q). (2.2.3) Since W is assumed to be indecomposable, it follows that for any φ End Q (Q) there exists λ K such that: W = W λ. (2.2.4) 10

18 Furthermore, either φ is nilpotent (if λ = 0) or φ is an isomorphism (if λ 0). In particular, any non-isomorphism is nilpotent. Claim: The set I = {φ End Q (W ) φ is nilpotent} forms a two-sided ideal. Let φ 1, φ 2 I and suppose φ 1 + φ 2 / I. Then φ 1 + φ 2 is invertible, so set α = φ 1 (φ 1 + φ 2 ) 1 and β = φ 2 (φ 1 + φ 2 ) 1. Since both φ 1 and φ 2 are nilpotent, there exists m N such that α m = β m = 0. Also, it is clear that α + β = Id W, so in particular they commute. Thus, we have: Id W = (α + β) 2m = 2m i=0 ( ) 2m α i β 2m i = 0. m Since this is a contradiction, it follows that φ 1 + φ 2 I i.e. I is closed under addition. Let φ I and f End Q (W ). If φ f or f φ is not in I, then φ is surjective or injective respectively. But since each component of φ is a linear map on vector spaces of equal dimension, either of these cases imply φ is an isomorphism, which is a contradiction. Thus I is closed under multiplication by elements of End Q (W ). Since I is a two sided ideal, Lemma 2 implies that End Q (W ) is local. In particular, rep(q) is a Krull-Schmidt category. 2.3 Exact sequences of quiver representations Definition 9: Let V, W Rep(Q), with V W. The quotient representation W/V is defined by: (W/V )(i) = W (i)/v (i), i Q 0 ; (W/V )(a) : w + V (ta) W (a)(w) + V (h(a), a Q 1. 11

19 Since V is a subrepresentation of W, we have W (a)(v (ta)) V (ha), thus (W/V )(a) is well-defined. Definition 10: Let Q be a quiver, V, W Rep(Q), and φ Hom Q (V, W ). Then: (a) the kernel of φ is defined as ker φ := (ker φ(i)) i Q0 ; (b) the image of φ is defined as im φ := (im φ(i)) i Q0 ; (c) the cokernel of φ is defined as coker(φ) := W/ im φ. Remark 3: For any a Q 1 we have: V (a)(ker φ(ta)) ker φ(ha); W (a)(im φ(ta)) im φ(ha). Thus by Definition 3, the image and the kernel of φ are both subrepresentations of V and W respectively, with the maps along each arrow given by the obvious restrictions. Definition 11: A sequence of morphisms of representations f 1 f 2 f n 1 V 1 V2... Vn 1 Vn is called exact if im f i = ker f i+1 for each i 1. A short exact sequence is an exact sequence of the form: 0 V f U g W 0 (2.3.1) hold: We call (2.3.1) a split short exact sequence if one of the following conditions 12

20 There exists s Hom Q (W, U) such that g s = Id W ; There exists r Hom Q (U, V ) such that r f = Id V. Note that if (2.3.1) splits then U = V W. And if U, V, W rep(q), then (2.3.1) splits if and only if U = V W. Definition 12: Let P, I Rep(Q) and consider the left exact functors Hom Q (P, ) and Hom Q (, I). We say P is projective if Hom Q (P, ) is right exact. We say I is injective if Hom Q (, I) is right exact. For any quiver Q, let Q op be the quiver with the same vertex set as Q, but with all arrows reversed. With this in mind, we introduce two important classes of representations. Definition 13: Let Q be an arbitary quiver (not necessarily acyclic) and let i Q 0. (a) We define P i rep(q), the projective representation at vertex i, as follows: for each j Q 0, P i (j) is the K-vector space with a basis consisting of all oriented paths p from i to j; for each a Q 1, P i (a) : P i (ta) P i (ha) sends a path p P i (ta) to ap P i (ha). (b) We define I i rep(q), the injective representation at vertex i, as follows: for each j Q 0, I i (j) := P op i (j) ; for each a Q 1, I i (a) := P op i (a op ). 13

21 (* denotes the dual space) Remark 4: i. For any V Rep(Q) and i Q 0, we have the canonical isomorphisms: Hom Q (P i, V ) = Hom Q (V, I i ) = V (i). ii. For each i Q 0, P i is indecomposable and projective and I i is indecomposable and injective. iii. If Q is acyclic, then {P i } i Q0 and {I i } i Q0 form a complete set of finitedimensional, pairwise non-isomorphic indecomposable projective and injective representations, respectively. 2.4 Path algebras Definition 14: Let Q be a finite quiver. The path algebra of Q, denoted KQ, is the K-algebra with basis consisting of all oriented paths in Q, include the trivial paths e i, i Q 0. Multiplication is defined by concatenation of paths. Example 5: Let Q = A 3 = 1 p 2 q 3. The set of paths is {e 1, e 2, e 3, p, q, qp}. In particular, KQ is isomorphic to the algebra of 3 3 upper triangular matrices. Example 6: Let Q be the Jordan quiver, i.e.: a Q = Then KQ = K[a], the polyonomial ring in one variable. 14

22 Proposition 2: The path algebra KQ is finite-dimensional if and only if Q has no oriented cycles. Definition 15: Let A be a K-algebra. We denote by Mod(A) the category of (left) A-modules, and by mod(a) the category of finite-dimensional (left) A-modules. Theorem 4: ([2], Corollary 1.7) There exists an equivalence of categories between Rep(Q) and Mod(KQ) that restricts to an equivalence of categories between rep(q) and mod(kq). Proof. We can explicitly construct the two natural transformations: Mod(KQ) F Rep(Q). F Let M be a KQ-module. We define F (M) as follows: i Q 0, F (M)(i) = e i M a Q 1, F (M)(a): e ta M e ha M, with x ax = e ha a x e ha M To define F, let M Rep(Q); say M = (M(i), M(a)) i Q0,a Q 1. Then: F (M) := i Q 0 M(i). Consider the following maps π i and τ i : F (M) = j Q 0 M j π i M(i) τ i F (M). 15

23 Now define the action of e i on F (M) by: e i m = (τ i π i )(m), m F (M). a If p = 1 a hp 2 a... n tp is an oriented path, define p m = τ hp M(a 1 )... M(a n ) π tp (m), m F (M). It can be shown that this action of KQ on F (M) defines a KQ-module structure on F (M) and in fact F, F give rise to an equivalence of categories. From this point on we identify mod(kq) and rep(q) and use the same symbol to denote a KQ-module and its corresponding Q-representation. Theorem 5 (Ringel s Projective Resolution): [17] Let Q be a quiver and M mod(kq). Then the sequence: 0 a Q 1 (P ha K M(ta)) f g (P i K M(i)) M 0 (2.4.1) i Q 0 is exact where f(p m) = pa m p am and g(p m) = pm. Moreover, this sequence is a projective resolution of M. An immediate consequence of Theorem 5 is that any V rep(q), has projective dimension at most one. Now, let W rep(q). We apply the functor Hom Q (, W ) 16

24 to obtaining: 0 Hom Q (V, W ) g Hom k (V (i), W (i)) f Hom k (V (ta), W (ha)) (2.4.2) i Q 0 a Q 1 Note that g (φ) = (φ(i)) i Q0 and f ((φ(i)) i Q0 ) = φ(ha) V (a) W (a) φ(ta). (So a tuple (φ(i)) i Q0 of linear maps φ(i): V (i) W (i), i Q 0, defines a morphism of representations precisely when it lies in ker(f ).) Definition 16: In the setup of (2.4.2), define Ext 1 Q(V, W ) := coker(f ). Thus, we have a long exact sequence 0 Hom Q (V, W ) g Hom K (V (i), W (i)) f Hom K (V (ta), W (ha)) Ext 1 Q(V, W ) 0 i Q 0 a Q 1 (2.4.3) Note that for any X rep(q), Theorem 5 implies Ext i Q(X, Y ) = 0 for all Y rep(q), i 2. Proposition 3: The category mod(kq) is hereditary, meaning that any submodule of a projective module is again projective. Proof. Let P be a projective module and X a submodule of P. Then we have the short exact sequence 0 X P P/X 0. Let Y be an arbitrary module, and apply the functor Hom Q (, Y ) to obtain the long exact sequence 17

25 0 Hom(P/X, Y ) Hom(P, Y ) Hom(X, Y ) Ext 1 (P/X, Y ) Ext 1 (P, Y ) Ext 1 (X, Y ) Ext 2 (P/X, Y )... But Ext 2 (P/X, Y ) = 0 and Ext 1 (P, Y ) = 0, so Ext 1 (X, Y ) = 0, thus X is projective. 2.5 Gabriel s Theorem One of the fundamental problems in representation theory is to classify the indecomposable representations of an algebra. While the following definitions are stated explicitely for path algebras of quivers, they may be stated more generally for any finite-dimensional algebra (see [22]). Definition 17: A quiver Q is called: a) representation-finite if there are only finitely many indecomposable representations of Q b) (infinite) tame if there are infinitely many indecomposable representations of Q, and all but finitely many occur in a finite number of one-parameter families. c) wild if there exists a representation embedding rep( ) rep(q). Since the representation theory of representation finite quivers and infinite-tame quivers is understood, it is common to refer to both types of quivers as tame quivers. Next we define and investigate some important combinatorial tools for Q that carry rich invariant and representation theoretic information. 18

26 Definition 18: Let Q be a quiver. (1) The Euler bilinear form of Q is the bilinear form, : Z Q 0 Z Q 0 Z, defined by (α, β) α, β = i Q 0 α(i)β(i) a Q 1 α(ta)β(ha) (2) The Tits quadratic form of Q, denoted q Q (or just q when Q is clear from context) is defined by q Q (α) = α, α. (3) The symmetric bilinear form of Q, (, ) : Z n Z n Z, is defined by (α, β) = α, β + β, α Remark 5: The symmetric bilinear form and the quadratic form depend only on Q, the underlying graph of Q. Indeed, let n ij = # of edges between i and j and ñ ij = 2n ii n ij i = j i j Then, for α, β Z Q 0, we have: (α, β) = 2 n α(i)β(i) i=1 q Q (α) = 1 n (α, α) = 2 i=1 1 i,j n α(i) 2 ñ ij α(i)β(j); 1 i j n n ij α(i)α(j). These formulas make it clear that q Q and (, ) depend only on the underlying graph Q of Q. 19

27 Lemma 3: Let V, W rep(q). Then dim V, dim W = dim K Hom Q (V, W ) dim K Ext 1 Q(V, W ). Proof. Let α = dim V and β = dim W. Then the claim follows immediately from (2.4.3) upon realizing that dim K Hom K (V (i), W (i)) = α(i)β(i) and i Q 0 i Q 0 dim K Hom k (V (ta), W (ha)) = α(ta)β(ha). Since the alternating sum of a Q 1 a Q 1 dimensions on exact sequences is zero, we have: dim K Hom Q (V, W ) dim K Ext 1 Q(V, W ) = i Q 0 α(i)β(i) a Q 1 α(ta)β(ha) = α, β Definition 19: a) The radical of the Tits quadratic form q is: rad(q) = {α Z n (α, β) = 0, β Z n } b) A dimension vector α Z n 0 is called sincere if α(i) > 0 for each i Q 0. c) For i Q 0, e i = dim S i, is called the simple root of Q at i. Lemma 4: Let Q be a connected quiver with underlying graph Q. Assume that there exists 0 δ Z 0 rad(q). Then: a) δ is sincere and q is positive semi-definite; 20

28 b) For α Z n, α rad(q) q(α) = 0 α = Qδ Z n. Proof. Let n = Q 0. Since δ rad(q) (δ, e i ) = 0 i Q 0 This happens if and only if 2δ(i) ñ ij δ(j) = 0, i Q 0. 1 j n (2 2n ii )δ(i) = j i n ij δ(j) 1 i n. (2.5.1) Assume for a contradiction that there exists 1 i 0 n such that δ(i 0 ) = 0. Then, using (2.5.1), we get: n i0 jδ(j) = 0 δ(j) = 0 if there exists a path between i 0 and j. j i 0 Since Q is connected, this implies that δ = 0 (contradiction). Thus δ is sincere. Next we show that q is positive definite. For any α Z n it can be shown that: q(α) = 1 i j n n ij δ(i)δ(j) 2 ( α(i) δ(i) α(j) ) 0 (2.5.2) δ(j) So if q(α) = 0, then it follows that α(i) δ(i) = α(j) for any vertices i and j with an δ(j) edge between them. But Q is connected, so the claim follows. Definition 20: A quiver Q is called: 21

29 1. Dynkin if Q is Dynkin of type A n, D n, E 6,7,8 : A n : n 1 n (n 1). n D n : n 2 n 1 (n 4 ) 4 E n : n 1 n (n = 6, 7, 8) 2. Euclidean if Q is Extended Dynkin of type Ãn, D n, Ẽ6,7,8; in this case, rad(q) = Zδ. Ã n : n 1 n (n 0); δ = D n : n 2 1 n 1 (n 4); δ = n Ẽ 6 : 4 δ = Ẽ 7 : δ =

30 Ẽ 8 : δ = Note that Ã0 is just the Jordan quiver in Example 6 and Ã1 has two vertices with two edges between them (called the Kronecker quiver if the arrows point in the same direction). The following theorem completely classifies the quivers of finite representation type, (infinite) tame representation type, and wild representation type. Part (i) was proven by Gabriel in [8] in A year later, Nazarova [15] had proven part (ii) of the theorem. Part (ii) was also proven around the same time by Donovan-Freislic in [7]. Theorem 6: Let Q be a quiver. Then the following statements hold. (i) Q is representation finite Q is Dynkin q is positive definite. If this is the case, then the map: isomorphism classes of indecomposable Q-representations is bijective. { α Z Q 0 0 q(α) = 1 }, defined by [V ] dim V, (ii) Q is (infinite) tame Q is Euclidean q is positive semi-definite. (iii) Q is wild Q is not Dynkin or Euclidean there exists α Z n 0, α 0, such that q(α) < 0 and (α, e i ) 0 for all i Q 0. 23

31 Remark 6: When Q is an Extended Dynkin quiver, we have rad(q) = δz, where δ is the unique imaginary Schur root of Q (see Definition 20). For each Euclidean quiver, there exists an infinite one-parameter family of pairwise non-isomorphic Schur representations of dimension δ. We will see in Chapter 6 that these representations are simple in the (sub) category of regular representations. 24

32 CHAPTER 3 Geometric Invariant Theory 3.1 Background on Invariant Theory Affine varieties Definition 21: Let n Z 1. The affine n-space is A n = K n, consisting of all n- tuples a = (a 1,..., a n ) with a i K for each i. The affine coordinate ring of A n is denoted by K[A n ] := K[x 1,..., x n ]. For any ideal I in K[A n ], define V(I) = {a X : f(a) =, f I}. This gives rise to the Zariski topology of A n, where the closed subsets of A n are of the form V(I) for some ideal I K[A n ]. Conversely, given any subset X A n, define I(X) = {f K[x 1,..., x n ] f(x) = 0, x X}. The famous Hilbert s Nullstellensatz tells us that: (1) I(V(a)) = a for any ideal a of K[x 1,..., x n ]; (2) V(I(X)) = X for any subset X of A n. Definition 22: Let X is be a topological space. a) X is irreducible if whenever X = Y 1 Y 2 with Y 1, Y 2 X (closed) then Y 1 = X or Y 2 = X. b) The (combinatorial) dimension of X is defined as: dim X := sup{n F 0 F 1... F n with each F i closed irreducible subset of X}. 25

33 c) For x X, the local dimension of X at x is: dim x X = inf{dim U x U X, with U open }. Definition 23: Let X be a topological space and Y X. a) Y is locally closed if Y = U W for some open set U X, and some closed subset W X. (This is equivalent to Y being open in its closure.) b) Y is constructible if Y is a finite union of locally closed subsets. Definition 24: Let X be a locally closed subset of A n. The algebra of regular functions on X, denoted O(X), is defined as the set of all functions φ : X K such that for all x X, there exists an open neighborhood U x and two functions f, g K[x 1,..., x n ] such that: g(u) 0 for all u U x ; φ = f Ux X g Ux X. Proposition 4: Let X be a closed subset of A n. Then: O(X) = {F X F K[x 1,..., x n ]} = K[x 1,..., x n ]/I(X). Definition 25: A quasi-affine variety is a locally closed subset of A n together with its Zariski topology, and the knowledge of O(U) for every open subset U X. Definition 26: Let X and Y be two quasi-affine varieties. A map φ : X Y is 26

34 called a morphism of varieties if f φ O(X) for all f O(Y ). Note that morphisms are continuous with respect to the Zariski topology. A morphism φ is called an isomorphism if it is bijective and φ 1 is also a morphism. Definition 27: An affine variety is a (quasi-affine) variety that is isomorphic to a closed (but not necessarily irreducible) subset of some affine space A n. For an affine variety X, we denote its algebra of regular functions O(X) by K[X]. Remark 7: Let X be an affine variety. 1. X is irreducible if and only if I(X) is a prime ideal if and only if O(X) is an integral domain. 2. When X is irreducible, dim X is equal to the Krull dimension of O(X) Linear Algebraic Groups Recall that GL n (K) = {A Mat n n (K) det A 0} is a principle open subset of A n2 = Mat n n (K). However, GL n (K) can also be viewed as {(A, λ) A n2 +1 det A λ = 1}, a closed subset of A n2 +1. Thus GL n (K) is an affine variety with coordinate ring { f O(GL n (K)) = det l GLn(K) } l 0, f K[x i,j : 1 i, j n]. Definition 28: A linear algebraic group is a closed subgroup of some ambient gen- 27

35 eral linear group GL n (K). Definition 29: a) Let G be a linear algebraic group, and X a variety. We call X a G-variety if X is equipped with a group action α : G X X that is also a morphism of varieties. b) Given Y X, with X a G-variety, we say Y is G-stable if gy Y for all g G. The following proposition lays out some basic objects and their properties that will be of interest to us. Proposition 5: [14] Let G be a linear algebraic group, X a G-variety, Y X, and x X. Then: (i) X G := {x X gx = x, g G} is closed; (ii) Stab G (Y ) := {g G gy = y, y Y } is a closed subgroup of G; (iii) Gx := {gx g G}, the G orbit of x, is a locally closed subset of X; moreover, the irreducible components of Gx have the same dimension; (iv) if Y is G-stable, so is Y ; (v) dim G = dim Stab G (x) + dim Gx; (vi) for any y Gx Gx, we have dim Gy < dim Gx. In other words, (Gx) is a union of orbits of smaller dimension; (vii) The orbit closure Gx contains a closed orbit. 28

36 3.1.3 Hilbert s Fourteenth Problem & Affine Quotient Varieties In order to define quotient varieties (in the affine case), we need to work with linearly reductive groups which we now recall: Definition 30: Let G be a linear algebraic group. 1. A G-module (or a linear representation of G) is a pair (V, ρ) where V is a finite dimensional vector space and ρ : G GL(V ) is a homomorphism of groups. 2. A finite-dimensional rational G-module is a G-module (V, ρ) with ρ : G GL(V ) a morphism of algebraic groups. 3. A locally finite-dimensional rational G-module is a (possibly infinitedimensional) G-module (V, ρ) with the property that for any v V there exists a finite-dimensional G-invariant subspace W K V such that v W and G GL(W ), g ρ(g) W, g G is a morphism of algebraic groups. 4. A G-invariant vector subspace of a rational G-module V is a called a G- submodule of V. For convenience, whenever working with a rational G-module, we only specify its underlying vector space. If V and W are two rational G-modules, a linear map φ: V W is called a G-homomorphism if φ is G-equivariant, i.e. φ(gv) = gφ(v) for all v V. We will denote the set of all G-homomorphisms by Hom G (V, W ). Definition 31: Let G be a linear algebraic group and V a rational G-module. 29

37 (a) V is called simple if V 0 and the only G-invariant subspaces of V are {0} and V. (b) V is called semi-simple if V has a decomposition V = V 1 V 2... V m into simple G-subrepresentations V i G V. Definition 32: A linear algebraic group G is called linearly reductive if every finite-dimensional rational G-module is semi-simple. Proposition 6: [16] (i) The groups GL(n) and SL(n) linearly reductive. (ii) Products of linearly reductive groups are linearly reductive. (iii) If G is linearly reductive and H G, then G/H is also linearly reductive. The following theorem, classically known as Hilbert s Finiteness Theorem, is one of the most fundamental results in Invariant Theory. This is problem fourteen in Hilbert s famous list of twenty-three problems announced at the International Congress of Mathematicians at Paris in Theorem 7: [16] Let G be a linearly reductive group and X a G-variety. Then K[X] G is a finitely generated K-algebra. This theorem tells us that K[X] G is an affine K-algebra and hence there exists an affine variety, denoted by X//G, such that K[X//G] = K[X] G. 30

38 Moreover, the inclusion K[X] G K[X] corresponds to a dominant morphism of varieties π : X X//G. We call (X//G, π), or simply X//G, the affine quotient variety of X by the action of G. We are now ready to list the key features of affine quotient varieties: Theorem 8: Let G be a linearly reductive algebraic group, and π X : X X//G the algebraic quotient. Then the following hold: (i) π is a G-invariant, surjective homomorphism; (ii) if Y X is closed and G-invariant, then π(y ) is closed in X//G; (iii) if Z 1, Z 2 are two disjoint, closed, G-invariant subsets of X, then π(z 1 ) π(z 2 ) = ; (iv) each fiber of X//G contains a unique closed orbit; (v) if x 1, x 2 X, then π(x 1 ) = π(x 2 ) if and only if Gx 1 Gx 2 ; (vi) if U X//G is open, then π : O(U) O(π 1 (U)) G is an isomorphism of algebras. Remark 8: Theorem 8(iv) tells us that the points of X//G are in one-to-one correspondence with the closed G orbits in X. This will be important later on. Definition 33: Let G be a linear algebraic group. 31

39 a) A character of G is a homomorphism of algebraic groups χ : G K that is also a morphism of varieties. For a character χ, let G χ = ker(χ). b) A one-parameter subgroup (1-psg) of G is a homomorphism of algebraic groups λ : K G that is also a morphism of varieties. Throughout, we use X (G) to indicate the characters of G and X (G) the oneparameter subgroups of G. Remark 9: Given χ X (GL(β)), λ X (GL(β)), we have: χ λ: K K. Thus χ λ = t m for some m Z. Throughout, we will denote this m by (χ, λ) = m. Now, let X be an affine G-variety, λ X (G), and x X. Define the regular morphism λ x : K X by t λ(t) x If X K l, then we can write λ x (t) = (p 1 (t),..., p l (t)) with each p i (t) a regular function on K, i.e. p i (t) K[t, t 1 ]. With this in mind, we define the following limit: lim λ(t)x = t 0 (p 1 (0),... p l (0)) if p i K[t], 1 i l otherwise The following two results establish important connections between 1-parameter subgroups and closed orbits. Theorem 10 will play a fundamental role in Section

40 Theorem 9: [16] Let G be a linearly reductive group, V a finite-dimensional G- module and v V. Then, the following hold: 1. Gv contains a unique closed G-orbit, call it C. 2. There exists a 1-parameter subgroup λ X (G) such that lim t 0 λ(t)v C. Theorem 10: Let G be a linearly reductive group and V a rational G-module with v V. Then the following are equivalent: 1. 0 Gv 2. (Hilbert-Humford Criterion) There exists a 1-psg λ X (G) such that lim t 0 λ(t)v = For all homogenous f K[V ] G with positive degree we have f(v) = Semi-invariants for quivers The Weight Space Decomposition Let G be a linear algebraic group, V a rational G-module, and χ X (G). The weight space of semi-invariants of weight χ is: SI(V, G) χ = {v V g v = χ(g)v, g G} Theorem 11 (Decomposition Theorem in Weight Spaces): Let G be a linearly reductive group and V be a locally finite-dimensional rational representation of G. 33

41 Then: V [G,G] = SI(V, G) χ, χ X (G) where [G, G] is the commutator subgroup of G. Proof. Since K is abelian, for any χ X (G) we have: χ([g, h]) = χ(ghg 1 h 1 ) = χ(g)χ(h)χ(g) 1 χ(h) 1 = 1. So χ ([G, G]) = {1} and hence any v SI(V, G) χ is [G, G] invariant. This shows that χ X (G) SI(V, G) χ V [G,G] Next, let v V [G,G] and W V a finite-dimensional rational G -submodule of V such that v W. Then v W [G,G], and W [G,G] is a rational G/[G, G]-module. But since G is linearly reductive and [G, G] G, it follows from Proposition 6 that G/[G, G] is linearly reductive. Thus W [G,G] = n i=1 W i with each W i a simple G/[G, G]-module. But since G/[G, G] is an abelian group and K is algebraically closed, each W i is onedimensional; hence, we can write each W i = w i K. Note that for each g G, there exists a g K such that g w i = a g w i w i K. Define χ i : G K by χ(g) = a g. Then χ i X (G), and we have W [G,G] n SI(V, G) χi SI(V, G) χ. i=1 χ X (G) 34

42 3.2.2 The Space of β-dimensional Quiver Representations So far we have worked in the general case of a linearly reductive group acting on an affine variety. Now we return our attention to quivers and their representations. Let Q be an arbitrary quiver and β Z Q 0 0 a dimension vector of Q. The representation space of β-dimensional Q-representations is the affine space rep(q, β): = The change of base group a Q 1 Hom K ( K β(ta), K β(ha) ) = GL(β) := i Q 0 GL(β(i), K) a Q 1 Mat β(ha) β(ta) (K) acts on rep(q, β by simultaneous conjugation. In other words, for any g = (g(i)) i Q0 GL(β) and any V = (V (a)) a Q1 rep(q, β): (g V )(a) = g(ha) V (a) g 1 (ta), a Q 1 Example 7: Let Q = A 3 := and β = (2, 3, 2). Consider the representation V = K 2 [ ] K 3 [ ] K 2 rep(q, β) and g = ( [ ] ) [ ], 0 2 2, [ ] GL(β). Then 35

43 g V = K 2 [ ] [ ] [ ] 1 K 3 [ ] [ ] [ ] K 2 i.e. g V = K 2 [ ] [ K ] K 2 Remark 10: For any g GL(β) and V rep(q, β), we have gv = V, though gv and V in are (in general) different points in the affine variety rep(q, β). There is a bijective correspondence between the GL(β)-orbits in rep(q, β) and the isomorphism classes of β-dimensional representations of Q. In particular, two β-dimensional representations are isomorphic if and only if they are in the same orbit. Proposition 7: Let V rep(q, β). Then: a) dim GL(β) = dim GL(β)V + dim K End Q (V ); b) codim(gl(β)v ) = dim Ext 1 Q(V, V ). Proof. (a) We have: Stab GL(β) (V ) = {φ = (φ i ) i Q0 GL(β) φ V = V } = {φ GL(β) φ(ha)v (a) = V (a)φ(ta), a Q 1 } = {φ Hom Q (V, V ) det φ(i) 0, i Q 0 } End Q (V ) Note that the last condition is an open condition, and in general if U X, with X irreducible and U open, then dim U = dim X. So the first claim follows from Proposition 5(5). 36

44 ( ) (b) By definition, codim GL(β)V rep(q, β) = dim rep(q, β) dim GL(β)V. But dim rep(q, β) = a Q 1 β(ta)β(ha) and by part (a), dim GL(β)V = dim GL(β) dim K End Q (V ). So we have: dim rep(q, β) dim GL(β)V = a Q 1 β(ta)β(ha) i Q 0 β(i) 2 + dim K End Q (V ) = q(β) + dim K End Q (V ) = (dim K End Q (V ) dim K Ext 1 (V, V )) + dim K End Q (V, V ) = dim K Ext 1 Q(V, V ). Recall that the action of GL(β) on K[rep(Q, β)] is given by: (g f)(v ) = f(g 1 V ), V rep(q, β), f K[rep(Q, β)], g GL(β). We are now ready to define the main objects of study in quiver invariant theory: Definition 34: a) The ring of invariants on rep(q, β) is I(Q, β) := K[rep(Q, β)] GL(β) = {f K[rep(Q, β)] g f = f, g GL(β)}. 37

45 b) The ring of semi-invariants on rep(q, β) is SI(Q, β) := K[rep(Q, β)] SL(β). Theorem 12: Let Q be a quiver, β Z Q 0 0, and V, W rep(q, β). Then the following are equivalent: 1. There exists λ X (GL(β)) such that lim t 0 λ(t)v = W exists. 2. There exists a finite filtration of sub-representations of V, F.(V ) : 0 = F m+1 (V ) F m (V )... F 0 (V ) = V such that the associated graded representation gr F.(V ) := m F i (V )/F i+1 (V ) = W. i=0 Proof. (1 2) Let λ X (GL(β)). For each i Q 0, K acts on V (i) = K β(i) via λ. In other words, given t K and v V (i) = K β(i), we have t v = λ(t)(i) v. Then V (i) has a weight space decomposition with respect to this action, V (i) = k Z V k (i), where V k (i) = {v V (i) t v = t k v, t K } Note that the action on the left is the action induced by λ, and the action on the 38

46 right is just the scalar multiplication. Now, for an arrow a Q 1, we can view V (a) as a β(ha) β(ta) matrix over K whose (l, k)th entry is the linear map V k,l (a) : V k (ta) τ k V (ta) V (a) V (ha) π l Vl (ha) Recall that if G acts on V and W, then G acts on Hom Q (V, W ) by (g f)(v) = g f(g 1 v). So in this case we have (λ(t) V k,l )(a)(v) = t l V k,l (a)(t k v) = t l k V k,l (a)(v) Putting everything together we have: lim λ(t)v t 0 exists lim t 0 λ(t)v lim t 0 t l k V k,l (a) V k,l (a) = 0 exists component-wise exists component wise for all k > l Note that V k,l (a) ( = 0 for all ) k > l implies V (a) is a lower triangular matrix. So in particular, V (a) V k (ta) V k (ha) for all k > l. But this happens if and k m k m ( ) only if V k (i) is a subrepresentation of V, for each m. k m i Q 0 ( ) Defining F m (V ) = V k (i) we have shown that lim t 0 λ(t)v exists if k m i Q 0 and only if F m (V ) is a subrepresentation of V, for all m Z. When this is the case, lim t 0 λ(t)v = gr F.(V ). The other direction follows from a similar construction of a one-parameter sub- 39

47 group. Lemma 5: If M mod(kq) is a semi-simple module, then M = gr F.(M) for any finite filtration F.(M) of M. Proof. We proceed by induction on the length of the filtration. If n = 0, there is nothing to prove. If n = 1, then our filtration is 0 = F 2 F 1 M. So gr F.(M) = M/F 1 F1 = M (since M is semi-simple, every submodule of M has a complement). Now suppose our claim holds for filtrations of length k. If we have a filtration of length k + 1, then the induced filtration on F 1 is of length k, so by induction hypothesis gr F.(F1 ) = F 1. Furthermore, gr F.(M) = grf.(f1 ) M/F1 = F 1 M/F1 = M. The next result shows how representation theoretical conditions can describe geometric conditions. Theorem 13: Let Q be an arbitrary quiver, β a dimension vector of Q, V rep(q, β). Then GL(β)(V ) = GL(β)(V ) in rep(q, β) if and only if V is a semisimple representation of Q. Proof. ( ) Consider a Jordan-Holder filtration of V, F.(V ) : 0 = F m+1 V F m (V )... F 0 (V ) = V By Theorem 12(1), there exists a 1-psg λ X (GL(β)) such that lim t 0 λ(t)v = gr F.(V ). Note that λ(t)v GL(β)V and lim t 0 λ(t)v GL(β)V. Thus gr F.(V ) 40

48 GL(β)V = GL(β)V, i.e. V = gr F.(V ). And since gr F.(V ) is semi-simple by construction, it follows that V is semi-simple. ( ) Let C be the unique closed orbit contained in GL(β)V. It suffices to show C = GL(β)V. By Theorem 9, there exists a 1-psg λ X (GL(β)) such that lim t 0 λ(t)v C. Using Theorem 12, we know that there exists a filtration F.(V ) on V such that gr F.(V ) C. Since V = gr F.(V ) by Lemma 5, we have V C. Thus GL(β)V = GL(β)V. Corollary 1: Let Q be an acyclic quiver and β a dimension vector. Then S := i Q 0 S β(i) i is the unique representation in rep(q, β) whose GL(β)-orbit is closed. Proof. Since Q is acyclic, by Proposition 1, the simple representations of Q are precisely the S i. Hence the only semi-simple β-dimensional representation is S. Remark 11: The corollary above combined with Theorem 13 implies that K[rep(Q, β)] GL(β) = K whenever Q is acyclic. On the other hand, in the acyclic case, the SL(β) action on rep(q, β) leads to very interesting algebras of semi-invariants. The rich combinatorial and geometric structure of these algebras of semi-invariants has found important applications in various seemingly unrelated areas, such Littlewood- Richardson coefficients and eigenvalue problems for Hermitian matrices (see [9]). It follows from Theorem 11 that SI(Q, β) has a weight space decomposition, namely: SI(Q, β) = SI(Q, β) χ χ X (GL(β)) 41

49 with SI(Q, β) χ = {f k[rep Q, β] g f = χ(g)f, g GL(β)}. Note that any θ Z Q 0 defines a rational character χ θ : GL(β) K by χ θ ((g(x)) x Q0 ) = x Q 0 det g(x) θ(x). (3.2.1) In this way, we get a natural epimorphism Z Q 0 X (GL(β)). We refer to the rational characters of GL(β) as integral weights of Q. In case β is a sincere dimension vector, this epimorphism is an isomorphism which allows us to identify Z Q 0 with X (GL(β)). From now on, for any θ Z Q 0, we set: SI(Q, β) θ ={f k[rep Q, β] g f = χ θ (g) f g GL(β)} ( ) ={f k[rep Q, β] g f = det (g(i)) θ(i) f, g GL(β)}. i Q 0 Lemma 6: Assume SI(Q, β) θ 0. Then θ(β) = 0. Proof. Let f SI(Q, β) θ be a non-zero semi-invariant and let V 0 rep(q, β) such that f(v 0 ) 0. It is easy to see that f(v 0 ) = i Q 0 det ( tid (β(i)) ) θ(i) f(v0 ) for all t K. This implies that: 1 = t i Q 0 β(i)θ(i), t K which is equivalent to θ(β) = i Q 0 β(i)θ(i) = 0, (3.2.2) 42

50 giving us a linear homogeneous restraint on θ. Recall that: { } GL(β) θ = ker(χ θ ) = (g(i)) i Q0 GL(β) det (g(i)) θ(i) = 1 i Q 0 From now on, throughout the remaining of the thesis, we assume that our quivers are acyclic. Lemma 7: Let θ be a weight such that the corresponding rational character χ θ X (GL(β)) is not trivial. If there exists N 1 such that SI(Q, β) Nθ 0, then K[rep(Q, β)] GL(β) θ = n 0 SI(Q, β) nθ. Proof. Given f SI(Q, β) nθ, for any n Z, we have g f = (nθ)(g) f = ( i Q 0 det (g(i)) θ(i) ) n f = f, g GL(β) θ. So, n 0 SI(Q, β) nθ K[rep(Q, β)] GL(β) θ. For the other inclusion, let f K[rep(Q, β)] GL(β) θ K[rep(Q, β)] SL(β) = SI(Q, β) τ. τ X (GL(β)) So write f = τ f τ with f τ SI(Q, β) τ. Now, assume for a contradiction that there exists τ 0 such that f τ0 0 and τ 0 is not an integer multiple of θ. In other words, 43

51 there exists g 0 ker χ θ such that τ 0 (g 0 ) 1 (see Remark 12). We have f τ = f = g 0 f = g 0 f τ = τ τ τ τ(g 0 ) f τ So f τ0 = τ 0 (g 0 )f τ0. But f τ0 0, thus 1 = τ 0 (g 0 ), a contradiction. This shows K[rep(Q, β)] GL(β) θ n Z SI(Q, β) nθ, and hence K[rep(Q, β)] GL(β) θ n Z SI(Q, β) nθ. Claim: SI(Q, β) nθ = 0 for all n N. Since there exists N 1 such that SI(Q, β) Nθ 0, so choose 0 f SI(Q, β) Nθ. Assume there exists n Z 0 such that there exists 0 µ SI(Q, β) nθ. Then by Remark 11 we get that: µ N f n SI(Q, β) 0 = I(Q, β) = K. So µ is a constant polynomial. But the only constant semi-invariant polynomial of weight nθ is the zero polynomial. Thus µ = 0 and this proves the claim. It now follows from the claim above that K[rep(Q, β)] GL(β) θ has the desired weight space decomposition. Remark 12: In general, for two characters χ 1, χ 2 GL(β) θ, χ 2 = χ m 1 ker χ 1 ker χ 2 44

52 3.2.3 Stability Weights for Quiver Representations Let Q be a quiver, β a dimension vector, and θ an integral weight of Q. Recall that GL(β) θ = ker(χ θ ) GL(β). Definition 35: Let W rep(q, β). a) We say that W is θ-semi-stable if there exist n Z 1 and f SI(Q, β) nθ such that f(w ) 0. b) We say that W is θ-stable if W is θ-semi-stable, and GL(β) θ W is a closed orbit of dimension dim GL(β) 2. Remark 13: Let W rep(q, β)be a θ-semi-stable representation; in particular, θ(β) = 0 by Lemma 6. Recall that λ K { (λidβ(i) } )i Q0, also denoted by K, acts trivially on rep(q, β) and K ker(χ θ ) because θ(β) = 0. Therefore G θ = ker(χ θ )/K is well-defined, and furthermore has dimension GL(β) 2 (assuming that χ θ is not the trivial character). These remarks show that a θ-semi-stable representation W rep(q, β) becomes θ-stable if, in addition, the G θ -orbit of W is closed of dimension dim G θ (assuming that χ θ is not trivial). The following two results (Theorems 14 and 15) provide a numerical description of semi-stability and stability for quiver representations. Theorem 14 (King s Criterion for semi-stability): For W rep(q, β), the following are equivalent: i. W is θ-semi-stable. 45

53 ii. θ(dim W ) = 0 and θ(dim W ) 0 for every subrepresentation W W ; iii. θ(β) = 0, and for every 1-PSG λ : K GL(β) for which lim t 0 λ(t) W exists we have (θ, λ) 0. Proof. Without loss of generality, assume β is sincere. (3 2) Let W W be a subrepresentation and let us consider the 1-psg induced by W. Specifically, for each i Q 0, fix a decomposition of W (i) = W (i) W (i) and for each a Q 1, write W (a) = W (a) X(a) 0 (W/W ) (a) For each t K, define λ(t) as follows: For each i Q 0, set λ(t)(i) = t Id W (i) 0 0 Id W (i) GL(β)(i) It is clear that λ X (GL(β)); moreover, for each t K, we have: (λ(t) W ) (a) = t Id W (ha) 0 0 Id W (ha) = W (a) tx(a) 0 W/W (a) W (a) t 1 Id W (ta) 0 0 Id W (ta) Hence lim t 0 λ(t)w = W W/W. In particular, this limit exists, and so by 46

54 assumption (θ, λ) 0. Setting β = dim W, we have: (θ, λ)(t) = det (λ(t)(i)) θ(i) = i Q 0 i Q 0 det tid β (i) 0 0 Id (β β )(i) θ(i) So we have (θ, λ)(t) = t i β (i)θ(i) which shows that (θ, λ) = i β θ(i) = θ(dim W ) 0. (2 3) Given λ X (GL(β)) such that lim t 0 λ(t)w exists, we need to show (θ, λ) 0. But since the limit exists, by Theorem 12 there exists a finite filtration of subrepresentation of W whose associated grade module is isomorphic to lim t 0 λ(t)w. Let us construct the filtration. At each vertex i we have a weight space decomposition: W (i) = k Z W k (i) with W k (i) = {w W (i) t w = t k w, t K } As previously shown, the collection of subspaces F m (W )(i) = W k (i), i k m Q 0, forms a subrepresentation, F m (W ) of W, and {F m (W )} m Z is a filtration whose associated graded module is isomorphic to the limit in question. Fix N such that W m (i) = 0 for all m N, i Q 0. Then (θ, λ)(t) := i Q 0 det t N+1 Id WN+1 (i) 0 0 t N+2 Id WN+2 (i)... θ(i) 47

55 = t m N+1 m(dim Wm(i))θ(i) i Q 0 =t i Q 0 m N+1 θ(i) m dim Wm(i) Thus: (θ, λ) = i Q 0 θ(i) m dim W m (i) m N+1 = θ(i) i Q 0 N dim W N+1 (i)+ dim W N+1 (i) +N dim W N+2 (i)+ dim W N+2 (i)+ dim W N+2 (i) +N dim W N+3 (i)+ dim W N+3 (i)+ dim W N+3 (i)+ dim W N+3 (i) = i Q 0 θ(i) (N dim W i + dim F N+1 (W )(i) + dim F N+2 (W )(i)...) =N (θ(i)β(i) + i Q 0 =N θ(β) + m N+1 m N+1 =0 + θ(dim F m (W ) 0. θ(dim F m (W ) i Q 0 θ(i) dim F m (i) Therefore (θ, λ) 0. (1 3) Let λ X (GL(β)) such that lim t 0 λ(t)w = W 0 rep(q, β). Choose f SI(Q, β) nθ, with n 1, such that f(w ) 0. Then lim f (λ(t)w ) = f(w 0 ). t 0 48