Asymptotic Degeneration of Representations of Quivers V. Strassen University of Konstanz January 5, 999 Abstract We dene asymptotic degeneration of ni

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1 Universitat Konstanz Asymptotic Degeneration of Representations of Quivers Volker Strassen Konstanzer Schriften in athematik und Informatik Nr. 77, November 998 ISSN 430{3558 c Fakultat fur athematik und Informatik Universitat Konstanz Fach D 88, Konstanz, Germany preprints@informatik.uni{konstanz.de WWW:

2 Asymptotic Degeneration of Representations of Quivers V. Strassen University of Konstanz January 5, 999 Abstract We dene asymptotic degeneration of nilpotent representations of an arbitrary nite quiver, using large tensor powers and small direct sums, and characterize this notion by a simple and eective criterion. Introduction We start with a nite quiver Q and study degenerations of Q-modules, i.e. of nite dimensional representations of Q over C, in the sense of orbit closure. (See [9], [3], [7], [2]; [], [4] for denitions.) We write U V to express the fact that U is a degeneration of V. Building on work of Abeasis and Del Fra [] and Riedtmann [4], Bongartz [4], [5] has given practicable necessary and sucient conditions for degeneration when Q is a tame quiver, i.e., when the underlying undirected graph is an extended Dynkin diagram. There seems to be little hope for results of a similar quality in the case of wild quivers (in spite of Zwara's recent result [6]). Let Q be a nite quiver, Q 0 its sets of points and Q its set of arrows. A Q-module T is called trivial, when T (l) = 0 for all l 2 Q. We call Q-modules U and V equivalent, when there exist trivial modules S and T such that U S ' V T. The set 2 (Q) of equivalence classes of Q-modules is a commutative semiring with unit element, where addition and multiplication are induced by direct sum and tensor product of modules. (The tensor product of U and V is dened by (U V )() := U() V () for 2 Q 0 and (U V )(l) := U(l) V (l) for l 2 Q.) Degeneration of modules induces a partial order in (Q): p q :() 9U 2 p; V 2 q U V: This partial order is compatible with addition and multiplication in the sense that p q implies p + r q + r and pr q r for any p; q; r 2 (Q). In short: (Q) is an ordered commutative semiring with unit element. 3 A Q-module V is nilpotent, when there exists a positive integer such that V (w) = 0 for all paths w of length at least. This is preserved under equivalence. Every Q-module is nilpotent unless Q contains directed cycles. The trivial modules are exactly the semisimple

3 nilpotent modules. Submodules, homomorphic images, direct sums and tensor products of nilpotent modules are again nilpotent. In fact, the tensor product of a nilpotent module with an arbitrary Q-module is nilpotent. Therefore the set N (Q) of equivalence classes of nilpotent Q-modules is an ideal of the semiring (Q). In particular N (Q) is an ordered commutative semiring, possibly lacking a unit element. An important additional property of N (Q) is that we have 0 q for all q 2 N (Q). 4 First example: Consider the Jordan quiver Q consisting of a single point together with a single loop l. A nilpotent Q-module V consists of a nite dimensional vector space V () together with a nilpotent linear endomorphimsm V (l) of V (). U is a degeneration of V when the closure of the conjugacy class of V (l) contains an endomorphism isomorphic to U(l). The semiring N (Q) can be identied with the semiring of functions f: Z >0! Z 0 which have nite support and are convex in the sense that f(j?)+f(j +)?2f(j) 0 holds for all j >. The isomorphism is given by [V ] 7! (j 7! R(V (l) j )), where [V ] denotes the equivalence class of V and R denotes the rank function for linear maps between nite dimensional vector spaces. 5 a result of Gerstenhaber [] (see also Kraft{Procesi [3]) the partial order in N (Q) becomes, under the above identication, the pointwise order of functions. Second example: Let Q be the quiver with Q 0 := f; : : : ; ng and Q := f(i; i+) : i < ng, i.e., the quiver of type A n with all arrows pointing in the same direction. Then N (Q) = (Q) may be identied with the semiring of interval functions f: f(i; j) : i < j ng! Z 0, satisfying f(i? ; j) f(i; j), f(i; j + ) f(i; j) and f(i? ; j) + f(i; j + ) f(i; j) + f(i? ; j + ), whenever these inequalities are meaningful. By The isomorphism is given by [V ] 7! ((i; j) 7! R(V (w i;j ))), where w i;j denotes the unique directed path from i to j. By a result of Abeasis and Del Fra [] the partial order in N (Q) becomes, under the given identication, the pointwise order of interval functions. Unfortunately, such neat descriptions of the ordered semiring N (Q) seem to be rare. In the spirit of [5] we therefore introduce asymptotic degeneration : For p; q 2 N (Q) we set p q :() 9k 2 N 8N pn N k q N ; () where `8N ' is a shorthand for `9N 0 8N N 0 '. It is easy to see that is a preorder in N (Q) compatible with multiplication. It is also compatible with addition, as follows, e.g., from Theorem below. Of course implies. In the two examples above and coincide, as is always the case when (N (Q); ) is isomorphic to a semiring of functions with pointwise order. The following result shows that asymptotic degeneration can be characterized in a simple way. Recall that R denotes the rank function for linear maps. By a path in Q we always mean a directed path. q if and only if the Theorem Let Q be a quiver, p; q 2 N (Q) and U 2 p, V 2 q. Then p following two conditions are satised:. R(U(w)) R(V (w)) for any path w of positive length. 2. U(v) = U(w) whenever 2 C and v; w are paths of positive length with the same starting point and the same end point and such that V (v) = V (w). Theorem will be proved in Section 3. (The necessity of the conditions is easy to see.) 2

4 Let us call a quiver Q economical, if for any ; 2 Q 0 there is at most one path from to. Such a quiver does not contain directed cycles, hence N (Q) = (Q). Take an economical quiver Q. We dene asymptotic equivalence of classes p; q 2 (Q) by p q :() p q & q p: Then is a congruence relation in (Q) and therefore (Q)= is an ordered commutative semiring with unit element. Let (Q) be the (nite) set of paths of Q of positive length and let F((Q); N) be the semiring of nonnegative integral valued functions on (Q). We have a map ': (Q)=?! F((Q); N); which sends the asymptotic equivalence class of [V ] 2 (Q) to the function w 7! R(V (w)). Corollary Let Q be economical. Then ' is an imbedding of ordered semirings. Borrowing the language of [5], we may call (Q) an asymptotic spectrum of (Q). 6 Corollary 2 Let Q be economical. Then the following conditions are equivalent:. and coincide. 2. The ordered semiring ((Q); ) is isomorphic to a semiring of functions with pointwise order. An interesting problem consists in characterizing the image of ' in Corollary in a way similar to its characterization in the special case of example 2. The quiver of example is not economical. Nevertheless, Corollaries and 2 hold in this case. This indicates that we have not stated the corollaries in their most general form. 2 A Generalization Let A be a nite dimensional associative C -algebra with a multiplicative basis. This means that A has a distinguished basis B such that the product of any two elements of B is either 0 or again an element of B and such that rad A = lin B for some B B. Note that the unit element of A does not necessarily belong to B. By an A-module we mean a nite dimensional complex vector space together with a multiplicative linear map A! End C V, a 7! a V. We do not assume V = id V, i.e. we allow nonunital modules. Clearly V is a projection and a module endomorphism and therefore produces a direct decomposition of V into the unital submodule V u := im( V ) and the submodule V 0 := ker( V ), which is annihilated by A. This decomposition is functorial. We may and will identify an A-module structure on the vector space V with its restriction to the basis B. Then such a structure is given by a map B! End C V, b 7! b V such that b V c V = (b c) V whenever b c 2 B, and b V c V = 0 otherwise. This map will be called the representation map of the module. Besides the direct sum of two modules U and V we have their tensor product U V, dened by the representation map b 7! b U b V. Note that 2 A need not be represented by 3

5 U V. Thus the tensor product of two unital modules may be nonunital. 7 A module V is called semisimple when the unital submodule V u is semisimple in the usual sense. This is the case if and only if the representation map of V vanishes on B. We call A-modules U and V equivalent, when there exist semisimple modules S and T such that U S ' V T. The set (A) of equivalence classes of A-modules is a commutative semiring (possibly without unit element), where addition and multiplication are induced by direct sum and tensor product, respectively. Degeneration of modules is dened using the closure of the conjugacy class of the representation map (see [2] II.3, [5], [6]). We write U V to indicate that U is a degeneration of V. For p; q 2 (A) we dene p q :() 9U 2 p; V 2 q U V: Then is a partial order 8 in (A) compatible with addition and multiplication and such that 0 p for all p 2 (A) 9. Asymptotic degeneration is dened by p q :() 9k 2 N 8N pn N k q N : (2) It is easy to see that is a preorder in (A) compatible with multiplication. It is also compatible with addition, as follows from Theorem 2 below. We are going to characterize in a simple and eective way. Before doing this we consider yet another order: Let U; V be A-modules. We say that U is a restriction of V, and write U V, when U is a quotient of a submodule of V. We dene restriction and asymptotic restriction in (A) by and p q :() 9U 2 p; V 2 q U V p < q :() 9k 2 N 8N pn N k q N ; (3) respectively. It is easy to see that is a preorder in (A), compatible with addition and multiplication, and that implies 0. Hence is a partial order. It follows that < is a preorder in (A) compatible with multiplication (and by Theorem 2 also with addition), and that < implies. Theorem 2 Let A ba a nite dimensional C -algebra with a multiplicative basis B such that rad A = lin B, where B B. Take p; q 2 (A) and U 2 p, V 2 q. The following three conditions are equivalent: p < q; (4) p q; (5) 8b 2 B R(b U ) R(b V ) 8b; c 2 B ; 2 C b V = c V =) b U = c U : (6) 4

6 Theorem 2 will be proved in Section 3. We remark that the theorem also holds when the right hand sides of (2) and (3) are replaced by the apparently weaker conditions and 8" > 0 8N p N 2 b"n c q N 8" > 0 8N p N 2 b"n c q N ; respectively. (This will follow from the proof of Theorem 2.) Hence these relaxations have no eect on the resulting notions of asymptotic degeneration and asymptotic restriction. 3 Proofs For p; q 2 (A) and U 2 p; V 2 q we have and p q () 9k 8N 9S; T S U N T p < q () 9k 8N 9T U N T N k N k V N ; (7) V N ; (8) where it is understood that k; N are positive integers and S; T semisimple modules. First we prove three lemmas. Lemma Let A be a nite dimensional C -algebra and let U,V be A-modules. Then the following conditions are equivalent: 9r 2 N; T semisimple U T r V; (9) ann V \ rad A ann U: (0) When the conditions are satised one may choose r dim U dim A. The estimate of r is easily improved, but we will not need this. Proof: Suppose that (9) holds. Since rad A ann T and since the annihilator increases when a module is being replaced by a submodule or a quotient we have ann V \ rad A = ann(t r V ) \ rad A ann U \ rad A ann U: Conversely, suppose that (0) holds. Take a basis (x ; : : : ; x m ) of U and elements y ; : : : ; y n of T n V such that ann V = ann(y i= i) and n T dim A. This is possible since we may choose the y j j inductively with strictly decreasing dim ann(y i= i). Dene n y := y : : : y n 2 5 V:

7 Then and therefore Since L m U ' ' ' m m m m m m m m Ax i A= ann(x i ) ann V = ann(y); dquotientc A= ann U dquotientc A=(ann V \ rad A) (A= ann V A= rad A) A= ann(y) Ay n m V m A= rad A m A= rad A dquotient, by (0)c dsubmodulec A= rad A dsubmodulec. A= rad A is semisimple, this implies (9) with r = m n dim U dim A. 2 The next lemma is well known. Lemma 2 Suppose that E is a nite dimensional complex vector space and that z ; : : : ; z q 2 E are pairwise linearly independent nonzero vectors. Then for suciently large the vectors z ; : : : ; zq 2 E are linearly independent. Proof: By symmetry it suces to show that for suciently large there is a linear form f 2 (E ) such that f(z ) 6= 0; 8i > f(zi ) = 0: Polarization implies that this is equivalent to the existence of a homogeneous polynomial function F on E of degree with the property F (z ) 6= 0; 8i > F (z i ) = 0: The existence of such a form follows from the fact that f[z 2 ]; : : : ; [z q ]g is a Zariski closed subset of the projective space P(E) not containing [z ]. 2 6

8 Lemma 3 Let A be a nite dimensional C -algebra with a multiplicative basis B, where rad A = lin B, and let V be an A-module. For suciently large 2 N the following is true: If N is a positive integer and W is isomorphic to a submodule of V N then ann(v W ) \ rad A is linearly generated by ann(w ) \ B together with all b? +N c such that b; c 2 B and b? c 2 ann V. Proof: Clearly ann(w ) \ B ann(v W ) \ rad A. Since V W is isomorphic to a submodule of V ( +N ), the elements b? +N c with b; c 2 B and b? c 2 ann V also belong to ann(v W ) \ rad A. d(b? +N c) V (+N) = b V (+N)? +N c V (+N) = (b V ) ( +N )? (c V ) ( +N ) = 0.c This holds for arbitrary. Now take so large as to ensure the conclusion of Lemma 2 for E = End V and for every list z ; : : : ; z q of pairwise linearly independent elements from fb V : b 2 B g. Let L denote the linear hull of ann(w ) \ B together with all b? +N c such that b; c 2 B and b? c 2 ann V. We have just seen that L ann(v W ) \ rad A P s Pand we have to prove equality. By way of contradiction let i= ib i 2 ann(v W )\rad A, s i= ib i =2 L be a counterexample of minimal length s (where i 2 C and b i 2 B ). Clearly s > 0, and by the minimality of s we have i 6= 0 and b i 62 ann W. oreover, the (b i ) V are pairwise linearly independent: Otherwise (b ) V = (b 2 ) V, say, so that b? b 2 2 ann V and therefore b? +N b 2 2 L. But then b + 2 b 2 may be replaced by b + 2 b 2? (b? +N b 2 ) = ( +N + 2 )b 2, contradicting the minimality of s. Thus by the choice of the endomorphisms (b i ) V = ((b i ) V ) are linearly independent. Since b i 62 ann W, there is a linear form y 2 (End(W )) such that i := y((b i ) W ) 6= 0 i= i (b i ) V (b i ) W = 0. for all i. Now P s i= ib i 2 ann(v W ), which means that P s P s Applying id V y to this equation we get i= i i (b i ) V = 0. But the (b i ) V are linearly independent, hence all i i = 0. This is a contradiction, since s > 0, i 6= 0 and i 6= 0. 2 Proof of Theorem 2: (4))(5): This has already been observed just before of the statement of Theorem 2. (5))(6): In view of (7) we have for some k 2 N and for N R(b U ) N = R(b U N ) N k R(b V N ) = N k R(b V ) N ; since the rank decreases under degeneration and b annihilates T. Taking N-th roots and letting N grow to innity we obtain R(b U ) R(b V ). We also have for N b V = c V =) b V N = N c V N =) b U N = N c U N =) (b U ) N = N (c U ) N ; since the annihilator of modules increases under degeneration and b and c annihilate S. We assume without loss of generality b U 6= 0 and c U 6= 0 (hence 6= 0) and choose y 2 (End C U) such that y(b U ) = 6= 0, y(c U ) = 6= 0. Applying y N to (b U ) N = N (c U ) N and y N id End C U to (b U ) (N +) = N + (c U ) (N +) we obtain N = N N and N b U = N + N c U. This implies b U = c U. 7

9 (6))(4): Set m := dim U and n := dim V. For the moment we view V simply as a vector space (rather than an A-module) and use some classical information on the decomposition of V N into simple GL(V )-modules, due to Frobenius, Young, Schur and Weyl (see Fulton-Harris [8], Theorem 6.3 and (4.)): V N ' 2P n(n ) f S (V ); () where P n (N) is the set of partitions of N into at most n parts, S (V ) denotes a simple GL(V )- module with highest weight and f is the dimension of the simple S N -module corresponding to. We have where l i := i + n? i, and dim S (V ) = Y i<jn l i? l j j? i ; (2) N!Q i<j f = (l i? l j ) : l! : : : l n! We will not use this last equation explicitly. But note that f is typically of exponential size in N. In contrast, the dimension of S (V ) as well as the size of P n (N) are polynomially bounded in N: jp n (N)j N n ; dim S (V ) N n2 : (3) How does () relate to the A-module structure of V N? Any g 2 GL(V ) acts on V N as the tensor power g N. By denition, GL(V )-submodules of V N are therefore stabilized by such g N. Since GL(V ) is dense in End(V ), a continuity argument shows that GL(V )- submodules of V N are also stabilized by tensor powers of arbitrary linear endomorphisms of V. In particular this is true for b V N = (b V ) N, where b 2 B. Hence GL(V )-submodules of V N are automatically A-submodules. A similar continuity argument yields that isomorphic GL(V )-submodules of V N are also isomorphic as A-modules. By transport of structure the right hand side of () is an A-module and () is an isomorphism of A-modules. By what we have just seen the direct sum on the right hand side of () is a decomposition into A- submodules. (The newborn A-modules S (V ) need not be simple or indecomposable.) Now x such that the conclusion of Lemma 3 holds. In view of (8) it suces to show 9k 8N 9T U ( +N ) T N k V ( +N ) ; (4) where T is understood to be a semisimple module. Besides () and (3) we will need the isomorphism analogous to () and the estimates U ( +N ) ' 2P m( +N ) f S (U) (5) jp m ( + N)j ( + N) m ; dim S (U) ( + N) m2 (6) 8

10 analogous to (3). In view of the A-module isomorphisms () and (5) we may reformulate (4) as 9k 8N 9T 2P m( +N ) f S (U) T N k V 2P n(n ) f S (V ) A : By (6) this is a consequence of 9k 8N 8 2 P m ( + N) 9T f S (U) T N k V f S (V ) A 2P n(n ) with the same stipulation for T. This in turn is implied by 9k; l 8N 8 2 P m ( + N) 9T S (U) T N k 0 V L L duse f 2P n(n ); N l f f S (V ) L N L l f L2P S n(n ) (V ).c Setting it suces to show W = W l;n; := 2Pn(N) Nlf f S (V ); 2Pn(N) Nlf f C S (V ) A : 9l 8N 8 2 P m ( + N) ann(v W ) \ rad A ann S (U): (7) dlemma then yields with 9r; T S (U) T r (V W ) r dim S (U) dim A ( + N) m2 dim A by (6).c We check the inclusion in (7) on the set of linear generators of the left hand side as given by Lemma 3 (which is applicable by our choice of ). First consider a generator of the form b? +N c, where b; c 2 B and b?c 2 ann V. By (6) we have b?c 2 ann U, hence b? +N c 2 ann U ( +N ) and therefore b? +N c 2 ann S (U). Next consider a generator of the form b 2 ann(w ) \ B. For any we have V N ' 2P n(n ) f 9 S (V ) = X Y;

11 where X := 2Pn(N) Nlf f f S (V ); Y := 2Pn(N) Nlf <f f S (V ): Since b annihilates W and since X and W are composed of the same A-modules S (V ) (with possibly dierent positive multiplicities), b also annihilates X and we conclude from (3) and the denition of Y R(b V ) N = R(b N V ) = R(b V N ) dim Y N?l f N n2 +n : (8) If (arguing by contradiction) b 62 ann S (U) for some, we may estimate R(b U ) +N = R(b U (+N)) R(bL f S(U )) f : (9) By assumption (6) we have R(b U ) R(b V ). Together with (8) and (9) this gives f (N?l f N n2 +n ) +=N : ( +N ) In view of the trivial estimate f dim U and large N. This proves (7) and the theorem. 2 Proof of Theorem : m +N this is impossible for l > n 2 + n Both U and V are -nilpotent for some positive integer in the sense that U(w) = V (w) = 0 for all paths w of length. Without loss of generality the semiring N (Q) in the statement of the theorem may therefore be replaced by the subsemiring N (Q) of all -nilpotent equivalence classes of Q-modules. A well known equivalence of categories ([2], Theorem.5) induces an isomorphism of semirings N (Q)! (A), where A is the path algebra of Q divided by the ideal spanned by all paths of length at least, and with the set of residue classes of all paths of length at most? as a multiplicative basis. (The radical of A is spanned by the residue classes of paths of positive length. Tensor products of -nilpotent Q-modules and of A-modules correspond up to equivalence.) This isomorphism respects degeneration (Bongartz [4]) and therefore asymptotic degeneration. Now Theorem follows from Theorem 2. 2 Acknowledgement I thank K. Bongartz, H. Kraft and. Nusken for having called my attention to some relevant literature, and. Franz for a critical reading of the manuscript. Annotations. Bongartz [6] uses a kind of inverse notation, that does not suit the purposes of the present paper. 0

12 2. Code equivalence classes by suitable representatives. 3. A warning: By an example of J. Carlson (Riedtmann [4], p. 282) there are p; q 2 (Q) and U 2 p, V 2 q such that p q, U and V have the same dimension vector, but U is not a degeneration of V. 4. Degenerate V 2 q along a Jordan-Holder-series to a semisimple module p (see [2], Proposition 4.3) and observe that p is nilpotent, since q is. Hence p is trivial. 5. This follows from three observations:. An equivalence class is specied by a numerical partition, interpreted as the sequence of sizes of nontrivial nilpotent Jordan blocks arranged in nonincreasing order. 2. A single nilpotent Jordan block of size m 2 is mapped to the function f m : j 7! minfm? j; 0g. 3. Any convex function f: Z >0! Z 0 with nite support is a unique linear combination of such f m with nonnegative integral coecients. 6. Note that the image of ' separates points. 7. This is the reason for allowing nonunital modules. 8. Antisymmetry: As has rst been observed by Gabriel [0] (see also Kraft [2], II,4), the Jordan-Holder multiplicities of modules are preserved under degeneration. This holds in the non-unital case as well. Now assume p q and q p and choose U 2 p and V 2 q. Then U S V T and V T 0 U S 0 for certain semisimple modules S; S 0 ; T; T 0. Without loss of generality S = S 0. Then V T and V T 0 have the same Jordan-Holder multiplicities, hence T = T 0, hence U S ' V T. 9. Degenerate U 2 p along a Jordan{Holder series as in Kraft [2], II,4. 0. If U 2 p, V 2 q, W 2 r and if 0! U! V! W! 0 is exact, then p + r q and therefore p q and r q. References [] Abeasis, S., Del Fra, A., Degenerations for the representations of a quiver of type A m, J. Algebra 93, (985) [2] Auslander,., Reiten, I., Smal, O., Representation Theory of Artin Algebras, Cambridnge Studies in Advanced athematics 36 (995) [3] Bernstein, I. N., Gelfand, I.., Ponomarev, V. A., Coxeter functors and a theorem of Gabriel, Uspechi at. Nauk 28, 9-33 (973) [4] Bongartz, K., A geometric version of the orita equivalence, J. Algebra 39, (99)

13 [5] Bongartz, K., Degenerations for representations of tame quivers, Ann. Sci. Ecole Norm. Sup. 28, (995) [6] Bongartz, K., Some geometric aspects of representation theory, in: Algebras and odules I, CS conference proceedings vol. 23, ed. by Reiten, I., Smal, S. O. and Solborg,., p. -27 (998) [7] Dlab, V., Ringel, C.., Indecomposable representations of graphs and algebras, emoirs Amer. ath. Soc. 73 (976) [8] Fulton, W., Harris, J., Representation Theory, Graduate Texts in athematics 29, Springer Verlag (99) [9] Gabriel, P., Unzerlegbare Darstellungen I, man. math. 6, 7-03 (972) [0] Gabriel, P., Finite representation type is open, Springer Lecture Notes 488, (975) [] Gerstenhaber,, On nilalgebras and linear varieties of nilpotent matrices III, Ann. ath. 70, No. (959) [2] Kraft, H., Geometric ethods in Representation Theory, Springer Lecture Notes 944, (982) [3] Kraft, H., Procesi, C., Closures of conjugacy classes of matrices are normal, Invent. math. 53, (979) [4] Riedtmann, C., Degenerations for representations of quivers with relations, Ann. Sci. Ecole Norm. Sup. 4, (986) [5] Strassen, V., The asymptotic spectrum of tensors, J. reine angew. ath. 384, (988) [6] Zwara, G., Degenerations of nite dimensional modules are given by extensions, to appear in Compositio ath. ( address: 2