Maximal Green Sequences of Exceptional Finite Mutation Type Quivers
SSymmetry, Integrability and Geometry: Methods and Applications SIGMA (2014), 089, 5 pages Maximal Green Sequencesof Exceptional Finite Mutation Type Quivers (cid:63)
Ahmet I. SEVENMiddle East Technical University, Department of Mathematics, 06800, Ankara, Turkey
E-mail: [email protected]
Received June 18, 2014, in final form August 15, 2014; Published online August 19, 2014http://dx.doi.org/10.3842/SIGMA.2014.089
Abstract.
Maximal green sequences are particular sequences of mutations of quiverswhich were introduced by Keller in the context of quantum dilogarithm identities and in-dependently by Cecotti–C´ordova–Vafa in the context of supersymmetric gauge theory. Theexistence of maximal green sequences for exceptional finite mutation type quivers has beenshown by Alim–Cecotti–C´ordova–Espahbodi–Rastogi–Vafa except for the quiver X . In thispaper we show that the quiver X does not have any maximal green sequences. We alsogeneralize the idea of the proof to give sufficient conditions for the non-existence of maximalgreen sequences for an arbitrary quiver. Key words: skew-symmetrizable matrices; maximal green sequences; mutation classes
Maximal green sequences are particular sequences of mutations of quivers. They were usedin [9] to study the refined Donaldson–Thomas invariants and quantum dilogarithm identities.Moreover, the same sequences appeared in theoretical physics where they yield the completespectrum of a BPS particle, see [5, Section 4.2]. The existence of maximal green sequences forexceptional finite mutation type quivers has been shown in [1] except for the quiver X . Inthis paper, we show that the quiver X does not have any maximal green sequences. We alsogive some general sufficient conditions for the non-existence of maximal green sequences for anarbitrary quiver.To be more specific, we need some terminology. Formally, a quiver is a pair Q = ( Q , Q )where Q is a finite set of vertices and Q is a set of arrows between them. It is represented asa directed graph with the set of vertices Q and a directed edge for each arrow. We considerquivers with no loops or 2-cycles and represent a quiver Q with vertices 1 , . . . , n , by the uniquelyassociated skew-symmetric matrix B = B Q defined as follows: the entry B i,j > B i,j many arrows from j to i ; if i and j are not connected to each other by an edgethen B i,j = 0. We will also consider more general skew-symmetrizable matrices: recall thatan n × n integer matrix B is skew-symmetrizable if there is a diagonal matrix D with positivediagonal entries such that DB is skew-symmetric. To define the notion of a green sequence,we consider pairs ( c , B ), where B is a skew-symmetrizable integer matrix and c = ( c , . . . , c n )such that each c i = ( c , . . . , c n ) ∈ Z n is non-zero. Motivated by the structural theory of clusteralgebras, we call such a pair ( c , B ) a Y -seed. Then, for k = 1 , . . . , n and any Y -seed ( c , B ) suchthat all entries of c k are non-negative or all are non-positive, the Y -seed mutation µ k transforms( c , B ) into the Y -seed µ k ( c , B ) = ( c (cid:48) , B (cid:48) ) defined as follows [8, equation (5.9)], where we usethe notation [ b ] + = max( b, (cid:63) a r X i v : . [ m a t h . C O ] A ug A.I. Seven • the entries of the exchange matrix B (cid:48) = ( B (cid:48) ij ) are given by B (cid:48) ij = (cid:40) − B ij if i = k or j = k , B ij + [ B ik ] + [ B kj ] + − [ − B ik ] + [ − B kj ] + otherwise; (1) • the tuple c (cid:48) = ( c (cid:48) , . . . , c (cid:48) n ) is given by c (cid:48) i = (cid:40) − c i if i = k, c i + [sgn( c k ) B k,i ] + c k if i (cid:54) = k. (2)The matrix B (cid:48) is skew-symmetrizable with the same choice of D . We also use the notation B (cid:48) = µ k ( B ) (in (1)) and call the transformation B (cid:55)→ B (cid:48) the matrix mutation . This operationis involutive, so it defines a mutation-equivalence relation on skew-symmetrizable matrices.We use the Y -seeds in association with the vertices of a regular tree. To be more precise,let T n be an n -regular tree whose edges are labeled by the numbers 1 , . . . , n , so that the n edgesemanating from each vertex receive different labels. We write t k − t (cid:48) to indicate that vertices t, t (cid:48) ∈ T n are joined by an edge labeled by k . Let us fix an initial seed at a vertex t in T n and assign the (initial) Y -seed ( c , B ), where c is the tuple of standard basis. This definesa Y -seed pattern on T n , i.e. an assignment of a Y -seed ( c t , B t ) to every vertex t ∈ T n , such thatthe seeds assigned to the endpoints of any edge t k − t (cid:48) are obtained from each other by the seedmutation µ k ; we call ( c t , B t ) a Y -seed with respect to the initial Y -seed ( c , B ). We write: c t = c = ( c , . . . , c n ) , B t = B = ( B ij ) . We refer to B as the exchange matrix and c as the c -vector tuple of the Y -seed. These vectorshave the following sign coherence property [7]:each vector c j has either all entries nonnegative or all entries nonpositive. (3)Note that this property is conjectural if B is a general non-skew-symmetric (but skew-symmet-rizable) matrix. It implies, in particular, that the Y -seed mutation in (2) is defined for any Y -seed ( c t , B t ), furthermore c t is a basis of Z n [10, Proposition 1.3]. We also write c j > c j <
0) if all entries are non-negative (resp. non-positive).Now we can recall the notion of a green sequence [3]:
Definition 1.
Let B be a skew-symmetrizable n × n matrix. A green sequence for B isa sequence i = ( i , . . . , i l ) such that, for any 1 ≤ k ≤ l with ( c , B ) = µ i k − ◦ · · · ◦ µ i ( c , B ),we have c i k >
0, i.e. each coordinate of c i k is greater than or equal to 0; here if k = 1, thenwe take ( c , B ) = ( c , B ). A green sequence for a quiver is a green sequence for the associatedskew-symmetric matrix.A green sequence i = ( i , . . . , i l ) is maximal if, for ( c , B ) = µ i l ◦ · · · ◦ µ i ( c , B ), we have c k < k = 1 , . . . , n .In this paper, we study the maximal green sequences for the quivers which are mutation-equivalent to the quiver X (Fig. 1). Our result is the following: Theorem 1.
Suppose that Q is mutation-equivalent to the quiver X ( so Q is one of the quiversin Fig. . Then Q does not have any maximal green sequences. We prove the theorem using the following general statement, which can be easily checked togive a sufficient condition for the non-existence of maximal green sequences:aximal Green Sequences of Exceptional Finite Mutation Type Quivers 3 • (cid:31) (cid:31) • (cid:31) (cid:31) (cid:31) (cid:31) • (cid:63) (cid:63) (cid:63) (cid:63) • (cid:111) (cid:111) (cid:63) (cid:63) (cid:31) (cid:31) • (cid:111) (cid:111) • (cid:63) (cid:63) • (cid:111) (cid:111) (cid:111) (cid:111) • (cid:22) (cid:22) (cid:127) (cid:127) • (cid:111) (cid:111) (cid:127) (cid:127) • (cid:31) (cid:31) (cid:47) (cid:47) • (cid:95) (cid:95) (cid:47) (cid:47) (cid:127) (cid:127) • (cid:95) (cid:95) (cid:106) (cid:106) • (cid:53) (cid:53) (cid:47) (cid:47) • (cid:95) (cid:95) (cid:63) (cid:63) Figure 1.
Quivers which are mutation-equivalent to X ; the first one is the quiver X , see [6]. Proposition 1.
Let B be a skew-symmetrizable initial exchange matrix. Suppose that thereis a vector u > such that, for any Y -seed ( c , B ) with respect to the initial seed ( c , B ) , thecoordinates of u with respect to c are non-negative. Then, under assumption (3) , the matrix B does not have any maximal green sequences. We establish such a vector for the quiver X : Proposition 2.
Suppose that Q is a quiver which is mutation-equivalent to X , so Q isone of the quivers in Fig. , and B is the corresponding skew-symmetric matrix. Let u =( a , a , . . . , a ) be the vector defined as follows: ( ∗ ) if Q is the quiver X ( so Q is the f irst quiver in Fig. , then the coordinatecorresponding to the “center” is equal , and the rest is equal to ; if Q is not thequiver X ( so Q is the second quiver in Fig. , then all coordinates are equal to .Then, for any Y -seed ( c , B ) with respect to the initial seed ( c , B ) , the coordinates of u withrespect to c is of the same form as in ( ∗ ) . In particular, the coordinates of u with respect to c are positive. ( The vector u is a radical vector for B , i.e. B u = 0 . In fact, any radical vector for B isa multiple of u. )We generalize this statement to an arbitrary quiver as follows: Theorem 2.
Let B be a skew-symmetrizable initial exchange matrix and suppose that u > is a radical vector for B , i.e. B u = 0 . Suppose also that, for any Y -seed ( c , B ) with respectto the initial seed ( c , B ) , the coordinates of u with respect to c are non-negative. Then, underassumption (3) , for any B which is mutation-equivalent to B , the matrix B does not have anymaximal green sequences. We prove our results in Section 2. For related applications of maximal green sequences, werefer the reader to [4] and [11].
Let us first note how the coordinates of a vector change under the mutation operation, whichcan be easily checked using the definitions (assuming (3)):
Proposition 3.
Suppose that ( c , B ) is a Y -seed with respect to an initial Y -seed. Supposealso that the coordinate vector of u with respect to c is ( a , . . . , a n ) . Let ( c (cid:48) , B (cid:48) ) = µ k ( c , B ) and ( a (cid:48) , . . . , a (cid:48) n ) be the coordinates of u with respect to c (cid:48) . Then a i = a (cid:48) i if i (cid:54) = k and a (cid:48) k = − a k + (cid:80) a i [sgn( c k ) B k,i ] + , where the sum is over all i (cid:54) = k . A.I. SevenAs we mentioned, in view of Proposition 1, Theorem 1 follows from Proposition 2. To proveProposition 2, it is enough to show that the coordinates of the vector u change as stated, i.e. showthat if the coordinates of u with respect to c are as in ( ∗ ), then for the Y -seed ( c (cid:48) , B (cid:48) ) = µ k ( c , B ),the coordinates with respect to c (cid:48) are also of the form in ( ∗ ). This can be checked easily usingthe formula in Proposition 3.To prove Theorem 2, let us first note the following property of the coordinates of the radicalvectors: Lemma 1.
Suppose that ( c , B ) is a Y -seed with respect to an initial Y -seed ( c , B ) and u isa radical vector for B . Suppose that the coordinate vector of u with respect to c is ( a , . . . , a n ) .Then, for any index k , we have the following: (cid:88) a i [sgn( c k ) B k,i ] + = (cid:88) a i [ − sgn( c k ) B k,i ] + , where the sum is over all i (cid:54) = k .In particular, for radical vectors, the formula in Proposition that describe the change ofcoordinates under mutation depends only on the exchange matrix, not on the c -vectors. To prove the lemma, suppose that D = diag( d , . . . , d n ) is a skew-symmetrizing matrixfor B , so it is also skew-symmetrizing for B , so DB = C is skew-symmetric, i.e. C i,k = d i B i,k = − d k B k,i = − C k,i for all i , k . Let u = ( a , . . . , a n ). Then u is a radical vector for B , so itis also a radical vector for C = DB , i.e. Cu = 0, which means that for any index k , we have (cid:80) a i [sgn( c k ) C k,i ] + = (cid:80) a i [sgn( c k ) C i,k ] + , which is equal to (cid:80) a i [ − sgn( c k ) C k,i ] + , where the sumis over all i (cid:54) = k . Then, writing C k,i = d k B k,i , we have (cid:88) a i [sgn( c k ) d k B k,i ] + = (cid:88) a i [ − sgn( c k ) d k B k,i ] + . Dividing both sides by d k >
0, we obtain the lemma.We will also need the following property of the radical vectors:
Lemma 2.
In the set-up of Theorem , let u denote the vector which represents u with respectto the basis c . Then u is a radical vector for B , i.e. Bu = 0 . To prove the lemma, let us note that u can be obtained from u by applying the formula inProposition 3 along with the mutations. Thus, to prove the lemma, it is enough to show that,for any k = 1 , . . . , n , we have the following:( ∗∗ ) if u = ( a , . . . , a n ) is a radical vector for B , then u (cid:48) is a radical vector for B (cid:48) = µ k ( B ), i.e. B (cid:48) u (cid:48) = 0, where u (cid:48) = ( a (cid:48) , . . . , a (cid:48) n ) is the vector as in Proposition 3.To show ( ∗∗ ), we write B (cid:48) in matrix notation as follows [2, Lemma 3.2]: for (cid:15) = sgn( c k ), wehave B (cid:48) = ( J n,k + E k ) B ( J n,k + F k ) , where • J n,k denotes the diagonal n × n matrix whose diagonal entries are all 1’s, except for − k th position; • E k is the n × n matrix whose only nonzero entries are e ik = [ − εb ik ] + ; • F k is the n × n matrix whose only nonzero entries are f kj = [ εb kj ] + .aximal Green Sequences of Exceptional Finite Mutation Type Quivers 5It follows from a direct check that ( J n,k + F k ) u (cid:48) = u . Then B (cid:48) u (cid:48) = ( J n,k + E k ) B ( J n,k + F k ) u (cid:48) =( J n,k + E k ) Bu = ( J n,k + E k )0 = 0. This completes the proof of the lemma.Let us now prove Theorem 2. For this, we first consider the Y -seed pattern defined by theinitial Y -seed ( c , B ) at the initial vertex t . Let us suppose that t is a vertex such thatthe corresponding Y seed ( c , B ) has the exchange matrix B . Then we can consider the Y -seedpattern defined by the initial Y -seed ( c , B ) at the initial vertex t (where c is the tuple ofstandard basis). Then we have the following: for any fixed vertex t of the n -regular tree T n ,the exchange matrices of the Y -seeds assigned by these patterns coincide because the pattern isformed by mutating at the labels of the n -regular tree T n and mutation is an involutive operationon matrices; let us denote these seeds by ( c (cid:48) , B (cid:48) ) and ( c (cid:48)(cid:48) , B (cid:48) ) respectively.On the other hand, let u denote the vector which represents u with respect to the basis c ,which can be obtained by applying the formula in Proposition 3 along with the mutations.Then u is a radical vector for B , i.e. Bu = 0 (Lemma 2). Furthermore, the coordinates of thevectors u and u with respect to the bases c (cid:48) and c (cid:48)(cid:48) respectively will coincide by Lemma 1(which says that for radical vectors the formula in Proposition 3 depends only on the exchangematrices, not on the c -vectors). In particular, the coordinates of u with respect to any basisof c -vectors are non-negative. Thus, by Proposition 1, the matrix B does not have any maximalgreen sequences. This completes the proof. Acknowledgements
The author’s research was supported in part by the Scientific and Technological Research Councilof Turkey (TUBITAK) grant
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