C -Vectors and Non-Self-Crossing Curves for Acyclic Quivers of Finite Type
CC -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLICQUIVERS OF FINITE TYPE SU JI HONG
Abstract.
Let Q be an acyclic quiver and k be an algebraically closed field. The indecomposableexceptional modules of the path algebra kQ have been widely studied. The real Schur roots ofthe root system associated to Q are the dimension vectors of the indecomposable exceptionalmodules. It has been shown in [3] that for acyclic quivers, the set of positive c -vectors and theset of real Schur roots coincide. To give a diagrammatic description of c -vectors, K-H. Lee andK. Lee conjectured that for acyclic quivers, the set of c-vectors and the set of roots correspondingto non-self-crossing admissible curves are equivalent as sets in [11]. In [5], A. Felikson and P.Tumarkin proved this conjecture for 2-complete quivers. In this paper, we prove a revised versionof Lee-Lee conjecture for acyclic quivers of type A , D , and E and E . introduction Let Q be an acyclic quiver and k be an algebraically closed field. The path algebra kQ is analgebra generated by all oriented paths of Q . The category of kQ modules is equivalent to thecatertory of quiver representations. Kac has shown that the dimension vectors of indecomposablequiver representations are the positive roots of the root system of the quiver Q [10]. One of theobjects of interest in relation to kQ is the set of indecomposable exceptional representations of kQ . The dimension vectors of these representations are called real Schur roots and the set of realSchur roots is a subset of positive roots. There are many different ways to describe a real Schurroot. In [15], Schofield showed a way to classify real Schur roots using subrepresentations and in[9], Igusa and Schiffler used the Coxeter element to classify real Schur roots. In [8], Hubery andKrause used non-crossing paritions to determine real Schur roots. Another way to describe a realSchur root comes from the cluster algebra associated to Q . A cluster algebra of Q is an algebrawith generators, called cluster variables, obtained from the quiver and the exchange relation.Fomin and Zelevinksy showed that any cluster variables can be expressed as Laurent polynomialsin the initial cluster variables in [7]. When a cluster variable is at its most reduced form, theexponent vector of the denominator monomial gives the denominator vector, d -vector, of thatcluster variable. Caldero and Keller proved that for an acyclic quiver, the set of real Schur rootsand the set of d -vectors of non-initial cluster variables are equivalent in [2].Another description of a real Schur root comes from the framed quiver of Q . In the framedquiver of Q , c -vectors describe the relations between the mutable vertices and the frozen vertices a r X i v : . [ m a t h . R T ] M a y SU JI HONG (more precise definition is given in Section 2). One of the big questions about c -vectors was thesign coherency of c -vectors. In [4], Derksen, Weyman, and Zelevinsky proved the sign coherencyof c -vectors. Thus the set of c -vectors is the disjoint union of the set of positive c -vectors and theset of negative c -vectors. Chavez showed that for an acyclic quiver, the set of positive c -vectorsand the set of real Schur roots coincide in [3].Although there are many ways to describe a real Schur root, none of the descriptions mentionedso far are diagrammatic. In cluster algebras, the diagrammatic descriptions of cluster variablesare useful. In [6], Fomin, Shapiro, and Thurston described the cluster complex using the taggedtriangulation and arcs. Also, Nakanishi and Stella gave diagrammatic descriptions of c -vectorsand d -vectors of quivers of finite type in [13]. To give more general geometric description of c -vectors of acyclic quivers, hence real Schur roots, K.-H. Lee and K. Lee formulated the followingconjecture. Conjecture 1.1. (Lee-Lee [11] ) For an acyclic quiver, the set of roots associated to non-self-crossing admissible curves and the set of real Schur roots coincide.
In [5], Felikson and Tumarkin gave an alternate but equivalent definition of non-self-crossingadmissible curves. Conjecture 1.1 follows from [1, Proposition 33] for the acyclic quivers offinite type. Given a real Schur root, there are multiple non-self-crossing admissible curves thatcorrespond to the root. To categorize these curves, K. Lee defined positive, non-decreasing, andstrictly increasing curves which will be defined in Section 2. These descriptions of curves led tothe following conjecture.
Conjecture 1.2. ( [12] ) For an acyclic quiver, the set of associated roots of non-decreasing non-self-crossing admissible curves and the set of real Schur roots coincide. More precise version of this conjecture is given in Conjecture 2.9. Note that Conjecture 1.2implies Conjecture 1.1. Felikson and Tumerkin’s result [5] implies Conjecture 1.2 for acyclic2-complete quivers, i.e., quivers with at least two arrows between every pair of vertices. In thispaper we prove Conjecture 1.2 for the acyclic quivers type
A, D, E and E . Theorem 1.3. (Revised Lee-Lee Conjecture for type
A, D, E and E ) Given a quiver of type A, D, E or E , the set of roots associated to non-decreasing non-self-crossing admissible curvesis the same as the set of real Schur roots. In particular, if an acyclic quiver is of type A , thenthe set of roots associated to strictly increasing non-self-crossing admissible curves is the same asthe set of real Schur roots. Acknowledgements.
The author would like to thank Kyunyong Lee for guidance and helpfuldiscussions and Son Nyguen for helpful suggestions. -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 3 Background
In this section, we define some of the objects of interest including the admissible curve.2.1. c -vectors and roots system. A quiver Q is a finite oriented graph without any oriented2-cycles and loops. We notate Q = ( Q , Q ) where Q = [ n ] = { , . . . , n } is the set of vertices of Q and Q is the set of oriented edges, which we call arrows, of Q . Let h, t : Q → Q be mapsgiven by h ( i → j ) = j and t ( i → j ) = i . The exchange matrix of Q is B = ( b ij ) ≤ i,j ≤ n where b ij is the number of arrows from the vertex i to the vertex j (if there are r arrows from the vertex j to the vertex i , then b ij = − r ). We mutate a quiver Q at a vertex i to obtain µ i ( Q ) by the stepsbelow:(1) For every path j → i → k , create an arrow j → k .(2) Reverse the orientations of all the arrows incident to i .(3) Delete any 2-cycles. Definition 2.1.
Let Q = ( Q , Q ) be a quiver and Q (cid:48) be a duplicate of Q . Then the framedquiver of Q is Q (cid:48) = ( Q ∪ Q (cid:48) , Q ∪ { i → i (cid:48) | for all i ∈ [ n ] } ) where mutation at a vertex i (cid:48) is notallowed for all i (cid:48) ∈ Q (cid:48) . Definition 2.2.
Given a quiver Q and a sequence of mutations w = µ i k · · · µ i , consider w ( Q (cid:48) ).A c -vector of Q is given by c i, (cid:48) c i, (cid:48) ... c i,n (cid:48) , where c i,j (cid:48) = r if i r arrows −−−−−→ j (cid:48) − r if i r arrows ←−−−−− j (cid:48) . Example 2.3.
Consider a framed quiver Q (cid:48) and µ µ µ ( Q (cid:48) ) below.1 2 31 (cid:48) (cid:48) (cid:48) µ µ µ (cid:48) (cid:48) (cid:48) The c -vectors are − − − , , . As mentioned before, these c -vectors are sign coherent, meaning that all the entries of a c -vectorare either non-negative or non-positive.For the rest of this paper, let Q be an acyclic quiver and B = ( b ij ) ≤ i,j ≤ n be the exchangematrix of Q . Then the generalized Cartan matrix C ( B ) is ( a ij ) ≤ i,j ≤ n where a ii = 2 , a ij = −| b ij | for i (cid:54) = j. SU JI HONG
Let ∆( B ) be the root system of the Kac-Moody algebra associated to C ( B ). Let α , . . . , α n be the simple roots associated to the vertices 1 , , . . . , n respectively. Any root can be viewedas Z − linear combination of the simple roots, i.e., for any root α , there exist β , . . . β n ∈ Z suchthat α = (cid:80) ni =1 β i α i . Given a full subquiver Q (cid:48) , the root system of Q (cid:48) is a subroot system of∆( B ). Thus some of the roots of Q come from roots of Q (cid:48) . All the roots are either positive ornegative. Thus ∆( B ) = ∆ + Q ∪ ∆ − Q where ∆ + Q is the set of all positive roots and ∆ − Q is the setof all negative roots. Of these roots, a real Schur root is a positive root that corresponds to thedimension vector of an indecomposable exceptional module of kQ . Chavez showed that the setof c -vectors coincides with the set of real Schur roots and their opposites in the root system in[3]. Since real Schur roots are positive, the set of positive c -vectors coincide with the set of realSchur roots.Given a quiver and a root α associated to the quiver, we can represent α with the underlyinggraph of the quiver by labeling the vertex i with β i for i ∈ [ n ]. For example, consider the followingquiver whose underlying graph is:1 2 . . . n Then for any root, α = (cid:80) ni =1 β i α i , we can represent α as α : β β · · · β n Definition 2.4.
We define the standard partial order on ∆ + Q as follows. Let α = (cid:80) ni =1 β i α i and α (cid:48) = (cid:80) ni =1 β (cid:48) i α i be roots of a quiver. Then α ≤ D α (cid:48) if and only if β i ≤ β (cid:48) i for all i ∈ [ n ] and α < D α (cid:48) if α ≤ D α (cid:48) and β i < β (cid:48) i for some i ∈ [ n ].We can define reflections over any roots in ∆( B ); in particular, a simple reflection s i is areflection over a simple root α i and is defined by s i ( α j ) = α j − a ij α i , for all i, j ∈ [ n ] . Admissible Curves.
K.-H. Lee and K. Lee defined the admissible curves on some space in[11]. In [5], A. Felikson and P. Tumarkin gave an equivalent definition of the admissible curveson a disc with n marked interior points and a boundary point called a basepoint. We define thedisk and the admissible curves in a slightly modified version in this section. The definition of theadmissible curve in this paper is equivalent to the one in [5]. Definition 2.5.
Let H = R × R ≥ and for all i ∈ [ n ], p i = ( i,
1) be a marked point and ρ i = { ( i, j ) | j ≥ } be a ray in H . Let b = (0 ,
0) be the base point and P = { p i | i ∈ [ n ] } be the setof all marked points. An admissible curve is a smooth continuous function γ : [0 , → H suchthat(a) γ (0) ∈ P and γ (1) = b ,(b) if γ ( x ) ∈ { b } ∪ P , then x ∈ { , } , and -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 5 (c) if γ ((0 , ∩ ρ i (cid:54) = ∅ , then γ and ρ i intersect transversely.An admissible curve is non-self-crossing if for all x, y ∈ [0 ,
1] such that γ ( x ) = γ ( y ), x = y . SeeFigure 1A for an example of a non-self-crossing admissible curve. (A) Example of an admissible curve (B) γ when n = 4 Figure 1
Let γ i be a curve defined as γ i ( x ) = ( i (1 − x ) , − x ) , for all i ∈ [ n ]. We can easily see that γ i is an admissible curve, as(a) γ i (0) = p i ∈ P and γ i (1) = b ,(b) if γ i ( x ) ∈ { b } ∪ P , then γ i ( x ) = ( i (1 − x ) , − x ) = (0 ,
0) or ( k,
1) for some k ∈ [ n ], whichmeans that x = 0 or x = 1,(c) γ i ((0 , ∩ ρ j = ∅ for all j. Thus γ i is an admissible curve and it does not cross itself.Consider the braid group B n = (cid:28) σ , . . . , σ n − (cid:12)(cid:12)(cid:12) σ i σ i +1 σ i = σ i +1 σ i σ i +1 for i ∈ [ n − σ i σ j = σ j σ i for | i − j | ≥ (cid:29) where σ i is shown below.Let Γ be the set of non-self-crossing admissible curves. We can define a braid group actionon Γ. For any γ ∈ Γ, consider a sufficiently small neighborhood, N , around p i and p i +1 . Then σ i ( γ ) is given by exchanging p i and p i +1 by pushing p i up, pushing p i +1 down, and pulling γ inthat neighborhood with them. For example, in Figure 2, we apply σ and σ − to the admissible SU JI HONG curve in Figure 1A. This is a well-defined action as γ ∩ N is not self-crossing and does not cross γ ∩ H \ N . We will use the braid group to give an algorithm for some of the curves. (A) γ and σ (B) σ ( γ ) (C) γ and σ − (D) σ − ( γ ) Figure 2
For i, j ∈ [ n ] such that i < j , define σ [ i,j ] = σ j − · · · σ i +1 σ i and σ [ j,i ] = σ i σ i +1 · · · σ j − . Then σ − i,j ] = σ − i σ − i +1 · · · σ − j − and σ − j,i ] = σ − j − · · · σ − i +1 σ − i . Consider γ i and σ [ i,j ] γ i in Figure 3. Then σ [ i,j ] γ i (0) = p j and as x increases, σ [ i,j ] γ i ( x ) crosses ρ j − , ρ j − , ..., ρ i in that order. (A) γ i (B) σ [ i,j ] ( γ i ) (C) σ [ j,i ] σ [ i,j ] ( γ i ) Figure 3 -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 7
Definition 2.6.
Let I γ = ( k i ) mi =0 be the sequence of indices of rays that intersect γ ( x ) as x increases. Let π be a permutation and γ ∈ Γ. Then the associated root of γ with respect to π is α π ( γ ) = s π ( k m ) s π ( k m − ) . . . s π ( k ) s π ( k ) α π ( k ) , if s π ( k m ) s π ( k m − ) . . . s π ( k ) s π ( k ) α π ( k ) is positive , − s π ( k m ) s π ( k m − ) . . . s π ( k ) s π ( k ) α π ( k ) , if s π ( k m ) s π ( k m − ) . . . s π ( k ) s π ( k ) α π ( k ) is negative . Note that α π ( γ ) is always positive. Given two sequences I = ( i , . . . , i k ) and J = ( j , . . . , j (cid:96) ), let I ⊕ J = ( i , . . . , i k , j , . . . , j (cid:96) ). Definition 2.7.
Given a permutation π and a non-self-crossing admissible curve γ , assume π ( I γ ) = ( i, j , . . . , j p ). Then γ is said to be • positive if s j k · · · s j α i are positive for all k ∈ [ p ], • non-decreasing if s j k · · · s j α i ≥ D s j k − · · · s j α i , for all k ∈ [ p ], and • strictly increasing if s j k · · · s j α i > D s j k − · · · s j α i , for all k ∈ [ p ].Let Γ π,p be the set of positive non-self-crossing curves, Γ π,nd be the set of non-decreasing non-self-crossing curves, and Γ π,s be the set of strictly increasing curves. Remark 2.8.
It is clear that Γ π,s ⊂ Γ π,nd ⊂ Γ π,p ⊂ Γ. Conjecture 2.9.
Let Q be an acyclic quiver with n vertices and Γ π,nd be the set of non-decreasingnon-self-crossing admissible curve. Let P Q be the set of permutation π such that if a > b , thenthere is no arrow from π ( a ) to π ( b ) . Then (cid:91) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } = { real Schur roots of Q } . Remark 2.10.
For the acyclic quivers of type
ADE , the set of real Schur roots coincide with theset of positive roots. Thus we use the terms real Schur roots with positive roots interchangeablyin this paper. By definition of the associated root of a non-self-crossing admissible curve, it isclear that α π ( γ ) is a positive root, hence real Schur root, for all γ ∈ Γ. Thus, for an acyclicquiver Q of type ADE, (cid:91) π ∈ P Q { α π ( γ ) | γ ∈ Γ } ⊆ { real Schur roots of Q } . Coxeter Transformation
From now on we consider finite types of ADE. In this section we consider a Coxeter transfor-mamtion of the Weyl group W of a root system and develop some useful tools to prove Conjec-ture 2.9. The Weyl group W of a root system is a Coxeter group with generators, s , . . . , s n , SU JI HONG and relators on them. Let c π = s π (1) s π (2) · · · s π ( n ) where π is a permutation of [ n ]. The el-ement c π ∈ W is called a Coxeter transformation. As W is finite, h = | c π | is finite. Let θ i = s π ( n ) s π ( n − · · · s π ( i +1) α π ( i ) . We can recognize θ i as the associated root of a non-self-crossingadmissible curve. Let γ be a non-self-crossing curve as below:As I γ = ( i, . . . , n ), α π ( γ ) = θ i . We will abuse the notation and let c π : Γ → Γ be defined bysending γ ∈ Γ to a non-self-crossing admissible curve obtained from γ by wrapping around allmarked points counter-clockwise at the end. For example, γ i and c π ( γ i ) are given below.Note that I ( c π ( γ )) = I ( γ ) ⊕ ( n, n − , . . . , α π ( c γ ) = s π (1) s π (2) · · · s π ( n ) α π ( γ ) up to asign. Thus { c kπ θ i | ≤ k < h } + = { α π ( c kπ γ ) | ≤ k < h } where { c kπ θ i | ≤ k < h } + is the subset of { c kπ θ i | ≤ k < h } of positive roots.Son Nguyen informed the author of the following observation. Proposition 3.1 (Bourbaki [1]) . Let ∆ be a root sytem and W be its Weyl group. Let π be apermutation of [ n ] , θ i = s π ( n ) s π ( n − · · · s π ( i +1) α i , and Ω i = { c kπ θ i | k = 0 , . . . , h − } . Then(1) Ω i ∩ Ω j = ∅ for all i (cid:54) = j , and(2) ∆ = (cid:83) ni =1 Ω i Recall that P Q = { π ∈ S n | if π ( a ) → π ( b ) ∈ Q , then a < b } . Corollary 3.2 (Lee-Lee Conejcture for finite acyclic case) . For an acyclic quiver Q of type ADE , (cid:91) π ∈ P Q { α π ( γ ) | γ ∈ Γ } = { real Schur roots of Q } = ∆ + Q . Proof.
The set of real Schur roots of Q in this case is the same as the set of positive roots. Since α π ( γ ) is a positive root, we know that (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ } ⊆ ∆ + Q . Note that by Proposition -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 9 { c kπ θ i | ≤ k < h } is the set of all roots. Thus the subset of positive roots, denoted by { c kπ θ i | ≤ k < h } + , is the same as ∆ + Q .∆ + Q = { c kπ θ i | ≤ k < h } + = { α π ( c kπ γ ) | ≤ k < h } ⊆ (cid:91) π ∈ P Q { α π ( γ ) | γ ∈ Γ } ⊆ ∆ + Q . Therefore (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ } = ∆ + Q (cid:3) Example 3.3.
Consider the quiver below. Note that α is one of the positive roots of this quiver.3 2 1Consider the following non-self-crossing admissible curves below.Note that α π ( γ ) = α , α π ( γ ) = s s α = α , and α π ( γ ) = s s s s α = α . The associatedroots for all three curves are α .Given a positive root, there are many curves in Γ that correspond to that root. We impose astricter restriction on the non-self-crossing admissible curves. The following is our main theorem. Theorem 3.4.
Let Q be an acyclic quiver of type A, D, E and E . Then (cid:91) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } = ∆ + Q . Furthermore, if Q is an acyclic quiver of type A , then (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,s } = ∆ + Q . Note that Proposition 3.1 does not imply this theorem. To prove this theorem, we first observea relationship between the root system of a quiver and the root system of a full subquiver. Thepermutations in P Q are maps between the indices of the marked points and the vertices. Thusgiven a full subquiver induced by V ⊂ Q , π | π − ( V ) is a map between the indices of the markedpoints to the vertices in V and π | π − ( V ) ∈ P Q . Lemma 3.5.
Let Q = ( Q , Q ) be an acyclic quiver and Q (cid:48) be the full subquiver induced by V ⊂ Q . Let ϕ : P Q → P Q (cid:48) be given by ϕ ( π ) = π | π − ( V ) . For π (cid:48) ∈ P (cid:48) Q , • { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,s } ⊆ { α π ( γ ) | γ ∈ Γ π,s } , • { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,nd } ⊆ { α π ( γ ) | γ ∈ Γ π,nd } , • { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,p } ⊆ { α π ( γ ) | γ ∈ Γ π,p } , where π is any permutation in ϕ − ( π (cid:48) ) . Proof.
Let α = α π (cid:48) ( γ (cid:48) ) for π (cid:48) ∈ P Q (cid:48) and some non-self-crossing curve γ . Let ( j , . . . , j p ) be asequence of vertices in V such that π (cid:48) ( I ( γ (cid:48) )) = ( j i ) pi =1 . Let π ∈ ϕ − ( π (cid:48) ). Then π − ( j i ) − π − ( j i +1 )and π (cid:48)− ( j i ) − π (cid:48)− ( j i +1 ) are both positive or both negative. Let γ be a non-self-crossing curvesuch that γ (0) = p π − ( j ) , crosses ρ π − ( j i ) in the order that γ (cid:48) crosses, and goes under the markedpoints p π − ( k ) for all k (cid:54)∈ V . Then π ( I ( γ )) = ( j i ) pi =1 .If γ (cid:48) ∈ Γ π (cid:48) ,s , then s j k · · · s j α i > D s j k − · · · s j α i for all k ∈ [ p ]. Thus γ ∈ Γ π,s . Similarly, if γ (cid:48) ∈ Γ π,nd , then γ ∈ Γ π,nd and if γ (cid:48) ∈ Γ π,p , then γ ∈ Γ π,p . (cid:3) If γ is a non-self-crossing admissible curve, then so is c π γ . Furthermore, as I ( c π γ ) = I ( γ ) ⊕ ( n, n − , . . . , , we can compare α and c π α to say more about c π γ . Lemma 3.6.
Let Q be an acyclic quiver, π ∈ P Q and c π be a Coxeter transformation of theWeyl group of ∆( B ) . If α ∈ { α π ( γ ) | γ ∈ Γ π,nd } and α < D c π α , then c π α ∈ { α π ( γ ) | γ ∈ Γ π,nd } .Similarly, if α ∈ { α π ( γ ) | γ ∈ Γ π,nd } and α < D c − π α , then c − π α ∈ { α π ( γ ) | γ ∈ Γ π,nd } .Proof. Given π ∈ P Q and γ ∈ Γ π,nd , let α = α π ( γ ). Assume that α < D c π α . As c π : Γ → Γ, c π γ is a non-self-crossing admsisible curve. To show that c π γ is non-decreasing, note that I c π γ = I γ ⊕ ( n, . . . , , γ ∈ Γ π,nd , it suffices to show that α ≤ D s π ( n ) α and s π ( k +1) s π ( k +2) · · · s π ( n ) α ≤ D s π ( k ) s π ( k +1) · · · s π ( n ) α for all k ∈ [ n − a = α or a = s π ( k +1) s π ( k +2) · · · s π ( n ) α . Given anyroot, a simple reflection affects the coefficient of the corresponding simple root only. Thus a and s π ( k ) a differ only by the coefficient of α π ( k ) . As each simple reflections appears only once in c π ,the coefficient of α π ( k ) in a matches the coefficient of α π ( k ) in α . Similarly, the coefficient of α π ( k ) in s π ( k ) a matches the coefficient of α π ( k ) in c π α . As α < D c π α , we know that a ≤ D s π ( k ) a .Therefore c π γ is non-decreasing.If α < D c − π α , then c − π γ is a non-self-crossing admissible curve and c − π α = s π ( n ) s π ( n − · · · s π (1) α .Using a similar argument as above, we can see that c − π γ ∈ Γ π,nd and c − π α ∈ { α π ( γ ) | γ ∈ Γ π,nd } . (cid:3) Type A In this section, we focus on quivers of type A , i.e., quivers whose underlying undirected graphis: n n − . . . 2 1 e n − e n − e e Let A n be the set of such quivers. As mentioned before, all positive roots of type A are realSchur roots. It is known that all the positive roots of type A are of the form m (cid:88) k = (cid:96) α k , where 1 ≤ (cid:96) ≤ m ≤ n. For certain π ∈ P Q , any root is the associated root of a strictly increasing non-self-crossingadmissible curve. We show this by providing an algorithm to construct such curves. Furthermore, -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 11 for any permutation π ∈ P Q , any root is the associated root of a non-decreasing non-self-crossingadmissible curve.4.1. Strictly Increasing Curves.
Let Q ∈ A n in this section. To show that for a certainpermutation π ∈ P Q , { α π ( γ ) | γ ∈ Γ π,s } = ∆ + Q , we first define a unimodal permutation. Definition 4.1.
A permutation π ∈ S n is said to be unimodal if there exists k ∈ [ n ] such thatfor all i < j ≤ k , π ( i ) < π ( j ) and for all k ≤ i < j , π ( i ) > π ( j ). In particular, π ( k ) = n . Let U n be the set of all unimodal permutations in S n .For a permutation π ∈ U n , let Q π be a quiver in A n such that if π − ( i ) < π − ( n ), then t ( e i ) = i and if π − ( i ) > π − ( n ), then h ( e i ) = i . Define ω : U n → A n by ω ( π ) = Q π .For example, ω (( )) is5 4 3 2 1 e e e e For any quiver Q ∈ A n , let E u = { i ∈ [ n − | t ( e i ) = i } and E d = { i ∈ [ n − | h ( e i ) = i } .Define a = min E u and a i = min( E u \ { a , . . . , a i − } ) for 2 ≤ i ≤ | E u | . Let b = max E d , and b i = max( E d \ { b , . . . , b i − } ) for 2 ≤ i ≤ n − | E u | −
1. Then define a unimodal permutation π Q as below π Q = (cid:16) ··· | E u | | E u | +1 | E u | +2 ··· na a ··· a | Eu | n b ··· b n −| Eu |− (cid:17) . Let ψ : A n → U n be a map defined by ψ : Q (cid:55)→ π Q . For example, given Q below,4 3 2 1the unimodal permutation given by ψ is ψ ( Q ) = ( ). Remark 4.2.
Note that given Q ∈ A n and π Q = ψ ( Q ), there are no arrows from the vertex π Q ( a ) to the vertex π Q ( b ) in Q if a > b . Thus π Q ∈ P Q . Lemma 4.3.
There exists a bijection between A n and U n .Proof. We claim that ω : U n → A n given above is a bijection. It suffices to show that ω and ψ are inverses. To show that ω ( ψ ( Q )) = Q , it suffices to show that the arrows, e i , have the sameorientations in both Q and ω ( ψ ( Q )). Let π Q = ψ ( Q ). Let c = π − Q ( n ) and a = π − Q ( i ) for some i ∈ [ n − t ( e i ) = i , then a < c . Thus in ω ( π Q ) , t ( e π Q ( a ) ) = i . Similarly, if h ( e i ) = i , then h ( e π Q ( a ) ) = i in ω ( π Q ). Thus ω ( π Q ) and Q have the same orientations for all edges, which means ω ( ψ ( Q )) = Q .To see that ψ ( ω ( π )) = π , let Q π = ω ( π ) and k = π − ( n ). Then E u = { i ∈ [ n − | t ( e i ) = i } = { i ∈ [ n − | π − ( i ) < π − ( n ) } = { π ( i ) | i < k } . As π is unimodal, π (1) < π (2) < · · · < π ( k − a = min E u = π (1), a = π (2) , . . . , and a k − = π ( k − E d = { π ( i ) | i > k } and b = π ( k + 1) , . . . , b n − k = π ( n ).Thus ψ ( ω ( π )) = π . Therefore ψ and ω are inverses. (cid:3) We will describe non-self-crossing admissible curves using the braid group action on { γ , . . . , γ n } .Recall σ [ i,j ] = σ j − · · · σ i +1 σ i and σ [ j,i ] = σ i σ i +1 · · · σ j − for i < j . Given π ∈ U n , letΣ i = σ − π − ( i )+1 ,π − ( i +1)] σ π − ( i ) if π − ( i ) < π − ( i + 1) ,σ [ π − ( i ) − ,π − ( i +1)] σ − π − ( i ) − if π − ( i ) > π − ( i + 1) . From here on, let i = p π ( i ) . To depict Σ i , consider π = ( ) . Then Σ γ π − (2) = σ [4 , σ − γ isNote that Σ γ π − (2) loops around 2, go below all the points between 2 and 3, and end at 3.Then α ψ ( Q ) (Σ γ π − (2) ) = σ ( α ) = α + α . Also there exists a sufficiently small neighborhoodaround 3 so that σ γ π − (2) is homotopy equivalent to γ π − (3) in that neighborhood also the bothcurves do not intersect ρ π − (3) ; see Figure 4. Thus Σ Σ γ π − (2) loops around 3 and going underall the other points and end at 4. Proposition 4.4.
Let Q be an acyclic quiver of type A and π ∈ P Q ∩ U n . Let α = (cid:80) mi = (cid:96) α i bea positive root of type A and let γ = Σ m − Σ m − · · · Σ (cid:96) γ π − ( (cid:96) ) . Then γ ∈ Γ π,s and α π ( γ ) = α .Furthermore, { α π ( γ ) | γ ∈ Γ π,s } = ∆ + Q . See Example 4.5 for an illustration of this Proposition.
Proof.
Let α = α (cid:96) + α (cid:96) +1 + · · · + α m and γ = Σ m − · · · Σ (cid:96) γ π − ( (cid:96) ) . We claim that π ( I γ ) =( m, m − , . . . , (cid:96) ). To prove this, we induct on m − (cid:96) . If m − (cid:96) = 0, then γ = γ π − ( (cid:96) ) . Thus π ( I γ ) = ( (cid:96) ). -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 13 (A) Σ γ (B) γ Figure 4.
In the neighborhood around 3, Σ γ homotopy equivalent to γ .Let γ (cid:48) = Σ m − · · · Σ (cid:96) γ π − ( (cid:96) ) . By the induction hypothesis, π ( I γ (cid:48) ) = ( m − , m − , . . . , (cid:96) ). Thereare two possible cases: π − ( m ) < π − ( m −
1) and π − ( m ) > π − ( m − π − ( m ) > π − ( m −
1) without loss of generality.Let a = π − ( m −
1) and b = π − ( m ). Then γ = Σ m − γ (cid:48) = σ − b − σ − b − · · · σ − a +1 σ a γ (cid:48) . By theinduction hypothesis, γ (cid:48) (0) = m −
1, which means that σ a ( γ (cid:48) )(0) = π ( a + 1). There are twopossibilities for π ( I σ a ( γ (cid:48) ) ): ( π ( a + 1) , m − , m − . . . , (cid:96) ) or ( π ( a + 1) , m − , . . . , (cid:96) ). Refer toFigure 5 for illustrations of these two cases. We claim that the second case is not possible. Byway of contradiction, assume π ( I σ a ( γ (cid:48) ) ) = ( π ( a + 1) , m − , . . . , (cid:96) ). Apply σ − a to σ a ( γ (cid:48) ) to obtain γ (cid:48) . Note that in this case γ (cid:48) crosses ρ a +1 . Since γ (cid:48) crosses ρ a +1 , (cid:96) ≤ π ( a + 1) ≤ m −
2. However, π is unimodal and π ( a ) = m − π ( b ) = m . For any a < j < b , m − < π ( j ) < m . Thus, π ( a + 1) > π ( a ) = m −
1, which contradicts that π ( a + 1) ≤ m −
2. Thus the second case doesnot happen and σ a ( γ (cid:48) ) crosses ρ a transversally. (A) Case 1: σ a ( γ (cid:48) ) crosses ρ a (B) Case 2: σ a ( γ (cid:48) ) does not cross ρ a (C) Case 1: σ − a ( σ a ( γ (cid:48) )) (D) Case 2: σ − a ( σ a ( γ (cid:48) )) Figure 5.
Two casesNow consider σ − a +1 σ a ( γ (cid:48) ). The curve goes below p a +1 since σ a ( γ (cid:48) ) crosses ρ a . Thus the curvedoes not cross ρ a +1 . Also σ − a +1 σ a ( γ (cid:48) ) ends at p a +2 . Similarly, σ − b − σ − b − . . . σ − a +1 σ a ( γ (cid:48) ) does not cross ρ i for all a < i ≤ b . Thus γ ends at p b = m and crosses ρ π − ( m − . Therefore π ( I γ ) =( m, m − , . . . , (cid:96) ).For a quiver in A n , s i ( α j ) = α j + α i if j = i − , i + 1 α j otherwise . Thus, s m − α m = α m − + α m and s m − ( α m − + α m ) = s m − ( α m − ) + s m − ( α m ) = α m − + α m − + α m . By continuing this process, we can see that s j s j − · · · s m − α m = (cid:80) mk = j α k . Note that (cid:80) mk = j − α k < D (cid:80) mk = j α k for all (cid:96) ≤ j < m . Therefore γ ∈ Γ π,s and α π ( γ ) = α .Futhermore, { α π ( γ ) | γ ∈ Γ π,s } ⊆ ∆ + Q as mentioned before. Since α π ( γ ) = α for any positiveroot α of type A , ∆ + Q = { α π ( γ ) | γ ∈ Γ π,s } . (cid:3) Example 4.5.
Consider the following quiver Q :6 5 4 3 2 1Then π Q = ( ) and γ = Σ Σ Σ Σ Σ γ is shown below.Thus π Q ( I γ ) = (6 , , , , ,
1) and α π Q ( γ ) = s s s s s α = (cid:80) i =1 α i . Note that this curve isnot given by c kπ θ i for any i and k . Thus Proposition 3.1 does not cover this curve.4.2. Non-decreasing curve.
Not all permutations in P Q has a strictly increasing non-self-crossing admissible curve for every positive root. However, for any permutation in P Q , there is anon-decreasing non-self-crossing admissible curve for every positive root. Proposition 4.6.
Let Q ∈ A n . For any π ∈ P Q , { α π ( γ ) | γ ∈ Γ π,nd } = ∆ + Q .Proof. As { α π ( γ ) | γ ∈ Γ π,nd } ⊆ ∆ + Q for any Q and π ∈ P Q , we just need to show the othercontainment. The positive roots of type A are of the form (cid:80) (cid:96)i = k α i , which is the highest root ofthe full subquiver induced by the vertices k, . . . , (cid:96). By Lemma 3.5, if Q (cid:48) is a full subquiver of Q ,then { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,nd } ⊆ { α π ( γ ) | γ ∈ Γ π,nd } for π (cid:48) ∈ P Q (cid:48) and π ∈ ϕ − ( π (cid:48) ). Thus it suffices toshow that the highest root is in { α π ( γ ) | γ ∈ Γ π,nd } for any Q ∈ A n and π ∈ P Q .Let’s induct on | Q | . If | Q | = 1, it is trivial. Assume that for a quiver in A n with less than n vertices, the highest root is in { α π ( γ ) | γ ∈ Γ π,nd } for any π . Let Q ∈ A n be a quvier with n vertices and α be the highest root. For any π ∈ P Q , c π α < D α . If c π α is positive, then itis of the form (cid:80) (cid:96)i = k α k where (cid:96) − k < n −
1. So c π α is the highest root of a full subquiver -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 15 of Q and by the induction hypothesis, c π α is in { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,nd } where π (cid:48) = ϕ ( π ). Thus c π α ∈ { α π ( γ ) | γ ∈ Γ π,nd } by Lemma 3.5 and α ∈ { α π ( γ ) | γ ∈ Γ π,nd } by Lemma 3.6.If c π α is negative, then c π α = − α π (1) as α is the highest root of Q . Thus α = s π ( n ) · · · s π (2) α .In this case, let γ be the curve shown belowThen α π ( γ ) = s π ( n ) · · · s π (2) α = α and γ ∈ Γ π,s ⊂ Γ π,nd . (cid:3) Type D In this section, we prove that for an acyclic quiver of type D and any permutation respectingthe orientations, there exists a non-decreasing non-self-crossing admissible curve for every root.Let D n be the set of type D quivers with n vertices, i.e. quivers whose underlying graph is n n − n − n − n − . . . D are either type A or one of thefollowing: ··· ··· , ··· ··· ··· Unlike the type A case, if Q is an acyclic quiver of type D , not all positivie roots are theassociated roots of strictly increasing non-self-crossing curves. Example 5.1.
Consider the quiver Q below. Then P Q = { ( ) , ( ) , ( ) } .5 4321Let α = . If α = α π ( γ ) for some π ∈ P Q and γ ∈ Γ π,s , then π ( I γ ) = ( k , k , k , k , k , , k i ’s are distinct and the full subquiver induced by { k , . . . , k i } is connected for all i ≤ π ∈ P Q .If α = (cid:80) ni =1 β i α i and β i = 0 for some i , then it suffices to view α as a root of a full subquiverby Lemma 3.5. Thus we only consider the roots of the form: ··· , ··· ··· Lemma 5.2.
Let Q be an acylic quiver of type ADE with n vertices. Let α be a positive root of Q with a unique index k ∈ [ n ] such that s k ( α ) > D α . If there is a vertex j that is adjacent to thevertex k such that s j ( α ) < D α , then c π α < D α or c − π α > D α for any π ∈ P Q .Proof. Let Q be an acyclic quiver of type ADE . Let α = (cid:80) ni =1 β i α i be a positive root of Q suchthat there exist two adjacent vertices j, k such that s k α > D α , s j α < D , and s i α ≤ D α for allother i . Note that as α is a positive root of finite type, s i α = α ± α i or α for any i . As j and k are adjacent, s k s j ( α ) = s k ( α − α j ) = s k ( α ) − s k ( α j ) = α + α k − ( α j + α k ) = α − α j . If i (cid:54) = k, j , then s i ( α − α j ) = s i α − s i α j ≤ D α − α j as s i α ≤ D α and s i α j ≥ D α j . Thus s i ( α − α j ) ≤ D α − α j . If π ( k ) < π ( j ), then c π α = s π (1) · · · s π ( n ) α ≤ D s k s j α < D α . If π ( k ) > π ( j ),then c − π α = s π ( n ) · · · s π (1) α ≤ D s k s j α < D α . (cid:3) Proposition 5.3.
Let Q ∈ D n . Then { α π ( γ ) | γ ∈ Γ π,nd } = ∆ + Q for any π ∈ P Q .Proof. Let Q ∈ D n and π be any permutation in P Q . It is known that { α π ( γ ) | γ ∈ Γ π,nd } ⊆ ∆ + Q .The positivie roots of Q are type A , ··· ··· , or ··· ··· ··· . By Lemma 3.5 andProposition 4.6 all the type A positive roots are in { α π ( γ ) | γ ∈ Γ π,nd } .If α = ··· ··· , then it suffices to view α as a root of full subquiver induced by verticeswhose corresponding simple roots have non-zero coefficients by Lemma 3.5. So α = ··· and s n α > D α , s n − α < D α , and s i ≤ D α for all other i . Thus by Lemma 5.2, c π α or c − π α is smallerthan α . Without loss of generality, assume that c π α < D α . Moreover c π α < D α − α n − . If c π α is positive, c π α is of type A as α − α n − is type A . Thus by Proposition 4.6 and Lemma 3.5, c π α ∈ { α π ( γ ) | γ ∈ Γ π,nd } . Then α ∈ { α π ( γ ) | γ ∈ Γ π,nd } by Lemma 3.6. If c π α is negative, then c π α = − α π (1) as h ( α ) = n . Then α = s π ( n ) · · · s π (2) α . Just as we have seen in the proof ofProposition 4.6, there is γ such that α π ( γ ) = s π ( n ) · · · s π (2) α = α and γ ∈ Γ π,s ⊂ Γ π,nd .If α has the form ··· ··· ··· , then induct on the height of α , h ( α ) = (cid:80) ni =1 β i , to showthat α ∈ { α π ( γ ) | γ ∈ Γ π,nd } . The smallest h ( α ) is 5, i.e., α = ··· . By Lemma 3.5, itsuffices to view α as the highest root of a quiver in D n with four vertices. Then c π α is type A or . By Proposition 4.6 and the above argument, c π α ∈ { α π ( γ ) | γ ∈ Γ π,nd } for any π ∈ P Q .Then α ∈ { α π ( γ ) | γ ∈ Γ π,nd } for any π ∈ P Q by Lemma 3.6.If h ( α ) > , assume that for all α (cid:48) such that h ( α (cid:48) ) < h ( α ), α (cid:48) ∈ { α π ( γ ) | γ ∈ Γ π,nd } . Let α = (cid:80) ni =1 β i α i . By Lemma 3.5, we can assume that β i > i . Let j be the smallest indexso that β j = 2. Note that s j +1 α = α + α j +1 , s j α = α − α j , and s i α < D α for all other i . Thusby Lemma 5.2, c π α < D α or c − π α < D α . As h ( α ) > n and each simple reflections decrease thecoefficient by at most 1, c π α and c − π α cannot be negative. By Lemma 3.6 and the inductionhypothesis, α ∈ { α π ( γ ) | γ ∈ Γ π,nd } . (cid:3) Example 5.4.
Let Q be the quiver below, π = ( ) ∈ P Q and α = . -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 17 c π α = . The set of indices with nonzero coefficient is { , , , , } . Notethat s s s s s c π α = and s s s s s s s s s c π α = α . Thus let γ be the curve below.Then π ( I γ ) = (3 , , , , , , , , , , , , , , α π ( γ ) = α , and γ ∈ Γ π,nd .6. Type E In this section, we consider the quivers of type E and E . For n ∈ { , } , let E n be a set ofquivers whose underlying graph is:2 · · · n − n n − n − E are of the form α + α or α + α where α is a positiveroot of a full subquiver of E . If there is a curve γ such that the associated root is α, then γ canbe extended a curve whose associated root is α + α or α + α . Lemma 6.1.
Let Q be a quiver in E n and α be a positive root of Q . Let Q (cid:48) be the full subquiverinduced by the vertices , . . . , n . If α − α ∈ { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,nd } where π (cid:48) ∈ P Q (cid:48) , then α ∈{ α π ( γ ) | γ ∈ Γ π,nd } for some π ∈ P Q . Similarly, if Q (cid:48) is a full subquiver induced by the vertices , , . . . , n and α − α ∈ { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,nd } where π (cid:48) ∈ P Q (cid:48) , then α is in { α π ( γ ) | γ ∈ Γ π,nd } forsome π ∈ P Q .Proof. First note that the vertices 1 and 2 are always either source or sink. If α − α ∈ { α π (cid:48) ( γ ) | γ ∈ Γ π (cid:48) ,nd } where π (cid:48) ∈ P Q (cid:48) , then α − α ∈ { α π ( γ ) | γ ∈ Γ π,nd } for any π ∈ ϕ − ( π (cid:48) ) by Lemma 3.5. Asthe vertex 1 is either source or sink, there are permutations in ϕ − ( π (cid:48) ) so that π (1) is either 1 or n . Let γ (cid:48) ∈ Γ π,nd such that α π ( γ ) = α − α and γ (cid:48) does not cross ρ . Consider γ obtainedfrom γ (cid:48) by looping around 1 at the end. Then α π ( γ ) = s ( α π ( γ (cid:48) )) = s ( α − α ) = α and as γ (cid:48) isnon-decreasing, γ is also non-decreasing.Similarly, there exists a permutation π ∈ P Q such that π − (2) is either 1 or n and γ ∈ Γ π,nd such that α π ( γ ) = α − α . We can extend γ to γ (cid:48) by looping around 2 to obtain γ (cid:48) . Then α π ( γ (cid:48) ) = s ( α − α ) = α . (cid:3) Type E . All roots of type E are known. Let Q ∈ E in this subsection. Any full subquiverof Q is either type A or type D . By Lemma 3.5, we only need to consider the following roots: , , Lemma 6.2.
Let Q ∈ E . If α is one of the roots above, then for any π ∈ P Q , c π ( α ) and c π ( α ) are positive. Furthermore, either c π ( α ) < D α or c − π ( α ) < D α .Proof. The height of all three roots are greater than 6. As c π is a composition of 6 simplereflections, for any root α above, h ( c π α ) > h ( c − π α ) > π . If α = , then s α > D α , s α < D α , and s i α ≤ D α for all other i . If α = , then s α > D α , s α < D α ,and s i α ≤ D α for all other i . Thus in either cases, c π α < D α or c − π α < D α by Lemma 5.2.If α = , then α is the largest positive root of type E . As c π α is a positive root that isdifferent than α , c π α < D α and c − π α < D α for any π ∈ P Q . (cid:3) Proposition 6.3.
Let Q ∈ E . Then (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } = ∆ + Q .Proof. The full subquivers of Q are type A or D . By Lemma 3.5 and Propositions 4.6 and 5.3,the positive roots of these full subquivers are in { α π ( γ ) | γ ∈ Γ π,nd } for any π ∈ P Q . Any rootthat is smaller than is type A , type D , or satisfy the condition of Lemma 6.1. Thus byLemma 3.5, 6.1, Propositions 4.6 and 5.3, such root is in { α π ( γ ) | γ ∈ Γ π,nd } for some π ∈ P Q . Wejust need to show that , , and are in (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } .Let π ∈ P Q such that { π − (1) , π − (2) } = { , } , { , } , or { , } . Such permutation exists asthe vertices 1 and 2 are always either source or sink. By Lemma 6.2, c π α or c − π α is smaller than α where α = . If { π − (1) , π − (2) } = { , } , then by the proof of 6.1, every root that issmaller than α is in { α π ( γ ) | γ ∈ Γ π,nd } . As c π α or c − π α < D α , the root α is in { α π ( γ ) | γ ∈ Γ π,nd } by Lemma 3.6 and 6.1. Similarly, the roots , are in { α π ( γ ) | γ ∈ Γ π,nd } .If { π − (1) , π − (2) } = { , } or { , } , then c π α = c (1 2) π α as s and s commute. Note thatevery root that is smaller than α is in { α π ( γ ) | γ ∈ Γ π,nd } or { α (1 2) π ( γ ) | γ ∈ Γ (1 2) π,nd } . Thus c π α = c (1 2) π α is in either { α π ( γ ) | γ ∈ Γ π,nd } or { α (1 2) π ( γ ) | γ ∈ Γ (1 2) π,nd } . Therefore α is in { α π ( γ ) | γ ∈ Γ π,nd } ∪ { α (1 2) π ( γ ) | γ ∈ Γ (1 2) π,nd } and so are the roots and . (cid:3) -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 19 Type E . Of the positive roots of E , many of them are either type A , D , or E . Also byLemma 6.1, many of the roots of type E are known to be in (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } if theroots below are in (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } .1 2 2 2 2 11 1 2 2 3 2 11 1 2 3 3 2 11 1 2 2 3 2 121 2 3 3 2 12 1 2 3 4 2 12 1 2 3 4 3 12 1 2 3 4 3 22 Lemma 6.4.
Let Q ∈ E and π ∈ P Q . If α ∈ { , , , , , } , then c π α and c − π α are positive and c π α < D α or c − π α < D α .Proof. As the height of each roots is greater than 7, c π α and c − π α are positive. The root α = is the largest root of type E . Thus c π α < D α for any π ∈ P Q . For other roots, itsuffices to show that for each root there is a unique k ∈ [7] such that s k α > D α and an adjacentvertex (cid:96) such that s (cid:96) α < D α due to Lemma 5.2.For instance, let α = . Note that s α = α + α , s α = α − α , s α = α − α , and s i α = α for other i . Similarly, all the other roots have a unique index such that s k α = α + α k and an adjacent vertex j such that s j α = α − α j . (cid:3) Note that the roots and reduce to a root that we know to be in (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } . To show the others, we first need to look at the roots and . Lemma 6.5.
Let Q ∈ E and α = . Then α ∈ (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } .Proof. Note that s α = α − α , s α = α + α , and s i α = α for i (cid:54) = 4 ,
6. As the vertex 4 is notadjacent to the vertex 6, we cannot apply Lemma 5.2. To show that α is the associated root ofa non-decreasing non-self-crossing admissible curve for some π ∈ P Q , we divide the quivers intodifferent cases. First, consider any quiver Q of the form below:2 3 4 7 5 16Due to the orientation of Q , for every π ∈ P Q , π − (4) < π − (7) < π − (6). Thus c π α ≤ D s s s α = . This root and all the smaller roots are in { α π ( γ ) | γ ∈ Γ π,nd } or { α (1 2) π ( γ ) | γ ∈ Γ (1 2) π,nd } for π ∈ P Q such that { π − (1) , π − (2) } = { , } , { , } , or { , } , just like intype E case. Thus by Lemma 3.6, α ∈ (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } .Similarly, if Q is a quiver with the following orientations π − (4) > π − (7) > π − (6) and c − π α < D α .Now consider a quiver of the orientation2 3 4 7 5 16Here c π α and c − π α are not comparable to α . Thus to find π ∈ P Q and γ ∈ Γ π,nd such that α π ( γ ) = α , we use Sage to find a permutation π and non-decreasing non-self-crossing admissiblecurve γ such that α π ( γ ) = α . Refer to Tables 1 and 2. The quivers of following orientation2 3 4 7 5 16are obtained from the quivers of orientation above by reversing all arrows. Then the associatedroot of the mirror images of the curves in the tables would be α . (cid:3) Lemma 6.6.
Let Q ∈ E and α = . Then α ∈ (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } .Proof. Similarly as the proof of Lemma 6.5, we look at different cases of quivers. Note that s α = α − α , s α = α − α , s α = α + α , s α = α + α , and s i α = α for other i . As s s s α = s s s α = α − α , if π − (7) < π − (4) and π − (7) < π − (6), then c π α ≤ D .Similarly, if π − (7) > π − (4) and π − (7) > π − (6), then c − π α ≤ D .Note that s s α = α − α and s s α = α − α . Thus if π − (3) < π − (4) and π − (7) < π − (6),then c π α ≤ D . Similarly, if π − (3) > π − (4) and π − (7) > π − (6), c − π α ≤ D .Now consider quivers of the form: -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 21 Quiver,
Q π ∈ P Q γ ∈ Γ π,nd · · · · · ·· ( ) ·· · ····· · · · · ·· ( ) ··· ····· · · · · ·· ( ) ·· ······ · · · · ·· ( ) ·· · ·· ··· · · · · ·· ( ) ········ · · · · ·· ( ) ··· ·· ··· · · · · ·· ( ) ·· ··· ··· · · · · ·· ( ) ····· ·· Table 1.
Given a quiver and a permutation in P Q , the given non-self-crossingadmissible curve γ is non-decreasing and α π ( γ )=1 2 3 3 2 11 .2 3 4 7 5 16 Quiver,
Q π ∈ P Q γ ∈ Γ π,nd · · · · · ·· ( ) · · · ····· · · · · ·· ( ) · ·· ····· · · · · ·· ( ) · · ···· ·· · · · · ·· ( ) · · · ·· ··· · · · · ·· ( ) · ······· · · · · ·· ( ) · ·· ·· ··· · · · · ·· ( ) · · ··· ··· · · · · ·· ( ) · ···· ·· Table 2.
Given a quiver and a permutation in P Q , the given non-self-crossingadmissible curve γ is non-decreasing and α π ( γ )=1 2 3 3 2 11 .Quivers of this form do not fit the descriptions above. Also c π α (cid:54) < D α and c − π α (cid:54) < D α . We useSage to find non-decreasing non-self-crossing admissible curves whose associated root is .Refer to Table 3 for these curves. (cid:3) Proposition 6.7. If Q ∈ E , then ∆ + Q = (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } . -VECTORS AND NON-SELF-CROSSING CURVES FOR ACYCLIC QUIVERS OF FINITE TYPE 23 Quiver,
Q π ∈ P Q γ ∈ Γ π,nd · · · · · ·· ( ) ·· ·· ···· · · · · ·· ( ) ···· ···· · · · · ·· ( ) ·· ·· · ··· · · · · ·· ( ) ···· · ··· · · · · ·· ( ) · · ·· ···· · · · · ·· ( ) · ··· ···· · · · · ·· ( ) · · ·· · ··· · · · · ·· ( ) · ··· · ·· Table 3.
Given a quiver and a permutation in P Q , the given non-self-crossingadmissible curve γ is non-decreasing and α π ( γ )=1 2 2 3 2 11 . Proof.
Let Q ∈ E and π be a permutation in P Q such that π − (1) = 1 or 7, π − (2) is themaximum or minimum value of { , . . . , } \ { π − (1) } , and π − (3) is the maximum or minimumvalue of { , . . . , } \ { π − (1) , π − (2) } . Such permutation exists as π ∈ P Q ∩ U n satisfies suchcondition. Note that (1 2) π ∈ P Q if and only if | π − (1) − π − (2) | = 1 and (1 3) π ∈ P Q if andonly if | π − (1) − π − (3) | = 1. Let R = { (1 2) π , (1 3) π , (1 3)(1 2) π, π } ∩ P Q . For all p ∈ R , c p α is constant as s commutes with s and s .By Lemma 3.5 and Propositions 4.6, 5.3 and 6.3, the positive roots of type A , D , and E are in (cid:83) p ∈ P Q { α p ( γ ) | γ ∈ Γ p,nd } . Also by Lemma 6.1, any root of the form α + α where α isa positive root of type D or of the form α + α where α is a positive root of type E is in (cid:83) p ∈ P Q { α p ( γ ) | γ ∈ Γ p,nd } . In particular, a root of the form α + α where α is type D is in { α p ( γ ) | γ ∈ Γ p,nd } for p ∈ P Q so that p − (1) = 1 or 7. Also a root of the form α + α where α is type E is in { α p ( γ ) | γ ∈ Γ p,nd } for p ∈ P Q so that p − (2) = 1 or 7 and p | p − ( { , ,... } ) fits the condition of the permutation given in the proof of Proposition 6.3. At least one of thepermutations in R satisfies such conditions; thus any root of type A , D , and E and the roots ofthe form α + α where α is type D and α + α where α is type E are in (cid:83) p ∈ R { α p ( γ ) | γ ∈ Γ p,nd } .By Lemma 6.5 and Lemma 6.6, , ∈ (cid:83) p ∈ P Q { α p ( γ ) | γ ∈ Γ p,nd } . In particular,they are in (cid:83) p ∈ R { α p ( γ ) | γ ∈ Γ p,nd } . To show that the roots in Lemma 6.4 are in (cid:83) p ∈ P Q { α p ( γ ) | γ ∈ Γ p,nd } , we induct on the height of the roots.The root with smallest height of the roots in Lemma 6.4 is . Either c π α or c − π α issmaller than which is in { α π ( γ ) | γ ∈ Γ π,nd } as π − (1) = 1 or 7. Thus α ∈ { α π ( γ ) | γ ∈ Γ (1 2) π,nd } .Let α be a root in Lemma 6.4. Assume that for any root with height less than h ( α ) is in (cid:83) p ∈ R { α p ( γ ) | γ ∈ Γ p,nd } . By Lemma 6.4, c p α or c − p α is smaller than α . Assume without loss ofgenerality, c p α < D α . As c p α is same for all p ∈ R , there is p ∈ R such that c p α ∈ { α p ( γ ) | γ ∈ Γ p,nd } and by Lemma 3.6, α ∈ { α p ( γ ) | γ ∈ Γ p,nd } . Therefore ∆ + Q = (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } . (cid:3) Theorem 6.8.
Let Q be an acyclic quiver of type A, D, E and E . Then (cid:91) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,nd } = ∆ + Q . Furthermore, if Q is an acyclic quiver of type A , then (cid:83) π ∈ P Q { α π ( γ ) | γ ∈ Γ π,s } = ∆ + Q .Proof. This theorem follows from Propositions 4.4, 4.6, 5.3, 6.3, and 6.7. (cid:3)
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