SEEDS AND WEIGHTED QUIVERS. 1. Seeds

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1 SEEDS AND WEIGHTED QUIVERS TSUKASA ISHIBASHI Abstract. In this note, we describe the correspondence between (skew-symmetrizable) seeds and weighted quivers. Also we review the construction of the seed associated with a word of simple roots of a semisimple complex Lie group. Hopefully this note helps the reader to clarify the relations between the works of Fomin- Zelevinsky and Fock-Goncharov. We define a seed following [FG06]. 1. Seeds Definition 1.1. A (skew-symmetrizable) seed i = (I, I 0, ε, d) consists of the following data: I is a finite set and I 0 I is a subset. The set I 0 is called the frozen subset. ε = (ε ij ) i,j I is a Q-valued matrix, called the exchange matrix, such that ε ij Z unless (i, j) I 0 I 0. d = (d i ) Z I >0 is a tuple of integers, called the multipliers, such that gcd(d i i I) = 1 and the matrix ε ij := ε ij d j is skew-symmetric. Remark 1.. (1) The exchange matrix defined above is related to the Fomin-Zelevinsky s exchange matrix B = (b ij ) [FZ0] by the rule ε ij = b ji. In particular d i b ij = ε ji d i = ε ji is skew-symmetric. () In [FG09, Zic16], a different definition is employed: the skew-symmetrization is given by ε ij = ε ij d 1 j. However, Definition 1.1 seems to be more suitable in comparison with the symmetrization of a Cartan matrix. Since these papers only treat with the rank seeds or skew-symmetric seeds, modifications are clear. For instance, in [Ip16, Le16], the definition above is employed where the higher rank cases are considered. A weighted quiver is a quiver with an integer weight on each vertex. Definition 1.3. For a seed i = (I, I 0, ε, d), we define a weighted quiver Q = Q i as follows: The set of vertices V (Q) := I. The weights are given by the multipliers d = (d i ). The structure matrix σ ij := #{arrows from i to j} #{arrows from j to i} is given by σ ij = ε ij gcd(d i, d j )/d i d j = d 1 i ε ij gcd(d i, d j ). ( ( ) ) 0 1 Example 1.4. Let i = (I, I 0, ε, d) := {1, },,, (1, ). Then ε = ( ) ( ) and σ =. Hence the corresponding weight quiver is given by Date: December,

2 TSUKASA ISHIBASHI Here the weights are indicated by the numbers encircled. We omit the weight 1. See Examples 1.7 and.3 for more examples. Definition 1.5 (seed mutations). For a seed i = (I, I 0, ε, d) and an index k I I 0, we define a new seed i = (I, I 0, ε, d) by the rule ε ij ε ij := ε ij + ε ik ε kj + ε ik ε kj otherwise. The map i i is called the mutation in the direction k, and we denote µ k : i i. The set i of seeds obtained from the initial seed i by finite compositions of mutations and seed isomorphisms is called the mutation class of i. From now on, we assume that there exists an integer m Z >0 such that d = (d i ) {1, m} I. Then the seed mutations are described by the corresponding weighted quivers as follows: Lemma 1.6 (quiver mutations). Let Q, Q be the weighted quivers associated with the seeds i and i, respectively. Then the structure matrix σ ij of Q is related to the structure matrix σ ij of Q by the equation σ ij σ ij = σ ij + σ ik σ kj + σ ik σ kj ij k otherwise, where k ij = m if d i = d j d k, and k ij = 1 otherwise. Proof. Multiply d 1 i gcd(d i, d j ) to the mutation rule of the matrix ε. Then we get σ ij σ ij = σ ij + σ ik σ kj + σ ik σ kj gcd(d i, d j ) d k otherwise. gcd(d i, d k ) gcd(d j, d k ) The last factor coincides with k ij. Example 1.7. Let Q be the weighted quiver given by Then the weighted quivers µ 1 (Q) and µ 3 (Q) are given by and 3 4, respectively.

3 SEEDS AND WEIGHTED QUIVERS 3. Seeds associated with double reduced words In this section we review the construction of the seed associated with a word of simple roots of a semisimple complex Lie group, following [FG06]. The construction can be generalized to Kac-Moody Lie groups, see [Wil13]. Let us fix the notations. Let G be a semisimple complex Lie group with trivial center. Let Π be the set of positive simple roots. The Cartan matrix is given by C := (, )/(, ), where, Π. Let d := (, )/ Z >0 be the multipliers. Here the inner product is normalized so that (, ) = for a short root Π. Then the matrix Ĉ := C d = (, ) is symmetric. Let Π denote the set of negative simple roots. Let W be the free semigroup generated by Π Π. An element of W is a word D = i1... ik, where ij Π Π. Remark.1. Our convention of the Cartan matrix is the transpose of that used in [FZ0] and their subsequent works. Compare with Remark 1.(1). In the following, we construct a seed J(D) for an element D W. Definition. (elementary seeds). For a simple positive root Π, a seed J() := (J(), J 0 (), ε(), d()) is given as follows. J() = J 0 () := (Π {}) {, + }, where ± is a new element. The exchange matrix ε() is given by ε() = 0 (, ), ε() + = ε() + = 1, ε() ± = ±( C )/, ε() ± = ( C )/ ( Π {}). The multipliers d() are given by d() ± := d, d() := d for Π {}. Note that J() is indeed a seed, since the matrix Ĉ := C d is symmetric. Halfintegers are allowed since all the indices are frozen. We call J() the elementary seed associated with the simple positive root. For a simple negative root ᾱ Π, the elementary seed J(ᾱ) is defined to be the chiral dual of the seed J(), J(ᾱ) := (J(), J 0 (), ε(), d()). Example.3. (1) Type A 1. The elementary seed J() associated with the unique simple root is given the quiver below. + () Type A. Let Π = {, }. The elementary seeds J(), J() are given by the quivers below. + + J() J() Here in the leftmost side is the Dynkin diagram of type A. Dashed arrows are half-arrows, whose corresponding entries in the structure matrix σ are 1/.

4 4 TSUKASA ISHIBASHI (3) Type A 3. Let Π = {,, }, where C = 0. The elementary seeds J(), J() and J() are given by the quivers below J() J() J() (4) Type B. ( Let Π = ){, }, where is the long root. The Cartan matrix is C B =. The elementary seeds J(), J() are given by the 1 quivers below. + J() + J() Definition.4 (the amalgamation of seeds). Let i 1 = (J, J 0, ε, d) and i = (I, I 0, η, c) be two seeds, L a set embedded into both J 0 and I 0 so that d(i) = c(i) for all i L. Then the amalgamation i = (K, K 0, ζ, b) of i 1 and i is the seed given by: K := I L J, K 0 := I L J 0, b := c L d. 0 if i I L and j J L 0 if i J L and j I L ζ ij := η ij if i I L or j J L ε ij if i J L or j I L η ij + ε ij if i, j L For an element D = i1... ik W, the seed J(D) is constructed by amalgamating the elementary seeds J( i1 ),, J( ik ) in this order. The vertices which are not extremal are mutable. See [FG06] for the formal definition. We give several examples below. Example.5. (1) Type A. Let D := W. The seed J(D) is given by the quiver below. x 0 x 1 x 0 x 1 x Here only the vertex x 1 is mutable. () Type A. Let D := ᾱ W. The seed J(D) is given by the quiver below. x 0 x 0 x 1 x

5 SEEDS AND WEIGHTED QUIVERS 5 Here only the vertex x 1 is mutable. (3) Type A 3. Let D := W. The seed J(D) is given by the quiver below. x 0 x 1 x 0 x 1 x x 0 x 1 x x 3 Here the vertices x 1, x and x 1 are mutable. (4) Type B. Let D := W. The seed J(D) is given by the weighted quiver below. x 0 x 1 x x 0 x 1 x Here the vertices x 1 and x 1 are mutable. Let u, v W (G). A double reduced word of (u, v) is a shuffle of reduced words of u and v. Here the word of u is written by the letters in Π. For example, D = ᾱ is a double reduced word of (u, v) = (, ) W (A ) W (A ). Let B +, B be opposite Borel subgroups corresponding to the bases Π and Π, respectively. Then we have the Bruhat decompostions G = u W (G) B +ub + = v W (G) B vb. Each intersection G u,v := B + ub + B vb is called a double Bruhat cell. Theorem.6 (Fock-Goncharov [FG06], Williams [Wil13]). Let D W be a double reduced word of (u, v) W (G) W (G). Then the double Bruhat cell G u,v has the structure of a cluster X -variety, G u,v = X J(D) (C) birationally. The cluster Poisson structure coincides with the restriction of the standard Lie-Poisson structure on G. References [FG06] V. V. Fock and A. B. Goncharov. Cluster X-varieties, amalgamation, and Poisson-Lie groups. In Algebraic geometry and number theory, volume 53 of Progr. Math., pages Birkhäuser Boston, Boston, MA, 006. [FG09] Vladimir V. Fock and Alexander B. Goncharov. Cluster ensembles, quantization and the dilogarithm. Ann. Sci. Éc. Norm. Supér. (4), 4(6): , 009. [FZ0] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations. J. Amer. Math. Soc., 15():497 59, 00. [Ip16] Ivan Chi-Ho Ip. Cluster realization of u q (g) and factorization of the universal r-matrix, 016. [Le16] Ian Le. Cluster structures on higher Teichmuller spaces for classical groups, 016. [Wil13] Harold Williams. Cluster ensembles and Kac-Moody groups. Adv. Math., 47:1 40, 013. [Zic16] Christian K. Zickert. Fock-Goncharov coordinates for rank two Lie groups, 016. Graduate School of Mathematical Sciences, the University of Tokyo, Komaba, Meguro, Tokyo , Japan address: ishiba@ms.u-tokyo.ac.jp