QUANTUM F -POLYNOMIALS IN THE THEORY OF CLUSTER ALGEBRAS. A dissertation presented by Thao Tran to The Department of Mathematics

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1 QUANTUM F -POLYNOMIALS IN THE THEORY OF CLUSTER ALGEBRAS A dissertation presented by Thao Tran to The Department of Mathematics In partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics Northeastern University Boston, Massachusetts April

2 2 QUANTUM F -POLYNOMIALS IN THE THEORY OF CLUSTER ALGEBRAS by Thao Tran Abstract of Dissertation Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the field of Mathematics in the Graduate School of Arts and Sciences of Northeastern University, April, 2010

3 Abstract: F -polynomials and g-vectors were defined by Fomin and Zelevinsky to give a formula which expresses cluster variables in a cluster algebra in terms of the initial cluster data. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra. In this thesis, we define and prove the existence of analogous quantum F -polynomials for quantum cluster algebras. We compute quantum F -polynomials and g-vectors for a certain class of cluster variables, which includes all cluster variables in type A n quantum cluster algebras. Finally, we give formulas for F -polynomials and quantum F -polynomials in classical types when the initial exchange matrix is acyclic. 3

4 4 Acknowledgements I would like to thank my advisor Andrei Zelevinsky for sharing his vast knowledge of mathematics and for giving me his continual support and guidance in the course of my research. I am also very grateful for my fellow graduate students in the math department who have provided me with encouragement and friendship over the years.

5 5 Dedication I dedicate this thesis to my parents and to my boyfriend David. This thesis would not have been possible without their unconditional love and support.

6 6 Contents Abstract of Dissertation 2 Acknowledgements 4 Dedication 5 1. Introduction 6 2. Cluster Algebras of Geometric Type 8 3. F -polynomials and g-vectors F -polynomials in Classical Types F -polynomials and g-vectors corresponding to trees F -polynomials in type A n F -polynomials in Classical Types for Acyclic Initial Exchange Matrix Type D n Projections of F -polynomials and g-vectors Type C n Type B n Quantum Cluster Algebras F-polynomials in Quantum Cluster Algebras Properties of Quantum F -polynomials Quantum F -polynomials corresponding to trees Quantum F -polynomials in Classical Types for Acyclic Initial Matrix Combinatorial realizations of cluster algebras of types B n, C n, D n Type D n Type B n Type C n 94 References 98

7 7 1. Introduction Cluster algebras were introduced by Fomin and Zelevinsky in [8] in order to study total positivity and canonical bases in semisimple groups. Let m n be positive integers. Roughly speaking, a cluster algebra is a subalgebra generated by a distinguished collection of generators called cluster variables inside of an ambient field F which is isomorphic to the field of rational functions in m independent variables. To obtain these cluster variables, one begins with an initial seed. A seed is a pair ( x, B) such that x is an m-tuple of elements from F with the first n terms being cluster variables and the remaining m n terms being coefficient variables, and such that B is an m n integer matrix whose top n n submatrix is skew-symmetrizable; the m-tuple x is called the extended cluster of the seed. Seed mutations are certain operations which transform a seed into another seed. In mutating from one seed ( x, B) to another seed ( x, B ), one cluster variable x is exchanged for another cluster variable x ; the elements x, x satisfy a certain exchange relation which is of the form xx = M + +M, where M +, M are monomials on disjoint subsets of variables from x {x}. The role of the matrix B is to dictate exactly what these monomials M +, M are. The set of cluster variables which generate a given cluster algebra is obtained by mutating the initial seed with all possible sequences of mutations and taking all cluster variables from the seeds that result. Quantum cluster algebras were defined by Berenstein and Zelevinsky in [1]. A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra with an additional generator q lying in the center. Under this deformation, each extended cluster (x 1,..., x m ) in the cluster algebra is replaced by the m-tuple (X 1,..., X m ), where the elements X 1,..., X m now quasi-commute, i.e., for each 1 i, j m, there exists λ ij Z such that X i X j = q λ ij X j X i. The exchange relations are also altered to allow for the fact that extended cluster elements must quasi-commute. In [10], Fomin and Zelevinsky defined F -polynomials and g-vectors corresponding to an initial n n skew-symmetrizable integer matrix B 0. They gave recurrence relations for the F -polynomials and g-vectors which essentially only depend on B 0. One of the main results of [10] was a formula expressing any cluster variable in terms of F -polynomials and g-vectors using only the initial cluster of A. In this thesis, we demonstrate the existence of quantum F -polynomials, which is an analogue of F -polynomials in the quantum cluster algebra setting. Quantum F -polynomials satisfy the property that any cluster variable in a quantum cluster algebra may be computed (up to a multiple

8 8 of a power of q) in a formula with the appropriate quantum F -polynomial and g-vector using only the initial extended cluster. It is conjectured (and proven in some cases) that this formula can be sharpened so that the multiple of q does not appear (see Theorem 7.1). By setting q = 1 in a quantum F -polynomial, we obtain the appropriate F -polynomial for (nonquantum) cluster algebras. A (quantum) cluster algebra is of finite type if the total set of cluster variables is finite. In [9], it was proven that finite type cluster algebras are classified by the same Cartan-Killing types as semisimple Lie algebras or finite root systems. It was proven in [1] that an identical classification also holds for quantum cluster algebras of finite type. In particular, finite type cluster algebras include those of classical types A n, B n, C n, and D n. In this thesis, we recall a formula for F -polynomials from [4] given for cluster variables corresponding to induced trees of the quiver which can be defined using the initial matrix B 0, and state and prove a formula for the corresponding quantum F -polynomials. In addition, we show that these formulas may be applied to find all F -polynomials and quantum F -polynomials in type A n. Furthermore, formulas for F -polynomials and for quantum F -polynomials are given in classical types for the case where the initial matrix is acyclic (Theorem 4.16 and Theorem 9.1). For F -polynomials, different formulas were given in the cases where the initial matrix is bipartite [10], acyclic [18], and in general in [14]. Also, a formula was given in [15] for F -polynomials and g-vectors for cluster algebras arising from surfaces with marked points as defined in [7], and from this formula, one may compute the F -polynomials corresponding to type A n and D n. Theorem 4.16 differs from these other formulas in the statement of the formula, the methods for proving it, or both. The formula for the finite type F -polynomials in this thesis gives a combinatorial recipe for determining which monomials occur with nonzero coefficient and what the coefficient of the monomial is in that case. The formula for type A n was already proven in [4] using a formula for F -polynomials given in terms of quiver representations. To prove the result in type D n here, the same formula is used. To finish the proof in the other types, we show that the F -polynomials can be obtained as certain projections of F -polynomials from type A and D. This thesis is based on two papers by the author, one of which will appear in the journal Algebras and Representation Theory [16] and the other of which has been submitted [17]. The organization of the thesis is as follows. In Section 2, we recall the definition of cluster algebras. In Section 3, we recall the definition of F -polynomials and g-vectors as well as the aforementioned formula for cluster

9 variables (Theorem 3.6). In Section 4, we recall the result about F -polynomials corresponding to induced trees as described above, and the results about F -polynomials in classical types are stated and proven. Section 5 is devoted to recalling the definition of quantum cluster algebras. In Section 6, (left) quantum F -polynomials are defined. Theorem 6.3 is devoted to proving their existence. Properties of quantum F -polynomials are given in Section 7. Proposition 7.4 shows how to easily compute right quantum F -polynomials once the left ones are known. A recurrence relation 9 for quantum F -polynomials is given in Theorem Section 8 gives quantum F -polynomials corresponding to induced trees. In Section 9, the formulas for quantum F -polynomials in classical types are stated and proven. 2. Cluster Algebras of Geometric Type Following [10, Section 2], we give the definition of a cluster algebra of geometric type as well as some properties of these cluster algebras. The proofs of any statements given in this section can be found in [10, Section 2]. Definition 2.1. Let J be a finite set of labels, and let Trop(u j : j J) be an abelian group (written multiplicatively) freely generated by the elements u j (j J). We define the addition in Trop(u j : j J) by (2.1) j u a j j j u b j j = j u min(a j,b j ) j, and call (Trop(u j : j J),, ) a tropical semifield. If J is empty, we obtain the trivial semifield consisting of a single element 1. The group ring of Trop(u j : j J) (as a multiplicative group) is the ring of Laurent polynomials in the variables u j. Fix two positive integers m, n with m n. Let P = Trop(x n+1,..., x m ), and let F be the field of rational functions in n independent variables with coefficients in QP, the field of fractions of the integral group ring ZP. (Note that the definition of F does not depend on the auxiliary addition in P.) The group ring ZP will be the ground ring for the cluster algebra A to be defined, and F will be the ambient field, with n being the rank of A. Definition 2.2. A labeled seed in F is a pair ( x, B) where x = (x 1,..., x m ), where x 1,..., x n are algebraically independent over QP, and F = QP(x 1,..., x n ), and

10 10 B is an m n integer matrix such that the submatrix B consisting of the top n rows and columns of B is skew-symmetrizable (i.e., DB is skew-symmetric for some n n diagonal matrix D with positive integer diagonal entries). We call x the extended cluster of the labeled seed, (x 1,..., x n ) the cluster, B the exchange matrix, and the matrix B the principal part of B. We fix some notation to be used throughout the thesis. For x Q, [x] + = max(x, 0); 1 if x < 0; sgn(x) = 0 if x = 0; 1 if x > 0; Also, for i, j Z, write [i, j] for the set {k Z : i k j}. In particular, [i, j] = if i > j. Definition 2.3. Let k [1, n]. We say that an m n matrix B is obtained from an m n matrix B = (b ij ) by matrix mutation in direction k if the entries of B are given by b ij if i = k or j = k; (2.2) b ij = b ij + sgn(b ik ) [b ik b kj ] + otherwise. Definition 2.4. Let ( x, B) be a labeled seed in F as in Definition 2.2, and write B = (b ij ). The seed mutation µ k in direction k transforms ( x, B) into the labeled seed µ k ( x, B) = ( x, B ), where x = (x 1,..., x m), where x j = x j for j k, and (2.3) x k = x 1 k ( m i=1 x [b ik] + i + m i=1 x [ b ik] + i B is obtained from B by matrix mutation in direction k. ), One may check that the pair ( x, B ) obtained is again a labeled seed. Furthermore, the seed mutation µ k is involutive, i.e., applying µ k to ( x, B) yields the original labeled seed ( x, B).

11 Definition 2.5. Let T n be the n-regular tree whose edges are labeled with 1,..., n in such a way that for each vertex, the n edges emanating from that vertex each receive different labels. Write k t t to indicate t, t T n are joined by an edge with label k. 11 Definition 2.6. A cluster pattern is an assignment of a labeled seed ( x t, B t ) to every vertex t T n k such that if t t, then the labeled seeds assigned to t, t may be obtained from one another by seed mutation in direction k. Write x t = (x 1;t,..., x m;t ), B t = (b t ij ), and denote by Bt the principal part of B t. Definition 2.7. For a given cluster pattern, write (2.4) X = {x j;t : j [1, n], t T n }. The elements of X are the cluster variables. The cluster algebra A associated to this cluster pattern is the ZP-subalgebra of F generated by all cluster variables. That is, A = ZP[X ]. 3. F -polynomials and g-vectors The reference for all of the information given in this section (except the definition and properties of extended g-vectors) is [10]. For this section, fix an n n skew-symmetrizable integer matrix B 0 and an initial vertex t 0 T n. Assume that any cluster algebra A in this section has initial exchange matrix B 0 with principal part B 0. Definition 3.1. We say that a cluster pattern t ( x t, B t ) (or its corresponding cluster algebra) has principal coefficients at t 0 if x t0 = (x 1,..., x 2n ) (i.e., m = 2n) and the exchange matrix at t 0 is the principal matrix corresponding to B 0 given by (3.1) B0 = B0 I where I is the n n identity matrix. Denote the corresponding cluster algebra by A = A (B 0, t 0 ). Definition 3.2. Let A = A (B 0, t 0 ) be the cluster algebra with principal coefficients, with labeled seed at t 0 written as (3.2) (3.3) x t0 = (x 1,..., x n, y 1,..., y n ) B 0 = (b 0 ij)

12 12 Let Q sf (z 1,..., z l ) denote the set of all rational functions in l independent variables z 1,..., z l which can be expressed as subtraction-free rational expression in z 1,..., z l. Observe that by iterating the exchange relations in (2.3), any cluster variable x l;t A can be expressed as a unique rational function in x 1,..., x n, y 1,..., y n given as a subtraction-free rational expression. Denote this rational function by (3.4) X l;t = X B0 ;t 0 l;t Q sf (x 1,..., x n, y 1,..., y n ). Let F l;t = F B0 ;t 0 l;t y i = u i for all i in X l;t : Q sf (u 1,..., u n ) denote the rational expression obtained by setting x i = 1 and (3.5) F l;t = X l;t (1,..., 1, u 1,..., u n ). Using [9, Proposition 11.2], which is a sharpened version of the Laurent phenomenon (see [8, Theorem 3.1]) for cluster algebras with principal coefficients, the next theorem shows that the F l;t functions are indeed polynomials: Theorem 3.3. Let A = A (B 0, t 0 ) be a cluster algebra with principal coefficients at t 0. Suppose that the initial seed in A is given as in (3.2). Then (3.6) (3.7) X l;t Z[x ±1 1,..., x±1 n, y 1,..., y n ], F l;t Z[u 1,..., u n ]. The F -polynomials may be computed using the following recurrence. Proposition 3.4. Let t B t = (b t ij ) (t T n) be the family of 2n n matrices associated with the cluster algebra A (B 0, t 0 ). Then the polynomials F l;t = F B0 ;t 0 l;t (u 1,..., u n ) are uniquely determined by the initial conditions (3.8) F l;t0 = 1 (l = 1,..., n), together with the recurrence relations (3.9) F l;t = F l;t for l k; (3.10) F k;t = n i=1 F [bt ik ] + i;t n j=1 u[bt n+j,k ] + j + n i=1 F [ bt ik ] + n i;t j=1 u[ bt n+j,k ] + j, F k;t

13 13 for every edge t k t such that t lies on the (unique) path from t 0 to t in T n. In [10, Section 6], the following Z n -grading of A = A (B 0, t 0 ) was introduced: Proposition 3.5. Define a Z n -grading of Z[x ±1 1,..., x±1 n, y 1 ±,..., y± n ] by (3.11) deg(x i ) = e i, deg(y j ) = b 0 j, where e 1,..., e n are the standard basis vectors for Z n and b 0 j is the jth column of B0. Under this Z n -grading, the cluster algebra A (B 0, t 0 ) is a Z n -graded subalgebra of Z[x ±1 1,..., x±1 n, y ± 1,..., y± n ], and every cluster variable in A (B 0, t 0 ) is a homogeneous element. The Z n -degree of the cluster variable x l;t A (B 0, t 0 ) will be denoted by g l;t = g B0 ;t l;t = Z n ; we refer to g l;t as the g-vector of x l;t. g 1. g n For any cluster algebra A such that the initial extended cluster at t 0 is given by x = (x 1,..., x m ) and the exchange matrix at t 0 is B 0 = (b 0 ij ), let (3.12) y j = m i=n+1 x b0 ij i, ŷ j = m i=1 x b0 ij i for j [1, n]. The next theorem shows that any cluster variable in A may be computed in terms of the initial extended cluster if the corresponding F -polynomial and g-vector are known. Theorem 3.6. Let A be a cluster algebra such that the principal part of the exchange matrix at t 0 is B 0. Using the notation just given for the initial seed at A, any cluster variable x l;t in A may be expressed as (3.13) x l;t = F B0 ;t 0 l;t (ŷ 1,..., ŷ n ) F B0 ;t 0 l;t P (y 1,..., y n ) xg1 1 xgn n,

14 14 where g B0 ;t 0 l;t = g 1. g n and P = Trop(x n+1,..., x m ). Furthermore, for j [1, n], u j does not divide F l;t. Thus, in the cluster algebra A = A (B 0, t 0 ), (3.14) x l;t = F B0 ;t 0 l;t (ŷ 1,..., ŷ n )x g 1 1 xgn n. Next, we introduce extended g-vectors which generalize the g-vectors to any cluster algebra A. Definition 3.7. Let A be any cluster algebra with initial extended cluster x = (x 1,..., x m ), and let l [1, n], t T n. Define g B 0 ;t 0 l;t cluster variable x l;t A satisfies = (g 1,..., g m ) to be the unique vector in Z m such that the (3.15) x l;t = F B0 ;t 0 l;t (ŷ 1,..., ŷ n )x g xgm m. For l [n + 1, m], define g B 0 ;t 0 l;t = e l Z m. We call g l;t = g B 0 ;t 0 l;t the extended g-vector. Observe that the existence of g l;t follows immediately from Theorem 3.6. Explicitly, the first n coordinates of g l;t are given by g B0 ;t 0 l;t, and the remaining m n coordinates are given by the exponents of x n+1,..., x m in the expression (3.16) 1 F B0 ;t 0 l;t P (y 1,..., y n ), where P = Trop(x n+1,..., x m ). In particular, g B0 ;t 0 l;t = g B 0 ;t 0 l;t when B 0 is principal. (Here, we extend g B0 ;t 0 l;t Z n to an element of Z 2n by appending n 0 s.) The next proposition gives a recurrence relation for the extended g-vectors. Let b j denote the jth column of the matrix B 0 (j [1, n]). Let B t = (b t ij ) denote the matrix obtained from B 0 by mutating the matrix from t 0 to vertex t T n, and let (b,t ij ) be the 2n n matrix obtained by mutating the principal matrix corresponding to B 0 from t 0 to t. Proposition 3.8. For l [1, m], t T n, g l;t is given by the initial conditions (3.17) g l;t0 = e l (l = 1,..., m)

15 15 together with the recurrence relations (3.18) (3.19) g l;t = g l;t for l k; m g k;t = g k;t + [ b t ik ] + g i;t i=1 n j=1 [ b,t n+j,k ] + b j, where the (unique) path from t 0 to t in T n ends with the edge t k t. Proof. Equation (3.17) is clear from the fact that F l;t0 = 1 for all l [1, n]. Let t k t in T n. If l k, then F l;t = F l;t, and (3.18) follows. Now consider g k;t. In section 3 of [10], an element Y k;t Q sf (u 1,..., u n ) was defined satisfying certain properties that we recall below. By [10, Proposition 3.12], (3.20) F k;t = Y k;t + 1 (Y k;t + 1) Trop(u1,...,u n) F 1 k;t n i=1 F [ bt ik ] + i;t, where the right hand side is computed in the field Q(u 1,..., u n ) of rational functions. By [10, (3.16)], the cluster variable x k;t A is given by (3.21) x k;t = (Y k;t + 1) F (ŷ 1,..., ŷ n ) (Y k;t + 1) P (y 1,..., y n ) x 1 k;t n i=1 x [ bt ik ] + i;t, where P = Trop(x n+1,..., x m ), and ŷ i, y i are elements of the ambient field F as defined at (3.12). By [10, (3.14)], (3.22) Y k;t P (y 1,..., y n ) = m i=n+1 x bt ik i F. By (3.22), (3.23) (Y k;t + 1) P (y 1,..., y n ) = m i=n+1 x min(0,bt ik ) i. Since min(0, b t ik ) = [ bt ik ] +, equation (3.21) may be rewritten as (3.24) x k;t = (Y k;t + 1)(ŷ 1,..., ŷ n )x 1 k;t m i=1 x [ bt ik ] + i;t,

16 16 By Definition 3.7, it follows from equation (3.24) that (3.25) x k;t = P k;t (ŷ 1,..., ŷ n )x g xg m, where (3.26) (3.27) P k;t = (Y k;t + 1)F 1 k;t (g 1,..., g m) = g k;t + n i=1 F [ bt ik ] + i;t, m [ b t ik ] + g i;t. i=1 In the particular case, where A has principal coefficients, (3.22) implies that (3.28) (Y k;t + 1) Trop(y1,...,y n)(y 1,..., y n ) = n j=1 y min(0,b,t j+n,k ) j Thus, (3.29) (Y k;t + 1) Trop(u1,...,u n) = n j=1 u min(0,b,t j+n,k ) j Q sf (u 1,..., u n ). Equation (3.20) may be rewritten as (3.30) F k;t = (Y k;t + 1)F 1 k;t n i=1 F [ bt ik ] + i;t n j=1 u [ b,t j+n,k ] + j It follows that (3.31) x k;t = F k;t (ŷ 1,..., ŷ n )x g xg m n j=1 ŷ [ b,t j+n,k ] + j, and equation (3.19) follows from Definition 3.7. Proposition 3.9 ([10, Proposition 6.6]). In the particular case when B 0 is principal and g B0 ;t 0 i;t = g B 0 ;t 0 i;t for all i [1, n], t T n, the recurrence (3.19) may be replaced by the following recurrence: (3.32) g k;t = g k;t + 2n i=1 [ɛb t ik ] +g i;t n j=1 [ɛb,t n+j,k ] + b j, for ɛ {+, }.

17 17 4. F -polynomials in Classical Types Formulas for F -polynomials and quantum F -polynomials will be given in terms of denominator vectors. We recall from [8] the definition of these vectors and the recurrence relations from which they may be computed. Let A be any cluster algebra whose initial exchange matrix has principal part B 0. Suppose A has initial extended cluster (x 1,..., x m ). By the Laurent phenomenon ([8, Theorem 3.1]), any cluster variable x j;t A may be expressed as (4.1) x j;t = N(x 1,..., x n ) x d , xdn n where N(x 1,..., x n ) is a polynomial with coefficients in Z[x ±1 n+1,..., x±1 m ] which is not divisible by any x i. Let d B0 ;t 0 j;t = d 1. d n The vectors d j;t = d B0 ;t 0 j;t, and call this the denominator vector of the cluster variable x j;t. are uniquely determined by the initial conditions (4.2) d B0 ;t 0 j;t 0 = e j (where e 1,..., e n are the standard basis vectors in Z n ) together with the recurrence relations implied by the exchange relation (2.3): for t k t in T n, we have d j;t if j k; (4.3) d j;t = ( n n ) d k;t + max [b t ik ] +d i;t, [ b t ik ] +d i;t if j = k i=1 Here, the max operation on vectors is performed component-wise. Observe that the denominator vector depends on B 0, t 0, t, j, not on the choice of coefficients. i= F -polynomials and g-vectors corresponding to trees. For any n n integer skewsymmetrizable matrix B = (b ij ), we may define a quiver Q(B) on the set of vertices [1, n], with b ij arrows from i to j if and only if b ij < 0. Let B 0 be any n n skew-symmetric integer matrix, and let Q 0 = Q(B 0 ). Fix T [1, n] such that the subgraph of Q 0 induced by T is a tree. In particular, there is at most one edge in Q 0 connecting any given pair of vertices in T. Without loss of generality, assume that T = [1, l] [1, n]. Furthermore, we may assume that the vertices of T

18 18 are labeled so that for each i [1, l], the subgraph of T induced by the set [1, i] is also a tree, and in that tree, the vertex i is a leaf. In this section, we associate to T a cluster variable in A (B 0, t 0 ) and give a formula for the corresponding F -polynomials which is known from [4]. For S [1, n], write (4.4) e S = i S e i Z n. The next two propositions are an immediate consequence of Proposition 5.7 and Corollary 5.8 of [4]: Proposition 4.1. There exists a cluster variable x T in A (B 0, t 0 ) such that the denominator vector of x T is e T. Furthermore, this cluster variable may be obtained from the initial cluster by mutating in directions k = 1,..., l and taking the lth cluster variable in the resulting cluster. Let t 1,..., t l in T n such that t i 1 i t i in T n for i = 1,..., l. Then the proposition implies that the cluster variable x i;ti A (B 0, t 0 ) is equal to x [1,i] for each i. Write F cl T = F B0 ;t 0 l;t l g T = g B0 ;t 0 l;t l for the nonquantum (or classical ) F -polynomial and g-vector corresponding to T, respectively. and Proposition 4.2. [4, Corollary 5.8] The F -polynomial corresponding to T is given by (4.5) F cl T (u 1,..., u n ) = S i S u i where the summation ranges over all subsets S T such that the following condition holds: (4.6) if j S and i T are such that j i in Q 0, then i S. For k [1, n], define (4.7) I in (k) = {i T : i k in Q 0 }, I out (k) = {j T : k j in Q 0 }. Proposition 4.3. [4, Remark 5.9] For k T, the kth component of g T is equal to (4.8) g k = I out (k) 1.

19 Next, a formula for the remaining components of the g-vector will be given in Proposition 4.5 in a particular situation. This formula was given in [4] in terms of representations of quivers with potentials. Let B 0 be an n n skew-symmetric matrix with entries from {0, 1, 1}. We recall some notation and definitions from [4], omitting certain technical details which are not needed here. A representation M (over C) of a quiver Q is given by assigning C-vector space M i to each vertex i [1, n] and assigning a b-tuple (ϕ (1) ji,..., ϕ(b) ji ) of linear maps ϕ(k) ji : M i M j to every arrow i j of multiplicity b = b ji. Let M be the quiver representation of Q 0 such that M i = C if i T, and M i = 0 otherwise. Also, for the arrow a : i j, let a M : M i M j be the identity map if i, j T, and let a M be the 0-map otherwise. For k [1, n], let 19 (4.9) M in (k) = M i, M out (k) = M j, i I in (k) j I out (k) where I in (k), I out (k) were defined at (4.7). A certain linear map γ k : M out (k) M in (k) was defined in [4]. This map is given by the matrix γ k = (γ k (i, j)), where the rows of the matrix are indexed by i I in (k), the columns are indexed by j I out (k), and γ k (i, j) is either a nonzero element of C if there exists a directed path from j to i in Q 0 with all vertices contained in T, and γ k (i, j) = 0 otherwise. Lemma 4.4. Let k [1, n]. The rank of γ k is equal to the maximum number r Z 0 for which there exist distinct j 1,..., j r I out (k) and distinct i 1,..., i r I in (k) such that there is a directed path from j s to i s in T for all s [1, r]. (In particular, rank(γ k ) does not depend on the specific nonzero values in the matrix γ k.) Proof. If r = 0, then γ k = 0, and the lemma holds. Assume for the remainder of the proof that r 1. Let j 1,..., j r, i 1,..., i r be vertices as in the statement of the lemma. We claim that there does not exist σ in the symmetric group S r, σ id, such that γ k (i σ(s), j s ) 0 for all s [1, r]. For the sake of contradiction, assume that such a σ exists. Let p be the order of σ in S r. Then there exists a directed path from j s to i σ(s) in T for all s [1, r], so it follows that there is a cycle in T containing the vertices i 1, j 1, i σ(1), j σ(1),..., j σ p 1 (1), i σ p (1) = i 1, which contradicts that fact that the subgraph of Q 0 induced by T is a tree.

20 20 Let Γ be the r r submatrix of γ k with rows indexed by i 1,..., i r, and columns indexed by j 1,..., j r. Then the determinant of Γ is r s=1 γ k(i s, j s ), which is nonzero since γ k (i s, j s ) 0 for all s [1, r]. This proves that rank(γ k ) r. Now let r = rank(γ k ), and suppose that Γ is an r r invertible submatrix of γ k with rows indexed by i 1,..., i r I in (k) and columns indexed by j 1,..., j r I out (k). In order for the determinant of Γ to be nonzero, there must exist σ S r such that γ iσ(s),j s 0 for all s [1, r ]. Thus, there is a directed path from j s to i σ(s) in T for each s [1, r ], which means that rank(γ k ) r. Proposition 4.5. The g-vector g T corresponding to T is given by (g 1,..., g n ), where (4.10) (4.11) g k = dim(ker(γ k )) dim(m k ) = I out (k) rank(γ k ) dim(m k ). for all k [1, n]. Proof. The proposition follows immediately from Theorem 5.1 and Proposition 5.7 of [4] F -polynomials in type A n. Let B 0 be an n n exchange matrix of type A n. Recall that such a matrix B 0 may be obtained via a sequence of matrix mutations from the n n matrix B = (b ij ), where (4.12) b ij = 1 if j = i if j = i 1 0 otherwise We will show that each cluster variable in the cluster algebra A (B 0, t 0 ) corresponds to an induced chain C, which means that Theorem 4.2 may be used to compute all F -polynomials for type A n. Also, we compute g-vectors for type A n using Proposition 4.5. Denote by Φ + (B 0 ) the collection of subsets C of [1, n] such that the subgraph of Q 0 induced by C is a chain. (A chain is an alternating sequence v 1, e 1, v 2,..., e p, v p of distinct vertices and edges such that the edge e i has vertices v i, v i+1 for i = 1,..., p 1, and there are no other edges connecting the vertices v 1,..., v p.)

21 Proposition 4.6. The cluster variables in A which are not in the initial cluster are in bijective correspondence with the elements of Φ + (B 0 ). To be more specific, if a cluster variable corresponds to the set C Φ + (B 0 ), then its denominator vector is e C = i C e i. 21 Remark 4.7. For the acyclic case in finite type, it was proven in [9] that the cluster variables not in the initial cluster are in bijective correspondence with the positive roots of the given type (see Section 4.3). In the acyclic case, Proposition 4.6 is a consequence of this result. More recently, Proposition 4.6 was independently stated and proven in [14]. To prove the proposition, we will use the following combinatorial description of cluster algebras of type A n given in [9]. In this realization, the cluster variables are in bijective correspondence with the diagonals of the regular (n + 3)-gon P n+3, and the clusters correspond to maximal sets of noncrossing diagonals, i.e., to triangulations of P n+3. (Note that two diagonals cross if they intersect in an interior point.) Let T = (β 1,..., β n ) be a list of pairwise distinct noncrossing diagonals of P n+3. Write (T ) for the corresponding set of triangles. In this setting, cluster mutations are encoded as follows. Let k [1, n]. To mutate T in direction k, let 1, 2 be the two triangles in (T ) which have β k as a side, and let a, b be the vertices opposite the side β k in each of these triangles. Then µ k (T ) is the list of diagonals obtained from T by replacing β k by the diagonal ab. One may associate to T a quiver Q(T ) on the set of vertices [1, n]. For the edges, let i, j [1, n], i j. If there is no triangle in T such that β i, β j are sides of the triangle, then there is no edge between i and j. Otherwise, suppose that the endpoints of β i are a, b, and the endpoints of β j are a, c. Then i j if a, b, c are in clockwise order, and i j if a, b, c are in counterclockwise order. The principal part of the exchange matrix B(T ) = (b ij ) corresponding to T is given by b ij = 0 if there is no edge between i and j, b ij = 1 if i j, and b ij = 1 if i j. Note that Q(B(T )) = Q(T ). In [3], Buan and Vatne characterize all type A n quivers (i.e., all quivers Q of the form Q = Q(T ) where T is a triangulation of P n+3 ). Lemma 4.8. [3, Proposition 2.4] Let Q be a quiver with vertex set [1, n]. Then Q is of type A n if and only (1)-(4) hold: (1) Any induced cycle in Q is an oriented 3-cycle. (In particular, there are no multiple edges.) (2) The degree of any vertex is at most 4.

22 22 (3) If a vertex i has degree 4, then two of the edges containing i are in a 3-cycle, and the other two edges containing i are in another 3-cycle. (4) If a vertex i has degree 3, then two of the edges containing i are in a 3-cycle, and the other edge does not belong to any 3-cycle. In particular, it follows that any induced tree in Q must be a chain. Let T 0 = {α 1,..., α n } be the triangulation of P n+3 corresponding to the cluster at t 0. Proposition 4.6 is an immediate consequence of the following lemma: Lemma 4.9. The diagonals of P n+3 which are not in T 0 are in bijective correspondence with the elements of Φ + (B 0 ). Under this correspondence, if β is a diagonal not in T 0, then the corresponding subset of [1, n] consists precisely of those i [1, n] for which β crosses α i. Furthermore, if C = {p 1,..., p j } Φ + (B 0 ) such that there is some edge between p i and p i+1 for i = 1,..., j 1, then the cluster variable corresponding to C can be obtained from the initial cluster by mutating in directions p 1,..., p j. The denominator vector of this cluster variable is e C. Proof. First, consider a diagonal β of P n+3 that is not in T 0. Let T be the set of diagonals in T 0 which β intersects. This set T may be constructed as follows: Start with one endpoint of β. This endpoint is the vertex of a triangle 1 in the initial triangulation such that β passes through the interior of the triangle. The diagonal β crosses another diagonal α p1 which is a side of the triangle 1. The diagonal α p1 is a side of another triangle 2 in the initial triangulation. Either β intersects a vertex of 2, in which case T = {α p1 }, or β crosses another side α p2 of 2. In the latter case, α p2 is the side of another triangle 3 in the initial triangulation, and it follows that either T = {α p1, α p2 }, or that α crosses another side α p3 α p2 of 2. Continuing this process, we get T = {α p1,..., α pj }, and triangles 1,..., j such that for each i = 1,..., j 1, the diagonal α pi is a side of the triangles i and i+1. It is clear that the subgraph of Q 0 induced by the vertices p 1,..., p j does not contain a cycle, since Lemma 4.8 implies that the vertices in any induced cycle in Q 0 correspond to the diagonals in a triangle in T 0, and β cannot cross all of the sides of a triangle. By Lemma 4.8, the subgraph of Q 0 induced by p 1,..., p j must be a chain. Next, consider a sequence p 1,..., p j of vertices from [1, n] such that the subgraph of Q 0 induced by these vertices is a chain (with an edge between any two consecutive vertices in the list). The goal is to find a diagonal β of P n+3 which crosses α p1,..., α pj and no other diagonals in T 0. If j = 1, then let 0, 1 be the triangles in (T 0 ) which have α p1 as a side. For j 2, any two

23 consecutive diagonals α pi, α pi+1 are the sides of a triangle i in the initial triangulation; also, there 23 exist triangles 0, j with 0 1, j j 1, such that α p1 is a side of 0, and α pj is a side of j. Let a 0 be the vertex of 0 which is opposite the side α p1. For i = 1,..., j, let a i be the vertex of i which is opposite the side α pi. We claim that β = a 0 a j is the desired diagonal. Observe that all of the triangles 0,..., j are distinct; otherwise, there would be three diagonals from the list α p1,..., α pj as sides of a triangle, which would mean that the vertices p 1,..., p j induce a cycle in Q 0. By construction, the total set of vertices from the triangles 0,..., j contains at most j + 3 vertices. Any triangulation of P = Conv( 0... j ) has at most j + 1 triangles, so it follows that 0,..., j is a triangulation of Conv( 0... j ). Thus, P = 0... j is convex. This means that the only diagonals from T 0 that β can intersect are in the list α p1,..., α pj. To show that these are exactly the diagonals from T 0 which β intersects, consider the diagonals of T 0 which β passes through as one moves from the endpoint a 0 to the other endpoint a j. If j = 1, then β intersects only the diagonal α p1. Otherwise, β intersects the interior of 1, and passes through the interior of another side of 1 different from α p1. This side must be α p2, since the side of 1 different from α p1 and α p2 is on the boundary of P. Continuing this argument, it is easy to show that β intersects α p1,..., α pj. Suppose that the triangulation T 0 is mutated in directions p 1,..., p j, and call the resulting sequence of triangulations T 1,..., T j. One verifies by induction that when T i 1 is mutated to T i, the diagonal α pi Proposition 4.1. is flipped to a 0 a i. The assertion about the denominator vector follows from Remark In [7], Fomin, Shapiro, and Thurston associate cluster algebras to punctured Riemann surfaces. The final assertion in Lemma 4.9 about the denominator vector is proven in greater generality for all triangulated surfaces in [7, Theorem 8.6]. Example Let T 0 be the triangulation of P 8 given in Figure 1. The quiver Q 0 = Q(T 0 ) of type A 5 is given below

24 24 h a 1 g b f 4 c e d Figure 1. Then the diagonals af and dg correspond to cluster variables in A (B 0, t 0 ) with denominator vectors e 1 + e 2 + e 3 + e 5 and e 3 + e 4, respectively. Proposition The g-vector of the cluster variable corresponding to the set C Φ + (B 0 ) is (g 1,..., g n ), where if k C, then g k = I out (k) 1; if k / C, the subgraph of Q 0 induced by C {k} is a chain, and k j in Q 0 for some j C, then g k = 1; otherwise, g k = 0. Remark Proposition 4.12 was independently stated and proven in [14], and generalized to other classical types. For finite type, formulas for (nonquantum) F -polynomials and g-vectors were given in [10] in the bipartite case and in [18] for the acyclic case (see Proposition 4.18 below). Proof. The first part follows from Proposition 4.3. Using Proposition 4.5 and the notation preceding it (with T replaced by C), it suffices to compute g k = dim(ker(γ k )) for k [1, n] C. Fix such an index k. First, assume that there exists j C such that k j in Q 0, and the subgraph of Q 0 induced by C {k} is a chain. Then M in (k) = 0, so ker(γ k ) = M out (k) = M j, which means that dim(ker(γ k )) = 1.

25 Next, suppose that there exists j C such that k j in Q 0, and the subgraph of Q 0 induced by C {k} is not a chain (which means that it contains a cycle by Lemma 4.8). By the same lemma, the only induced cycles in Q 0 are directed 3-cycles, so it follows that there exists p C such that k j p k in Q 0. It may also be deduced from the same lemma that there is no edge between k and another vertex j C {j, p}. Thus, M out (k) = M j and M in (k) = M p. Since γ k is an isomorphism between M j and M p, it follows that dim(ker(γ k )) = 0. Finally, suppose that there is no j C such that k j in Q 0. Then M out (k) = 0, so ker(γ k ) = F -polynomials in Classical Types for Acyclic Initial Exchange Matrix. Let B 0 = (b 0 ij ) be an acyclic n n exchange matrix of type A n, B n, C n, or D n. Write Q 0 = Q(B 0 ). (Recall that the matrix B 0 is acyclic if the quiver Q 0 is acyclic, i.e., it does not contain any directed cycles.) In particular, b ij = ±a ij for i j, where (a ij ) is the Cartan matrix of the corresponding type. Note that we will use the convention for Cartan matrices given in [13] which is different from the one given in [2]. Denote by Φ + = Φ + (B 0 ) the set of all denominator vectors of cluster variables in A (B 0, t 0 ) which do not occur in the initial cluster. It was proven in [18] that these cluster variables are in bijective correspondence with the positive roots corresponding to the type of B 0, and this correspondence is via denominator vectors. To be more precise, if a cluster variable corresponds to α = d i α i, where the simple roots are given by α 1,..., α n, then the denominator vector of the cluster variable is (d 1,..., d n ) Z n. The sets Φ + are given in each type by the following: Type A n : Φ + = {e i e j : 1 i j n} Type B n : Φ + = {e i +...+e j : 1 i j n} {e i +...+e j 1 +2e j e n : 1 i < j n} Type C n : Φ + = {e i e j : 1 i j n} {e i e j + + 2e n 1 + e n : 1 i < j < n} {2e i + + 2e n 1 + e n : 1 i n 1} Type D n : Φ + = {e i e j : 1 i j n} {e i e n 2 + e n : 1 i n 2} {e i e j 1 + 2e j e n 2 + e n 1 + e n : 1 i < j n 2}

26 26 Remark In the case that B 0 is of type A n, the underlying graph of the quiver Q 0 is simply the type A n Dynkin diagram, i.e., a path on the vertices 1,..., n. In this case, Φ + (B 0 ) as defined here can be obtained from the Φ + (B 0 ) defined in Section 4.2 by replacing each S [1, n] by e S. Let F d = F B0 ;t 0 d vector d. be the F -polynomial corresponding to the cluster variable with denominator Define a partial order on Z n by (4.13) a a if a a Z n 0. Let d = (d 1,..., d n ) Φ + (B 0 ) and e = (e 1,..., e n ) Z n such that 0 e d. Definition We call an arrow i j in Q 0 acceptable (with respect to the pair (d, e)) if e i e j [d i d j ] +. Also, an arrow i j is called critical (with respect to (d, e)) if either (d i, e i ) = (2, 1), (d j, e j ) = (1, 0), or (d j, e j ) = (2, 1), (d i, e i ) = (1, 1). Note that a critical arrow is always acceptable. Let S be the induced subgraph of Q 0 on the set of vertices {i : (d i, e i ) = (2, 1)}. For a connected component C of S, define ν(c) as the number of critical arrows having a vertex in C. Theorem Fix d = (d 1,..., d n ) Φ + (B 0 ), e = (e 1,..., e n ) Z n, and define S as above. The coefficient of the monomial u e uen n (1) 0 e d; (2) all arrows in Q 0 are acceptable; (3) ν(c) 1 for all components C of S; (4) if B 0 is of type C n, then in F d is nonzero if and only if (a) e n = 1, d n 1 = 2, and n n 1 in Q 0 imply that e n 1 = 2; (b) e n 1 1, d n = 1, and n 1 n in Q 0 imply that e n = 1. (5) If B 0 is of type B n, S consists of a single component which contains the vertex n 1, and there is a critical arrow in S, then S does not contain the vertex n.

27 If the conditions above are all satisfied, then the coefficient of u e uen n is 2 c, where c is the number of components C such that ν(c) = Remark If 0 d i 1 for all i [1, n], then S is empty, so the third condition is automatically satisfied. The formula for g-vectors in the next theorem will be useful in computing quantum F -polynomials for the classical types. Theorem [18] The g-vector g d corresponding to d = (d 1,..., d n ) Φ + (B 0 ) is given by (4.14) d i e i + d i [ b 0 ji] + e j. i [1,n] i,j [1,n] Proof. This follows from Theorems 1.8 and 1.10 of [18]. Example Let (4.15) B 0 = Then B 0 is of type B 4, and Q 0 is the quiver below: Suppose that d = e 1 + 2e 2 + 2e 3 + 2e 4. Note that the only possible critical arrow in Q 0 is 1 2, and this arrow is critical if e 1 = e 2 = 1. Let e = (e 1, e 2, e 3, e 4 ) Z 4. Then u e 1 1 ue 2 2 ue 3 3 ue 4 4 occurs with nonzero coefficient in F d if and only if 0 e d, the three inequalities e 1 e 2, e 2 e 3, and e 4 e 3 are satisfied, and e (1, 1, 1, 1). Using these conditions, it is easy to check that the

28 28 F -polynomial corresponding to d is (4.16) F e1 +2e 2 +2e 3 +2e 4 = u 1 u 2 2u 2 3u u 1 u 2 2u 3 u u 2 2u 2 3u u 1 u 2 2u 3 u 4 + u 1 u 2 u 3 u u 1 u 2 2u u 1 u 2 2u 4 + u 1 u u 1 u 2 u u 2 2u 2 3u u 2 2u 3 u 4 + 2u 2 u 3 u u 2 u u 2 2u 4 + u 2 2u u u u 2 u 3 u 4 + 2u 1 u 2 u 4 + u 1 u 2 + 4u 2 u 4 + 2u 4 + 2u Some other F -polynomials corresponding to B 0 are given below. (4.17) (4.18) (4.19) (4.20) F e2 = u F e2 +e 3 = u 2 u 3 + u F e2 +e 3 +e 4 = u 2 u 3 u 4 + u 2 u 4 + u 4 + u F e2 +2e 3 +2e 4 = u 2 u 2 3u u 2 u 3 u u 2 u 3 u 4 + u 3 u u 2 u u 2 u 4 + u 2 + u u The corresponding g-vectors are given below. (4.21) (4.22) (4.23) (4.24) (4.25) g e2 = e 1 e 2 + e 3 g e2 +e 3 = e 1 e 2 g e2 +e 3 +e 4 = e 1 e 2 + e 3 e 4 g e2 +2e 3 +2e 4 = e 1 e 2 + e 3 2e 4 g e1 +2e 2 +2e 3 +2e 4 = e 1 2e 2 + 2e 3 2e 4 For type A n, Theorem 4.16 follows from Proposition 4.2 and Proposition 4.6. The remainder of this section is devoted to proving Theorem 4.16 in the remaining types. The following easily verified proposition will be useful in checking if an arrow i j in Q 0 is acceptable. Proposition Let d = (d 1,..., d n ) Φ + (B 0 ), e = (e 1,..., e n ) Z n such that 0 e i d i for all i [1, n], and let i j be an arrow in Q 0. If at least one of the equalities d i = 0, d j = 0, e i = 0, or e j = d j holds, then i j is an acceptable arrow.

29 4.4. Type D n. In this subsection, let B 0 be an acyclic n n exchange matrix of type D n. The 29 underlying undirected graph of the quiver Q 0 = Q(B 0 ) is simply the Dynkin diagram of the corresponding type. We will use the formula for F -polynomials given in [4]. First, recall some terminology and notation for representations of quivers. The dimension of a representation M is the vector d = (d 1,..., d n ) Z n given by d i = dim(m i ). A subrepresentation N of a representation M is given by a collection of subspaces N i M i for each i [1, n] such that ϕ (k) ji (N i) N j for all i, j, k. For a representation M of dimension d and for an integer vector e = (e 1,..., e n ) such that 0 e d (i.e., 0 e i d i for all i), let Gr e (M) denote the variety of all subrepresentations of M of dimension e. Thus, Gr e (M) is a closed subvariety of the product of Grassmanians Gr ei (M i ). Denote by χ e (M) the Euler-Poincare characteristic of Gr e (M) (see [12, Section 4.5]). To compute F d, we use the following result adapted from [4]: Theorem Assume B 0 is as above. Let d Φ + (B 0 ), and let M be an indecomposable representation of Q 0 of dimension d. Then (4.26) F d = χ e (M)u e uen n, where the summation ranges over e = (e 1,..., e n ) Z n such that 0 e d. The following lemma will be useful in determining when there exist subrepresentations of M of dimension e: Lemma Let M and M be vector spaces of dimensions d and d, respectively, and ϕ : M M be a linear map of maximal possible rank min(d, d ). Let e and e be two integers such that 0 e d and 0 e d. Then the following conditions are equivalent: (1) There exist subspaces N M and N M such that dim N = e, dim N = e, and ϕ(n ) N. (2) e e [d d ] +. Proof. First, observe that (4.27) dim(ker ϕ) = d min(d, d ) = [d d ] +.

30 30 For the proof that (1) implies (2), (4.28) [d d ] + dim(ker ϕ N ) = e dim(im ϕ N ) e e. Now assume that (2) holds. Choose a subspace N of M of dimension e so that N Ker ϕ if e [d d ] +, or N Ker ϕ if e [d d ] +. Then (4.29) (4.30) dim(im ϕ N ) = e dim(ker ϕ N ) e [d d ] + if e [d d ] + = 0 otherwise. Since dim(im ϕ N ) e in either case, any e -dimensional subspace N of M which contains Im ϕ N will work. We may restrict attention to e Z n which satisfy the first two conditions given in Theorem (If these conditions do not hold, Theorem 4.21 together with Lemma 4.22 imply that the coefficient of u e uen n in F d equals 0.) If d i = 0 or 1 for all i, then Theorem 4.16 follows from Proposition 4.2. If d i > 1 for some i, then d = e p e r 1 + 2e r e n 2 + e n 1 + e n for some p, r Z satisfying 1 p < r n 2. An indecomposable representation M = (M i ) of dimension d can be selected so that for p i r 2 and for r i n 3, the map between M i and M i+1 (which is either ϕ i,i+1 or ϕ i+1,i ) is the identity map. If r 1 r in Q 0, then let V r = Im(ϕ r,r 1 ); otherwise, let V r = Ker(ϕ r 1,r ). For each value j = n 1 and j = n, if n 2 j in Q 0, then let V j = Ker(ϕ j,n 2 ); otherwise, let V j = Im(ϕ n 2,j ). Then V r, V n 1, and V n are three distinct one-dimensional subspaces of the two-dimensional space M n 2. To compute χ e (M), we will demonstrate how to construct all possible subrepresentations which have dimension vector e (if there are any), from which it will be easy to compute χ e (M). In order to verify that a given N = (N i ) (with dimension e) is a subrepresentation of M, we need to check that for each i, j [1, n] which are connected by an edge in Q 0, the following property is satisfied: (4.31) ϕ ji (N i ) N j if i j in Q 0, or ϕ ij (N j ) N i if j i in Q 0. Observe that for any i [1, n] S, there is only one possible subspace of M i of dimension e i. Consider a component C of S. If i, j are vertices in C and i j, then (4.31) implies that N i = N j. Thus, once a subspace of dimension 1 is chosen for one vertex of the component, all of the vertices

31 in that component must be assigned that same subspace. If i, j [p, n] such that i j in Q 0 and so that at least one of i, j is in [r, n 2] S, then it is easy to check that ϕ ji (N i ) N j. For example, if i [r, n 2] S, then e i = 0 or e i = 2. If e i = 0, then ϕ ji (N i ) = 0. If e i = 2, then the fact that i j is acceptable implies e j = d j, so N j = M j ϕ ji (N i ). One can also easily check that for if i j with i, j [1, r 1], then ϕ ji (N i ) N j. Thus, it only remains to verify for each proposed subrepresentation (N i ) that the property (4.31) holds for (i, j) = (r 1, r) if r S, and that the property holds for (i, j) = (n 2, n 1), (n 2, n) if n 2 S. Next, consider which 1-dimensional subspaces may be assigned to each component. Observe that when (i, j) = (r 1, r) is a critical pair, property (4.31) is satisfied for (i, j) by N if and only if N r = V r. Also, when (i, j) = (n 2, n) (resp. (i, j) = (n 2, n 1)) is a critical pair, property (4.31) is satisfied for (i, j) if and only if N n 2 = V n (resp. N n 2 = V n 1 ). Let C be a component of S. If ν(c) = 0, then any 1-dimension subspace of C 2 may be assigned to the vertices of C. Suppose ν(c) = 1. If (r 1, r) is a critical pair and r C, then the subspace on the vertices of C must by V r. If (n 2, n) (resp. (n 2, n 1)) is a critical pair and n 2 C, then the subspace on the vertices of C must V n (resp. V n 1 ). Finally, if ν(c) 2, then there is no 1-dimensional space which can be assigned to the vertices of C so that (4.31) is satisfied. Since the Euler-Poincare characteristic of the projective line P 1 is 2, it follows that the contribution to χ e (M) is a multiplicative factor of 0, 1, or 2 when ν(c) = 0, ν(c) = 1, or ν(c) 2, respectively. This concludes the proof of Theorem 4.16 in type D n Projections of F -polynomials and g-vectors. To prove Theorem 4.16 in the remaining types, we will show that F -polynomials and g-vectors of type B n and C n can be obtained as certain projections of F -polynomials and g-vectors of type D n+1 and A 2n 1, respectively. In this subsection, we prove a more general result (Theorem 4.24) concerning such projections. For this subsection, let B 0 = (b 0 ij ) be an arbitrary n n skew-symmetric integer matrix such that B0 is acyclic. The methods are similar to those used by Dupont in [5] for coefficient-free cluster algebras, but to give the desired results, we will need to work with cluster algebras with principal coefficients. We begin by recalling some terminology and notation from [5]. For σ S n, let σ S 2n be given by σ(i) = σ(i) and σ(i + n) = σ(i) + n for i [1, n]. (Thus, σ acts on the set [1, 2n] via σ.) If B = (b ij ) is in M 2n,n (Z), then define an action of σ on B by σ B = (b σ 1 i, σ 1 j). We may also define an action of σ on matrices B in M n,n (Z) in a similar way. Say that a subgroup G of S n is an automorphism group for B (resp. B) if σ B = B (resp. σb = B) for all σ G.