Q(x n+1,..., x m ) in n independent variables

Transcription

1 Cluster algebras of geometric type (Fomin / Zelevinsky) m n positive integers ambient field F of rational functions over Q(x n+1,..., x m ) in n independent variables Definition: A seed in F is a pair ( x, B) where 1. x = (x 1,..., x m ) form a free generating set for F. ( x is the extended cluster.) 2. B = (b ij ) is an m n integer matrix such that the submatrix B consisting of the top n rows of B (called the principal part) is skew-symmetrizable. B is the exchange matrix, x 1,..., x n are the cluster variables in the seed, x n+1,..., x m are the coefficient variables.

2 Definition: ( x, B) seed, B = (b ij ). For k [1, n], a seed mutation in direction k transforms ( x, B) into the pair ( x, µ k ( B)) given by: 1. x = x {x k } {x k }, where x k = x 1 k i [m] b ik >0 x b ik i + i [m] b ik <0 This is an exchange relation. x b ik i 2. (matrix mutation) µ k ( B) = (b ij ) is an m n matrix, where b ij = b ij if i = k or j = k b ij + b ik b kj + b kj b ik 2 otherwise.

3 Let T n be the n-regular tree with edges labeled by 1,..., n so that the n edges incident to a vertex receive different labels. Start with an initial seed ( x 0, B 0 ) at t 0 T n, and mutate to every vertex t T n. Denote the seed at t T n by ( x t, B t ), where x t = (x 1;t,..., x n;t, x n+1;t,..., x m;t ) B t = (b t ij ) The cluster algebra A( x 0, B 0 ) = A( B 0 ) is the Z[x ±1 n+1,..., x±1 m ]-subalgebra of F generated by all cluster variables.

4 Quantum Cluster Algebras (Berenstein / Zelevinsky) A quantum cluster algebra is a certain noncommutative deformation of a cluster algebra using an additional variable q. Ambient field is F q, the skew field of fractions of Z[q ±1, Y 1,..., Y m ], where Y 1,..., Y m are algebraically independent variables which are quasi-commutative, i.e. Y i Y j = q λ ijy j Y i for some integers λ ij.

5 Definition: A quantum seed in F q is a pair ( X, B), where 1. X = (X 1,..., X m ) is a free generating set such that X i X j = q λ ijx j X i for some λ ij Z. 2. B is an m n integer matrix which is compatible with Λ = (λ ij ), i.e. B T Λ = (D 0), where D is a diagonal matrix with positive diagonal entries.

6 Notation: If X 1,..., X m satisfy X i X j = q λ ijx j X i, then for c = (c 1,..., c m ) Z m, let X c = (q 1 2 i>j λ ijc i c j )X c 1 1 Xc m m Definition: ( X, B) quantum seed. For k [1, n], the quantum seed mutation in direction k transforms ( X, B) into ( X, µ k ( B)), where X = X {X k } {X k }, and X k = X e k+ b ik >0 b ike i + X e k b ik <0 b ike i The quantum cluster algebra A is the Z[q ±1 2, Y n+1 ±1,..., Y m ±1 ]-subalgebra of F q generated by all cluster variables. Note: Setting q = 1 in the quantum cluster algebra A with initial seed ( X 0, B 0 ) yields the cluster algebra A( B 0 ).

7 F-polynomials (Fomin / Zelevinsky) Fix an n n skew-symmetrizable matrix B 0 = (b ij ) Assume any cluster algebra below has the property that the exchange matrix B 0 = (b ij ) at t 0 has principal part B 0. Definition: The cluster algebra with principal coefficients is the cluster algebra with initial seed ( x = (x 1,..., x n, y 1,..., y n ), B 0 ), where B 0 = ( B 0 I n ) This cluster algebra is denoted by A = A (B 0, t 0 ) or A (B 0 ).

8 Definition: In A, any cluster variable x j;t can be expressed as a subtraction-free rational function in x 1,..., x n, y 1,..., y n. Denote this rational expression by X j;t = X B0 ;t 0 j;t Q sf (x 1,..., x n, y 1,..., y n ). Let F j;t = F B0 ;t 0 j;t := X j;t (1,..., 1, y 1,..., y n ) Q sf (y 1,..., y n ). F j;t is an F -polynomial. Proposition: F j;t Z[y 1,..., y n ].

9 Z n -grading on A Z[x ±1 1,..., x±1 n, y 1,..., y n ]: deg(x i ) = e i deg(y i ) = b 0 i (where b 0 i denotes the ith column of B 0 ). Proposition. Under this grading, all cluster variables in A are homogeneous elements. Write g j;t for the Z n degree of x j;t. This is called the g-vector.

10 Notation: In any cluster algebra A with initial extended cluster (x 1,..., x m ), write y j = ŷ j = m i=n+1 m i=1 x b ij i. x b ij i. If P is a subtraction-free polynomial in n variables, then let P T rop(xn+1,...,x m ) (y 1,..., y n ) denote the monomial in x n+1,..., x m obtained by plugging y 1,..., y n into P, and replacing regular addition (+) by tropical addition ( ) defined on pairs of Laurent monomials in the variables x n+1,..., x m : m i=n+1 x a i i m i=n+1 x b i i = m i=n+1 x min(a i,b i ) i

11 Theorem: Let A be a cluster algebra such that the extended cluster in the initial seed at t 0 is given by (x 1,..., x m ). Then the cluster variable x j;t A can be given by x j;t = F j;t (ŷ 1,..., ŷ n ) F j;t T rop(xn+1,...,x m ) (y 1,..., y n ) xg x g n n, where g j;t = (g 1,..., g n ). Remark: If A = A, then F j;t T rop(xn+1,...,x m ) (y 1,..., y n ) = 1.

12 Quantum F -polynomials Fix the following: a diagonal matrix D with positive diagonal entries d 1,..., d n such that DB 0 is skew-symmetric an n n skew-symmetric matrix Λ 0 = (λ ij ) Assume that all quantum cluster algebras below satisfy: the exchange matrix B 0 at t 0 has principal part B 0. the initial extended cluster is X 1,..., X m, and X i X j = q λ ijx j X i for all 1 i, j n

13 Proposition: There exists a unique quantization of A = A (B 0, t 0 ) such that the quasi-commutation relations of the initial cluster variables are given by Λ 0 and the compatibility condition is satisfied. Call this quantum cluster algebra A = A (B 0, D, Λ 0, t 0 ). Notation: For any quantum cluster algebra, define Ŷ j = X bj, where b j is the jth column of B 0. These elements satisfy the following quasi-commutation relations: Ŷ i Ŷ j = q d ib ij Ŷ j Ŷ i We will consider rational functions in quasi-commuting variables Z 1,..., Z n, with the quasi-commutation relations given by: Z i Z j = q d ib ij Z j Z i

14 Theorem / Definition: For each j [1, n], t T n, there exists a unique rational function F j;t in variables Z 1,..., Z n with coefficients in Z[q ±1 2] such that in A, X j;t = F j;t (Ŷ 1,..., Ŷ n )X g j;t. We call F j;t the quantum F -polynomial. Theorem. In any quantum cluster algebra A, the cluster variables X j;t can be expressed in the form X j;t = q λ j;tf j;t (Ŷ 1,..., Ŷ n )X h j;t, for some h j;t Z m, λ j;t 1 2 Z.

15 F -polynomials in classical types Let B 0 be an acyclic n n exchange matrix of type A n, B n, C n, or D n. Theorem (Fomin, Zelevinsky) The cluster variables not in the initial cluster are in bijective correspondence with the positive roots. Φ + : Positive roots in classical types Let α 1,..., α n be the simple roots. Type A n : α i α j (1 i j n) Type B n : α i α j (1 i j n), α i α n α j (1 i j n) Type C n : α i α j (1 i j n), α i α j 1 + 2α j α n (1 i < j n) Type D n : α i α j (1 i j n, (i, j) (n 1, n)), α i α n 2 + α n (1 i n 2), α i α j 1 + 2α j α n 2 + α n 1 + α n (1 i < j n 2)

16 Fix α = n i=1 a i α i Φ +. Write F α for the F -polynomial corresponding to α. Let Q 0 be the quiver on the vertices [1, n] with i j iff b ij < 0 (no multiple arrows). Let e = (e 1,..., e n ) Z n such that 0 e i a i for all i. Say an arrow i j in Q 0 is acceptable if e i e j max(a i a j, 0). An arrow i j is critical if either (a i, e i, a j, e j ) = (2, 1, 1, 0) or (2, 1, 1, 1). Let S be the induced subgraph of Q 0 on the vertices {i : (a i, e i ) = (2, 1)}. For a component C of S, let ν(c) be the number of critical arrows with a vertex in C.

17 Theorem 1 Let e = (e 1,..., e n ) Z n. The coefficient of u e u e n n in F α is nonzero if and only if 1. 0 e i a i for all i; 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 B n, then (a) e n = 1 and n n 1 in Q 0 implies e n 1 = 2; (b) e n 1 1 and n 1 n in Q 0 implies e n = 1. In this case, the coefficient of u e u e n n is 2 c, where c is the number of components C of S such that ν(c) = 0.

18 Theorem 2 The g-vector corresponding to α is given by i [1,n] a i e i + i [1,n] j [1,n] a i [ b ji ] + e j.

19 Notation: For e = (e 1,..., e n ) Z n, define Z e = q 1 2 ( 1 i<j n d jb ji e i e j ) Z e 1 1 Ze n n. Theorem 3 Let B 0 be of type A n or D n, and let D = di n. Then Z e occurs with nonzero coefficient in the quantum F -polynomial F α if and only if conditions (1)-(4) above are satisfied. In this case, the coefficient is q d 2 ( g α e) (q d 2 + q d 2) c = q d 2 ( g α e c) (1 + q d ) c, where g α = g B0 ;t 0 α is the g-vector corresponding to α. If the a i {0, 1} for all i, then c = 0 and g α e is equal to the number of components in the subgraph of Q 0 induced by {i [1, n] : e i = 1}.