Semicanonical basis generators of the cluster algebra of type A (1)

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1 Semicanonical basis geneatos of the cluste algeba of type A (1 1 Andei Zelevinsky Depatment of Mathematics Notheasten Univesity, Boston, USA andei@neu.edu Submitted: Jul 7, 006; Accepted: Dec 3, 006; Published: Jan 19, 007 Mathematics Subject Classification: 16S99 Abstact We study the cluste vaiables and imaginay elements of the semicanonical basis fo the coefficient-fee cluste algeba of affine type A (1 1. A closed fomula fo the Lauent expansions of these elements was given by P.Caldeo and the autho. As a by-poduct, thee was given a combinatoial intepetation of the Lauent polynomials in uestion, euivalent to the one obtained by G.Musike and J.Popp. The oiginal agument by P.Caldeo and the autho used a geometic intepetation of the Lauent polynomials due to P.Caldeo and F.Chapoton. This note povides a uick, self-contained and completely elementay altenative poof of the same esults. 1 Intoduction The (coefficient-fee cluste algeba A of type A (1 1 is a subing of the field Q(x 1, x geneated by the elements x m fo m Z satisfying the ecuence elations x m 1 x m+1 = x m + 1 (m Z. (1 This is the simplest cluste algeba of infinite type; it was studied in detail in [, 6]. Besides the geneatos x m (called cluste vaiables, A contains anothe impotant family of elements s 0, s 1,... defined ecusively by s 0 = 1, s 1 = x 0 x 3 x 1 x, s n = s 1 s n 1 s n (n. ( Reseach suppoted by NSF (DMS gant # and by a Humboldt Reseach Awad the electonic jounal of combinatoics 14 (007, #N4 1

2 As shown in [, 6], the elements s 1, s,... togethe with the cluste monomials x p mx m+1 fo all m Z and p, 0, fom a Z-basis of A efeed to as the semicanonical basis. As a special case of the Lauent phenomenon established in [3], A is contained in the Lauent polynomial ing Z[x ±1 1, x ±1 ]. In paticula, all x m and s n can be expessed as intege Lauent polynomials in x 1 and x. These Lauent polynomials wee explicitly computed in [] using thei geometic intepetation due to P. Caldeo and F. Chapoton [1]. As a by-poduct, thee was given a combinatoial intepetation of these Lauent polynomials, which can be easily seen to be euivalent to the one peviously obtained by G. Musike and J. Popp [5]. The pupose of this note is to give shot, self-contained and completely elementay poofs of the combinatoial intepetation and closed fomulas fo the Lauent polynomial expessions of the elements x m and s n. Results We stat by giving an explicit combinatoial expession fo each x m and s n, in paticula poving that they ae Lauent polynomials in x 1 and x with positive intege coefficients. By an obvious symmety of elations (1, each element x m is obtained fom x 3 m by the automophism of the ambient field Q(x 1, x intechanging x 1 and x. Thus, we estict ou attention to the elements x n+3 fo n 0. Following [, Remak 5.7] and [4, Example.15], we intoduce a family of Fibonacci polynomials F (w 1,..., w N given by F (w 1,..., w N = w k, (3 D whee D uns ove all totally disconnected subsets of {1,..., N}, i.e., those containing no two consecutive integes. In paticula, we have We also set k D F ( = 1, F (w 1 = w 1 + 1, F (w 1, w = w 1 + w + 1. f N = x N+1 1 x N F (w 1,..., w N wk =x k+1, (4 whee k stands fo the element of {1, } conguent to k modulo. In view of (3, each f N is a Lauent polynomial in x 1 and x with positive intege coefficients. In paticula, an easy check shows that f 0 = 1, f 1 = x + 1 x 1 = x 3, f = x 1 + x + 1 x 1 x = s 1. (5 Theoem.1 [, Fomula (5.16] Fo evey n 0, we have s n = f n, x n+3 = f n+1. (6 In paticula, all x m and s n ae Lauent polynomials in x 1 and x with positive intege coefficients. the electonic jounal of combinatoics 14 (007, #N4

3 Using the poof of Theoem.1, we deive the explicit fomulas fo the elements x m and s n. Theoem. [, Theoems 4.1, 5.] Fo evey n 0, we have x n+3 1 (x(n+1 + ( ( n n + 1 ( ( n n s n 3 Poof of Theoem.1 In view of (3, the Fibonacci polynomials satisfy the ecusion 1 x ; (7. (8 F (w 1,..., w N = F (w 1,..., w N 1 + w N F (w 1,..., w N (N. (9 Substituting this into (4 and cleaing the denominatos, we obtain x N f N = f N 1 + x N 1 f N (N. (10 Thus, to pove (6 by induction on n, it suffices to pove the following identities fo all n 0 (with the convention s 1 = 0: x 1 x n+3 = s n + x x n+ ; (11 x s n = x n+ + x 1 s n 1. (1 We deduce (11 and (1 fom ( and its analogue established in [6, fomula (5.13]: x m+1 = s 1 x m x m 1 (m Z. (13 (Fo the convenience of the eade, hee is the poof of (13. By (1, we have x m + x m x m 1 = x m 1 + x m + 1 x m x m 1 = x m 1 + x m+1 x m. So (x m 1 + x m+1 /x m is a constant independent of m; setting m = and using (, we see that this constant is s 1. We pove (11 and (1 by induction on n. Since both eualities hold fo n = 0 and n = 1, we can assume that they hold fo all n < p fo some p, and it suffices to pove them fo n = p. Combining the inductive assumption with ( and (13, we obtain x 1 x p+3 = x 1 (s 1 x p+ x p+1 = s 1 (s p 1 + x x p+1 (s p + x x p = (s 1 s p 1 s p + x (s 1 x p+1 x p = s p + x x p+, the electonic jounal of combinatoics 14 (007, #N4 3

4 and finishing the poof of Theoem.1. x s p = x (s 1 s p 1 s p = s 1 (x p+1 + x 1 s p (x p + x 1 s p 3 = (s 1 x p+1 x p + x 1 (s 1 s p s p 3 = x p+ + x 1 s p 1, 4 Poof of Theoem. Fomulas (7 and (8 follow fom (11 and (1 by induction on n. Indeed, assuming that, fo some n 1, fomulas (7 and (8 hold fo all the tems on the ight hand side of (11 and (1, we obtain and as desied. x n+3 = x 1 1 (s n + x x n+ 1 ( ( ( n n +(x (n+1 + ( ( n 1 n 1 x ( (x(n+1 + ( ( ( n n n ( (x (n+1 + ( ( n n + 1 s n = x 1 (x n+ + x 1 s n 1 1 x n (xn + 1 x n, ( ( n 1 n 1 x 1 + ( ( n 1 n 1 x (+1 1 ( ( ( n 1 n 1 n ( + 1 ( ( n n 1 x, 1 x the electonic jounal of combinatoics 14 (007, #N4 4

5 Refeences [1] P. Caldeo, F. Chapoton, Cluste algebas as Hall algebas of uive epesentations, Comment. Math. Helv. 81 (006, [] P. Caldeo, A. Zelevinsky, Lauent expansions in cluste algebas via uive epesentations, Moscow Math. J. 6 (006, [3] S. Fomin and A. Zelevinsky, Cluste algebas I: Foundations, J. Ame. Math. Soc. 15 (00, [4] S. Fomin, A. Zelevinsky, Y -systems and genealized associaheda, Ann. in Math. 158 (003, [5] G. Musike, J. Popp, Combinatoial intepetations fo ank-two cluste algebas of affine type, Electon. J. Combin. 14 (007, R15. [6] P. Sheman, A. Zelevinsky, Positivity and canonical bases in ank cluste algebas of finite and affine types, Moscow Math. J. 4 (004, the electonic jounal of combinatoics 14 (007, #N4 5