Introduction to cluster algebras. Andrei Zelevinsky (Northeastern University) MSRI/Evans Lecture, UC Berkeley, October 1, 2012

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1 Introduction to cluster algebras Andrei Zelevinsky (Northeastern University) MSRI/Evans Lecture, UC Berkeley, October, 202

2 Cluster algebras Discovered in 2000 (with S. Fomin) > 400 papers on arxiv Numerous conferences, summer schools, seminars, thematic programs, etc. (see links at the online Cluster Algebras Portal). Connections and applications across several disciplines Total positivity Representation theory String theory Statistical physics models Quiver representations Non-commutative geometry Teichmüller theory Hyperbolic geometry Discrete integrable systems Poisson geometry Tropical geometry Polyhedral combinatorics

3 Example [coefficient-free rank 2 cluster algebras]: For any positive integers b and c, the cluster algebra A(b, c) is the subring of F = Q(x, x 2 ) generated by the cluster variables x m (m Z) defined recursively by the exchange relations x m x m+ = { x b m + for m odd; x c m + for m even. Clusters: subsets {x m, x m+ } for m Z. Cluster monomials: elements x d m x d 2 m+ for m Z and d, d 2 Z 0.

4 Features of A(b, c) Laurent Phenomenon: A(b, c) Z[x ± m, x ± m+ ] for m Z. Sharper statement: A(b, c) = m Z Z[x ± m, x ± m+ ] = 2 m=0 Z[x ± m, x ± m+ ]. Finite type clasification: in A(b, c) the sequence (x m ) is periodic if and only if bc 3. More precisely, x m+p = x m, where: p = 5 for bc = ; p = 6 for bc = 2; p = 8 for bc = 3. For bc > 3 all x m are different. E.g., for b = c = get x 3 = x 2+ x, x 4 = x +x 2 + x x, x 2 5 = x + x, x 2 6 = x, x 7 = x 2 (goes back at least to Gauss). Good bases: for bc 3, the cluster monomials form a Z- basis in A(b, c). For bc > 3 this is not true, but they are linearly independent and belong to all good bases in A(b, c). Remark: the three finite type cases correspond naturally to the irreducible rank 2 root systems A 2, B 2, and G 2.

5 General setup Key ingredients: regular tree three discrete dynamical systems on a T n : n-regular tree with edges properly labeled by,..., n k (notation: t t ). Convenient to pick a root t 0 T n. T : t 0 t. T 2 : t t 0 t t 2 t 3. Z: a set with n involutions (mutations) µ,..., µ n. Z-pattern on T n : a Z-valued assignment t z t (t T n ) such k that z t = µ k (z t ) whenever t t. A Z-pattern on T n is uniquely determined by an (arbitrary) element z Z assigned to t 0. {z t : t T n } is the mutation-equivalence class of z = z t0.

6 Exchange matrices and their mutations Exchange matrix: a skew-symmetrizable integer n n matrix B = (b ij ). (Skew-symmetrizable: d i b ij = d j b ji for some d,..., d n > 0.) Exchange matrix mutations: µ k (B) = B = (b ij ) is given by b ij = b ij if i = k or j = k; b ij + [b ik ] + [b kj ] + [ b ik ] + [ b kj ] + otherwise, with the notation [b] + = max(b, 0). Pop out everywhere! Great applet by B. Keller (available from his homepage). Properties of µ k : () Preserves the set of exchange matrices. (2) Is an involution.

7 Quiver mutations {Skew-symmetric exchange matrices} {Quivers on vertices,..., n without loops and oriented 2-cycles} (B Q = Q(B) with [b ij ] + arrows from j to i). Matrix mutation translates to the 3-step procedure Q µ k (Q): () For every pair of arrows j k i, create a composite arrow j i. (2) Reverse all arrows at k. (3) Remove any maximal disjoint union of oriented 2-cycles (that could be created in Step ).

8 First classification: mutation-finite exchange matrices Classification obtained by A. Felikson - M. Shapiro - P. Tumarkin: arxiv: (skew-symmetric case); arxiv: (general case). Almost all (with exceptions) associated with triangulations of punctured Riemann surfaces. Used for the study of BPS quivers and spectra of N = 2 Quantum Field Theories by M. Alim - S. Cecotti - C. Cordova - S. Espahbodi - A. Rastogi - C. Vafa (arxiv:09.494, arxiv:2.3984)

9 Second classification: 2-finite mutation classes A mutation class S of exchange matrices is 2-finite if b ij b ji 3 for any B S and any i, j. Classification obtained by S. Fomin - A. Z. in [Cluster Algebras II]. Classified by Cartan-Killing types (or Dynkin diagrams). The Cartan counterpart of B is a generalized Cartan matrix A = A(B) = (a ij ) of the same size defined by a ij = 2 if i = j; b ij if i j. Theorem [CA II]. S is 2-finite A(B) is a Cartan matrix of finite type for some B S. Furthermore, the Cartan-Killing type of A(B) is uniquely determined by S. Finally, if A(B) = A(B ) is a Cartan matrix of finite type then B and B are mutationequivalent.

10 Y -seeds and their mutations Semifield (P,, +): abelian multiplicative group (P, ); addition: commutative, associative, distributive. A (labeled) Y -seed in a semifield P is a pair (y, B), where y = (y,..., y n ) is an n-tuple of elements of P; B = (b ij ) is an n n exchange matrix. The Y -seed mutation µ k : (y, B) (y, B ), where B = µ k (B); y y i = k if i = k; y i y [b ki] + k (y k + ) b ki if i k.

11 Y -seed patterns appear as: Discrete integrable systems (Y -systems) in theoretical physics Shear coordinates in Techmüller spaces [S. Fomin - D. Thurston, V. Fock - A. Goncharov, M. Gekhtman - M. Shapiro - A. Vainshtein] Fock-Goncharov varieties (cluster X-varieties) Wall-crossing formulas in Donaldson-Thomas/string theory [M. Kontsevich-Y. Soibelman, D. Gaiotto-G. Moore-A. Neitzke] The pentagram map and its generalizations

12 Seeds and their mutations (P,, ): semifield; ZP: integer group ring (ignores ); Q(P): field of fractions of ZP. F: ambient field isomorphic to Q(P)(u,..., u n ). A (labeled) seed in F: a triple (x, y, B), where (y, B) is a Y -seed in P, and x = (x,..., x n ) (a cluster ) is an n-tuple of elements of F forming a free generating set. The seed mutation µ k : (x, y, B) (x, y, B ), where (y, B ) = µ k (y, B), and x is obtained from x by replacing x k with x k = y k x [b ik ] + i + x [ b ik] + i. (y k )x k

13 Example: seeds in type A 2. Exchange matrices: B tr = ( ) r [ 0 0 ]. t y t x t t 0 y y 2 x x 2 t y (y 2 ) t 2 y (y 2 ) t 3 y y y 2 y 2 y y 2 y t 4 y y 2 y x x y 2 + y 2 x 2 (y 2 ) y y 2 y x y y 2 + y + x 2 x y 2 + y 2 (y y 2 y )x x 2 x 2 (y 2 ) x y y 2 + y + x 2 (y y 2 y )x x 2 y + x 2 x (y ) y x 2 y + x 2 x (y ) t 5 y 2 y x 2 x

14 Cluster algebra (x, y, B): initial seed attached to t 0. {x j;t : j =,..., n; t T n }: cluster variables. Cluster algebra A = A(x, y, B) = A(y, B): the ZP-subalgebra of the ambient field F generated by all cluster variables. Coordinate rings of many important varieties coming from Lie theory carry the CA structure: Grassmannians, (partial) flag varieties, Schubert varieties, double Bruhat cells,...

15 Finite type classification ([CA II]) A cluster algebra A is of finite type if it has only finitely many seeds (equivalently, finitely many cluster variables). Theorem [CA II]. A(y, B) is of finite type the mutation class of B is 2-finite. Thus cluster algebras of finite type are also classified by Cartan-Killing types. [ ] 0 b Example: n = 2, P = {}, B = B(b, c) = for some c 0 positive integers b and c. The corresponding cluster algebra is A(b, c) considered before.

16 c-vectors, g-vectors and F -polynomials ([CA IV]) Every pair (B; t 0 ) gives rise to: polynomials F j;t = F B;t 0 j;t Z[u,..., u n ] (F -polynomials); integer vectors c j;t = c B;t 0 j;t = (c j;t,..., c nj;t ) Z n (c-vectors) and g j;t = g B;t 0 j;t = (g j;t,..., g nj;t ) Z n (g-vectors). Properties: Each F j;t is not divisible by any u i, and is a subtraction-free rational expression in u,..., u n, hence can be evaluated in every semifield. y j;t = y c j;t y c nj;t n x j;t = x g j;t x g nj;t n Corollary. i F i;t P (y,..., y n ) b ij;t. F j;t F (ŷ,...,ŷ n ) F j;t P (y,...,y n ), where ŷ j = y j i x b ij i. x j;t ZP[x ±,..., x± n ] (Laurent phenomenon). y j;t Z[y ±,..., y± n ] provided b ij;t 0 for all i.

17 Example: c-vectors, g-vectors and F -polynomials in type A 2. t B t g ;t g 2;t F ;t F 2;t [ 0 t t [ t 2 [ [ 0 t 3 0 [ 0 t t 5 [ ] [ 0 ] [ ] 0 ] [ ] [ ] 0 0 ] [ ] [ ] 0 0 ] [ 0 ] [ 0 ] [ 0 ] [ ] ] [ ] ] [ ] 0 u 2 + u u 2 + u + u 2 + u u 2 + u + u + u +

18 One more example of Laurent Phenomenon: Somos-5 sequence The Somos-5 sequence is defined by the recurrence relations x m x m+5 = x m+ x m+4 + x m+2 x m+3 (m ) and the initial conditions x = = x 5 =. Unexpectedly, the terms are integers:,,,,, 2, 3, 5,, 37, 83, 274, 27, 66, 22833, 6573,.... Why??

19 Stronger statement: every x m is an integer Laurent polynomial in x,..., x 5. Enough to show: there is a (coefficient-free) cluster algebra with an initial cluster x = (x,..., x 5 ) having all the Somos-5 relations among its exchange relations, so that all x m are among its cluster variables. Claim: the initial (skew-symmetric) exchange matrix does the job! B = Indeed, the first column of B gives the desired Somos-5 relation, and one checks that σ(µ (B)) = B, where the operation σ moves the first column and the first row to the last place.,

20 Generalization [A. Fordy, R. Marsh] Let (a,..., a n ) be a palyndromic integer vector: a i = a n i for i =,..., n. Consider the sequence x, x 2,... given by the recurrence x m x m+n = n i= x [a n i] + m+i + i= x [ a i] + m+i (m =, 2,... ) with indeterminates x,..., x n as the initial terms (the Somos-5 recurrence appears as a special case for n = 5 and (a,..., a n ) = (,,, )). Then all the terms x m are integer Laurent polynomials in x,..., x n. The proof is the same: there is a skew-symmetric n n exchange matrix B with the first column [0, a,..., a n ], such that σ(µ (B)) = B. Exercise: Find B.

21 Conjectures from [CA IV] Constant term: Each polynomial F j;t has constant term. Sign-coherence for c-vectors: Each vector c j;t has either all components nonnegative, or all components nonpositive. Unimodularity: Z-basis in Z n. For every t T n, the g-vectors at t form a Parametrization by g-vectors: A cluster variable (more generally, cluster monomial) is uniquely determined by its g-vector. Tropical Langlands duality by Fock - Goncharov: g B;t 0 ij;t = c BT t ;t ji;t 0.

22 Theorem. [H. Derksen, J. Weyman, A. Z.] The above conjectures hold under the assumption that the exchange matrix B is skew-symmetric. The proof uses quivers with potentials and their representations. Other recent proofs: K. Nagao via Donaldson-Thomas theory, arxiv: P.-G. Plamondon via cluster categories, arxiv: All proofs are based on the correspondence (described above) B Q = Q(B) between skew-symmetric exchange matrices and quivers on vertices,..., n without loops and oriented 2-cycles.

23 Quiver Grassmannians Representation M of Q specified by: a finite-dimensional C-vector space M i attached to any vertex i; a linear map a = a M : M j M i attached to any arrow a : j i (notation: j = t(a), tail; i = h(a), head). Dimension vector : dim M = (dim M,..., dim M n ). For e Z n 0, denote by Gr e(m) the quiver Grassmannian of subrepresentations N M with dim N = e. Gr e (M) is a projective algebraic variety (not necessarily irreducible or smooth).

24 F -polynomial of a quiver representation χ(gr e (M)): the Euler-Poincaré characteristic of Gr e (M). F -polynomial of M: F M (u,..., u n ) = e χ(gr e (M)) n i= u e i i. F M has constant term. Thus the constant term conjecture is a consequence of Theorem. For B, t 0, j and t as above, there is an indecomposable representation M = M B;t 0 j;t of Q(B) such that F B;t 0 j;t = F M. Remark: One can show that the constant term conjecture implies (in an elementary way) all the other conjectures [T. Nakanishi, A. Z].

25 Open problem: Extend the proofs of conjectures to arbitrary skew-symmetrizable matrices. Partial results: L. Demonet, Mutations of group species with potentials and their representations. Applications to cluster algebras, arxiv: B. Nguefack, Modulated quivers with potentials and their Jacobian algebras, arxiv: D. Rupel, On Quantum Analogue of The Caldero-Chapoton Formula, arxiv: D. Labardini Fragoso, A. Z., Work in progress.