Introduction to cluster algebras. Andrei Zelevinsky (Northeastern University) MSRI/Evans Lecture, UC Berkeley, October 1, 2012
|
|
- Stanley Cain
- 6 years ago
- Views:
Transcription
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.
CLUSTER ALGEBRA ALFREDO NÁJERA CHÁVEZ
Séminaire Lotharingien de Combinatoire 69 (203), Article B69d ON THE c-vectors AND g-vectors OF THE MARKOV CLUSTER ALGEBRA ALFREDO NÁJERA CHÁVEZ Abstract. We describe the c-vectors and g-vectors of the
More informationBases for Cluster Algebras from Surfaces
Bases for Cluster Algebras from Surfaces Gregg Musiker (U. Minnesota), Ralf Schiffler (U. Conn.), and Lauren Williams (UC Berkeley) Bay Area Discrete Math Day Saint Mary s College of California November
More informationQuiver mutations. Tensor diagrams and cluster algebras
Maurice Auslander Distinguished Lectures April 20-21, 2013 Sergey Fomin (University of Michigan) Quiver mutations based on joint work with Andrei Zelevinsky Tensor diagrams and cluster algebras based on
More informationQUANTUM F -POLYNOMIALS IN THE THEORY OF CLUSTER ALGEBRAS. A dissertation presented by Thao Tran to The Department of Mathematics
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
More informationCombinatorics of Theta Bases of Cluster Algebras
Combinatorics of Theta Bases of Cluster Algebras Li Li (Oakland U), joint with Kyungyong Lee (University of Nebraska Lincoln/KIAS) and Ba Nguyen (Wayne State U) Jan 22 24, 2016 What is a Cluster Algebra?
More informationA SURVEY OF CLUSTER ALGEBRAS
A SURVEY OF CLUSTER ALGEBRAS MELODY CHAN Abstract. This is a concise expository survey of cluster algebras, introduced by S. Fomin and A. Zelevinsky in their four-part series of foundational papers [1],
More informationCluster algebras and derived categories
Cluster algebras and derived categories Bernhard Keller Abstract. This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras.
More informationQ(x n+1,..., x m ) in n independent variables
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
More informationIntroduction to cluster algebras. Sergey Fomin
Joint Introductory Workshop MSRI, Berkeley, August 2012 Introduction to cluster algebras Sergey Fomin (University of Michigan) Main references Cluster algebras I IV: J. Amer. Math. Soc. 15 (2002), with
More informationLecture 2: Cluster complexes and their parametrizations
Lecture 2: Cluster complexes and their parametrizations Nathan Reading NC State University Cluster Algebras and Cluster Combinatorics MSRI Summer Graduate Workshop, August 2011 Introduction The exchange
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: June 12, 2017 8 Upper Bounds and Lower Bounds In this section, we study the properties of upper bound and lower bound of cluster algebra.
More informationMutation classes of quivers with constant number of arrows and derived equivalences
Mutation classes of quivers with constant number of arrows and derived equivalences Sefi Ladkani University of Bonn http://www.math.uni-bonn.de/people/sefil/ 1 Motivation The BGP reflection is an operation
More informationCluster algebras and applications
Université Paris Diderot Paris 7 DMV Jahrestagung Köln, 22. September 2011 Context Lie theory canonical bases/total positivity [17] [25] Poisson geometry [13] higher Teichmüller th. [5] Cluster algebras
More informationarxiv: v5 [math.ra] 17 Sep 2013
UNIVERSAL GEOMETRIC CLUSTER ALGEBRAS NATHAN READING arxiv:1209.3987v5 [math.ra] 17 Sep 2013 Abstract. We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient
More informationSkew-symmetric cluster algebras of finite mutation type
J. Eur. Math. Soc. 14, 1135 1180 c European Mathematical Society 2012 DOI 10.4171/JEMS/329 Anna Felikson Michael Shapiro Pavel Tumarkin Skew-symmetric cluster algebras of finite mutation type Received
More informationA NEW COMBINATORIAL FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS
A NEW COMBINATORIAL FORMULA FOR CLUSTER MONOMIALS OF EQUIORIENTED TYPE A QUIVERS D. E. BAYLERAN, DARREN J. FINNIGAN, ALAA HAJ ALI, KYUNGYONG LEE, CHRIS M. LOCRICCHIO, MATTHEW R. MILLS, DANIEL PUIG-PEY
More information(Affine) generalized associahedra, root systems, and cluster algebras
(Affine) generalized associahedra, root systems, and cluster algebras Salvatore Stella (joint with Nathan Reading) INdAM - Marie Curie Actions fellow Dipartimento G. Castelnuovo Università di Roma La Sapienza
More informationLINEAR INDEPENDENCE OF CLUSTER MONOMIALS FOR SKEW-SYMMETRIC CLUSTER ALGEBRAS
LINEAR INDEPENDENCE OF CLUSTER MONOMIALS FOR SKEW-SYMMETRIC CLUSTER ALGEBRAS GIOVANNI CERULLI IRELLI, BERNHARD KELLER, DANIEL LABARDINI-FRAGOSO, AND PIERRE-GUY PLAMONDON Dedicated to Idun Reiten on the
More informationMaximal Green Sequences via Quiver Semi-Invariants
Maximal Green Sequences via Quiver Semi-Invariants Stephen Hermes Wellesley College, Wellesley, MA Maurice Auslander Distinguished Lectures and International Conference Woods Hole May 3, 2015 Preliminaries
More informationPOSITIVITY FOR CLUSTER ALGEBRAS FROM SURFACES
POSITIVITY FOR CLUSTER ALGEBRAS FROM SURFACES GREGG MUSIKER, RALF SCHIFFLER, AND LAUREN WILLIAMS Abstract We give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster
More informationQuiver mutation and derived equivalence
U.F.R. de Mathématiques et Institut de Mathématiques Université Paris Diderot Paris 7 Amsterdam, July 16, 2008, 5ECM History in a nutshell quiver mutation = elementary operation on quivers discovered in
More informationCALDERO-CHAPOTON ALGEBRAS
CALDERO-CHAPOTON ALGEBRAS GIOVANNI CERULLI IRELLI, DANIEL LABARDINI-FRAGOSO, AND JAN SCHRÖER Abstract. Motivated by the representation theory of quivers with potential introduced by Derksen, Weyman and
More informationCluster algebras and Markoff numbers
CaMUS 3, 19 26 Cluster algebras and Markoff numbers Xueyuan Peng and Jie Zhang Abstract We introduce Markoff numbers and reveal their connection to the cluster algebra associated to the once-punctured
More informationQuantizations of cluster algebras
Quantizations of cluster algebras Philipp Lampe Bielefeld University International Conference Mathematics Days in Sofia July 10, 2014, Sofia, Bulgaria Ph. Lampe (Bielefeld) Quantum cluster algebras July
More informationThe Greedy Basis Equals the Theta Basis A Rank Two Haiku
The Greedy Basis Equals the Theta Basis A Rank Two Haiku Man Wai Cheung (UCSD), Mark Gross (Cambridge), Greg Muller (Michigan), Gregg Musiker (University of Minnesota) *, Dylan Rupel (Notre Dame), Salvatore
More informationQuivers of Period 2. Mariya Sardarli Max Wimberley Heyi Zhu. November 26, 2014
Quivers of Period 2 Mariya Sardarli Max Wimberley Heyi Zhu ovember 26, 2014 Abstract A quiver with vertices labeled from 1,..., n is said to have period 2 if the quiver obtained by mutating at 1 and then
More informationProblem 1. Let f n (x, y), n Z, be the sequence of rational functions in two variables x and y given by the initial conditions.
18.217 Problem Set (due Monday, December 03, 2018) Solve as many problems as you want. Turn in your favorite solutions. You can also solve and turn any other claims that were given in class without proofs,
More informationThe geometry of cluster algebras
The geometry of cluster algebras Greg Muller February 17, 2013 Cluster algebras (the idea) A cluster algebra is a commutative ring generated by distinguished elements called cluster variables. The set
More informationSTRONGNESS OF COMPANION BASES FOR CLUSTER-TILTED ALGEBRAS OF FINITE TYPE
STRONGNESS OF COMPANION BASES FOR CLUSTER-TILTED ALGEBRAS OF FINITE TYPE KARIN BAUR AND ALIREZA NASR-ISFAHANI Abstract. For every cluster-tilted algebra of simply-laced Dynkin type we provide a companion
More informationLINEAR INDEPENDENCE OF CLUSTER MONOMIALS FOR SKEW-SYMMETRIC CLUSTER ALGEBRAS
LINEAR INDEPENDENCE OF CLUSTER MONOMIALS FOR SKEW-SYMMETRIC CLUSTER ALGEBRAS GIOVANNI CERULLI IRELLI, BERNHARD KELLER, DANIEL LABARDINI-FRAGOSO, AND PIERRE-GUY PLAMONDON Dedicated to Idun Reiten on the
More informationQuivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials
Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials Giovanni Cerulli Irelli and Daniel Labardini-Fragoso Abstract To each tagged triangulation
More informationExact WKB Analysis and Cluster Algebras
Exact WKB Analysis and Cluster Algebras Kohei Iwaki (RIMS, Kyoto University) (joint work with Tomoki Nakanishi) Winter School on Representation Theory January 21, 2015 1 / 22 Exact WKB analysis Schrödinger
More informationCombinatorial aspects of derived equivalence
Combinatorial aspects of derived equivalence Sefi Ladkani University of Bonn http://guests.mpim-bonn.mpg.de/sefil/ 1 What is the connection between... 2 The finite dimensional algebras arising from these
More informationCluster varieties for tree-shaped quivers and their cohomology
Cluster varieties for tree-shaped quivers and their cohomology Frédéric Chapoton CNRS & Université de Strasbourg Octobre 2016 Cluster algebras and the associated varieties Cluster algebras are commutative
More informationarxiv: v1 [math.rt] 4 Dec 2017
UNIFORMLY COLUMN SIGN-COHERENCE AND THE EXISTENCE OF MAXIMAL GREEN SEQUENCES PEIGEN CAO FANG LI arxiv:1712.00973v1 [math.rt] 4 Dec 2017 Abstract. In this paper, we prove that each matrix in M m n (Z 0
More informationTropicalizations of Positive Parts of Cluster Algebras The conjectures of Fock and Goncharov David Speyer
Tropicalizations of Positive Parts of Cluster Algebras The conjectures of Fock and Goncharov David Speyer This talk is based on arxiv:math/0311245, section 4. We saw before that tropicalizations look like
More informationSEEDS AND WEIGHTED QUIVERS. 1. Seeds
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
More informationOriented Exchange Graphs & Torsion Classes
1 / 25 Oriented Exchange Graphs & Torsion Classes Al Garver (joint with Thomas McConville) University of Minnesota Representation Theory and Related Topics Seminar - Northeastern University October 30,
More informationarxiv: v3 [math.ra] 17 Mar 2013
CLUSTER ALGEBRAS: AN INTRODUCTION arxiv:.v [math.ra] 7 Mar 0 LAUREN K. WILLIAMS Dedicated to Andrei Zelevinsky on the occasion of his 0th birthday Abstract. Cluster algebras are commutative rings with
More informationCluster algebras II: Finite type classification
Invent. math. 154, 63 121 (2003) DOI: 10.1007/s00222-003-0302-y Cluster algebras II: Finite type classification Sergey Fomin 1, Andrei Zelevinsky 2 1 Department of Mathematics, University of Michigan,
More informationarxiv:math/ v3 [math.rt] 21 Jan 2004
CLUSTER ALGEBRAS III: UPPER BOUNDS AND DOUBLE BRUHAT CELLS arxiv:math/0305434v3 [math.rt] 21 Jan 2004 ARKADY BERENSTEIN, SERGEY FOMIN, AND ANDREI ZELEVINSKY Abstract. We develop a new approach to cluster
More informationFACTORIAL CLUSTER ALGEBRAS
FACTORIAL CLUSTER ALGEBRAS CHRISTOF GEISS, BERNARD LECLERC, AND JAN SCHRÖER Abstract. We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements.
More informationLecture 3: Tropicalizations of Cluster Algebras Examples David Speyer
Lecture 3: Tropicalizations of Cluster Algebras Examples David Speyer Let A be a cluster algebra with B-matrix B. Let X be Spec A with all of the cluster variables inverted, and embed X into a torus by
More informationDimer models and cluster categories of Grassmannians
Dimer models and cluster categories of Grassmannians Karin Baur University of Graz Rome, October 18, 2016 1/17 Motivation Cluster algebra structure of Grassmannians Construction of cluster categories (k,n)
More informationQuantum cluster algebras
Advances in Mathematics 195 (2005) 405 455 www.elsevier.com/locate/aim Quantum cluster algebras Arkady Berenstein a, Andrei Zelevinsky b, a Department of Mathematics, University of Oregon, Eugene, OR 97403,
More informationCorrespondences between cluster structures
Correspondences between cluster structures by Christopher Fraser A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University
More informationMutations of non-integer quivers: finite mutation type. Anna Felikson
Mutations of non-integer quivers: finite mutation type Anna Felikson Durham University (joint works with Pavel Tumarkin and Philipp Lampe) Jerusalem, December 0, 08 . Mutations of non-integer quivers:
More informationQUIVERS AND LATTICES.
QUIVERS AND LATTICES. KEVIN MCGERTY We will discuss two classification results in quite different areas which turn out to have the same answer. This note is an slightly expanded version of the talk given
More informationSpectral networks at marginal stability, BPS quivers, and a new construction of wall-crossing invariants
Spectral networks at marginal stability, BPS quivers, and a new construction of wall-crossing invariants Pietro Longhi Uppsala University String-Math 2017 In collaboration with: Maxime Gabella, Chan Y.
More informationCluster Algebras and Compatible Poisson Structures
Cluster Algebras and Compatible Poisson Structures Poisson 2012, Utrecht July, 2012 (Poisson 2012, Utrecht) Cluster Algebras and Compatible Poisson Structures July, 2012 1 / 96 (Poisson 2012, Utrecht)
More informationCluster algebras of infinite rank
J. London Math. Soc. (2) 89 (2014) 337 363 C 2014 London Mathematical Society doi:10.1112/jlms/jdt064 Cluster algebras of infinite rank Jan E. Grabowski and Sira Gratz with an appendix by Michael Groechenig
More informationarxiv: v2 [math.co] 14 Jun 2018
DOMINANCE PHENOMENA: MUTATION, SCATTERING AND CLUSTER ALGEBRAS arxiv:1802.10107v2 [math.co 14 Jun 2018 NATHAN READING Abstract. An exchange matrix B dominates an exchange matrix B if the signs of corresponding
More informationCluster Algebras. Philipp Lampe
Cluster Algebras Philipp Lampe December 4, 2013 2 Contents 1 Informal introduction 5 1.1 Sequences of Laurent polynomials............................. 5 1.2 Exercises............................................
More informationQUIVERS WITH POTENTIALS AND THEIR REPRESENTATIONS II: APPLICATIONS TO CLUSTER ALGEBRAS
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 23, Number 3, July 2010, Pages 749 790 S 0894-0347(10)00662-4 Article electronically published on February 8, 2010 QUIVERS WITH POTENTIALS AND THEIR
More informationarxiv: v2 [math.qa] 13 Jul 2010
QUANTUM CLUSTER VARIABLES VIA SERRE POLYNOMIALS FAN QIN Abstract. For skew-symmetric acyclic quantum cluster algebras, we express the quantum F -polynomials and the quantum cluster monomials in terms of
More informationString Math Bonn, July S.C., arxiv: S.C., & M. Del Zotto, arxiv: N = 2 Gauge Theories, Half Hypers, and Quivers
N = 2 Gauge Theories, Half Hypers, and Quivers String Math Bonn, July 2012 S.C., arxiv:1203.6734. S.C., & M. Del Zotto, arxiv:1207.2275. Powerful methods to compute BPS spectra of 4d N = 2 theories: Geometric
More informationarxiv: v1 [math.rt] 15 Oct 2008
CLASSIFICATION OF FINITE-GROWTH GENERAL KAC-MOODY SUPERALGEBRAS arxiv:0810.2637v1 [math.rt] 15 Oct 2008 CRYSTAL HOYT AND VERA SERGANOVA Abstract. A contragredient Lie superalgebra is a superalgebra defined
More informationCluster algebras and cluster categories
Cluster algebras and cluster categories Lecture notes for the XVIII LATIN AMERICAN ALGEBRA COLLOQUIUM Sao Pedro - Brazil, Aug -8 009 Ralf Schiffler Contents 0 Introduction Lecture : Cluster algebras. Definition..............................
More informationQuadratic differentials as stability conditions. Tom Bridgeland (joint work with Ivan Smith)
Quadratic differentials as stability conditions Tom Bridgeland (joint work with Ivan Smith) Our main result identifies spaces of meromorphic quadratic differentials on Riemann surfaces with spaces of stability
More informationKac-Moody Algebras. Ana Ros Camacho June 28, 2010
Kac-Moody Algebras Ana Ros Camacho June 28, 2010 Abstract Talk for the seminar on Cohomology of Lie algebras, under the supervision of J-Prof. Christoph Wockel Contents 1 Motivation 1 2 Prerequisites 1
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: July 11, 2017 11 Cluster Algebra from Surfaces In this lecture, we will define and give a quick overview of some properties of cluster
More informationarxiv: v2 [math.gt] 15 Nov 2017
CLUSTER ALGEBRAS AND JONES POLYNOMIALS KYUNGYONG LEE AND RALF SCHIFFLER arxiv:1710.08063v2 [math.gt] 15 Nov 2017 Abstract. We present a new and very concrete connection between cluster algebras and knot
More informationCluster structures on open Richardson varieties and their quantizations
Cluster structures on open Richardson varieties and their quantizations Lie Theory and Its Applications in Physics XI, Varna, June 15-21, 2015 Milen Yakimov (Louisiana State Univ.) Joint work with Tom
More informationCLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES
CLUSTER ALGEBRAS, QUIVER REPRESENTATIONS AND TRIANGULATED CATEGORIES BERNHARD KELLER Abstract. This is an introduction to some aspects of Fomin-Zelevinsky s cluster algebras and their links with the representation
More informationTHE CLASSIFICATION PROBLEM REPRESENTATION-FINITE, TAME AND WILD QUIVERS
October 28, 2009 THE CLASSIFICATION PROBLEM REPRESENTATION-FINITE, TAME AND WILD QUIVERS Sabrina Gross, Robert Schaefer In the first part of this week s session of the seminar on Cluster Algebras by Prof.
More informationarxiv: v1 [math.co] 20 Dec 2016
F-POLYNOMIAL FORMULA FROM CONTINUED FRACTIONS MICHELLE RABIDEAU arxiv:1612.06845v1 [math.co] 20 Dec 2016 Abstract. For cluster algebras from surfaces, there is a known formula for cluster variables and
More informationCLUSTER EXPANSION FORMULAS AND PERFECT MATCHINGS
CLUSTER EXPANSION FORMULAS AND PERFECT MATCHINGS GREGG MUSIKER AND RALF SCHIFFLER Abstract. We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We
More informationCluster algebras from 2d gauge theories
Cluster algebras from 2d gauge theories Francesco Benini Simons Center for Geometry and Physics Stony Brook University Texas A&M University Heterotic Strings and (0,2) QFT 28 April 2014 with: D. Park,
More informationarxiv: v2 [math.co] 23 Jul 2015
Coxeter s frieze patterns at the crossroads of algebra, geometry and combinatorics arxiv:50305049v2 [mathco] 23 Jul 205 Sophie Morier-Genoud Sorbonne Universités, UPMC Univ Paris 06, UMR 7586, Institut
More information2-FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS
-FRIEZE PATTERNS AND THE CLUSTER STRUCTURE OF THE SPACE OF POLYGONS SOPHIE MORIER-GENOUD, VALENTIN OVSIENKO, AND SERGE TABACHNIKOV Abstract We study the space of -frieze patterns generalizing that of the
More informationQUIVERS WITH POTENTIALS ASSOCIATED TO TRIANGULATED SURFACES, PART III: TAGGED TRIANGULATIONS AND CLUSTER MONOMIALS
QUIVERS WITH POTENTIALS ASSOCIATED TO TRIANGULATED SURFACES, PART III: TAGGED TRIANGULATIONS AND CLUSTER MONOMIALS GIOVANNI CERULLI IRELLI DANIEL LABARDINI-FRAGOSO Abstract. To each tagged triangulation
More informationQuiver mutation and combinatorial DT-invariants
Discrete Mathematics and Theoretical Computer Science DMTCS vol. (subm.), by the authors, 1 1 arxiv:1709.03143v2 [math.co] 12 Sep 2017 Quiver mutation and combinatorial DT-invariants Bernhard Keller 1
More informationLecture Notes Introduction to Cluster Algebra
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Update: May 16, 2017 5 Review of Root Systems In this section, let us have a brief introduction to root system and finite Lie type classification
More informationDifference Painlevé equations from 5D gauge theories
Difference Painlevé equations from 5D gauge theories M. Bershtein based on joint paper with P. Gavrylenko and A. Marshakov arxiv:.006, to appear in JHEP February 08 M. Bershtein Difference Painlevé based
More informationCluster algebras and cluster monomials
Cluster algebras and cluster monomials Bernhard Keller Abstract. Cluster algebras were invented by Sergey Fomin and Andrei Zelevinsky at the beginning of the year 2000. Their motivations came from Lie
More informationCLUSTER AUTOMORPHISMS AND HYPERBOLIC CLUSTER ALGEBRAS IBRAHIM A SALEH
CLUSTER AUTOMORPHISMS AND HYPERBOLIC CLUSTER ALGEBRAS by IBRAHIM A SALEH B.A., Cairo University 1995 M.S., Cairo University 2002 M.S., Kansas State University 2008 AN ABSTRACT OF A DISSERTATION submitted
More informationarxiv: v3 [math.co] 12 Sep 2017
BASES FOR CLUSTER ALGEBRAS FROM ORBIFOLDS arxiv:1511.08023v3 [math.co] 12 Sep 2017 ANNA FELIKSON AND PAVEL TUMARKIN Abstract. We generalize the construction of the bracelet and bangle bases defined in
More informationCluster algebras and snake modules
(Lanzhou University and University of Connecticut) joint with Jian-Rong Li & Yan-Feng Luo A conference celebrating the 60th birthday of Prof Vyjayanthi Chari The Catholic University of America, Washington,
More informationAgain we return to a row of 1 s! Furthermore, all the entries are positive integers.
Lecture Notes Introduction to Cluster Algebra Ivan C.H. Ip Updated: April 4, 207 Examples. Conway-Coxeter frieze pattern Frieze is the wide central section part of an entablature, often seen in Greek temples,
More informationCluster structure of quantum Coxeter Toda system
Cluster structure of the quantum Coxeter Toda system Columbia University CAMP, Michigan State University May 10, 2018 Slides available at www.math.columbia.edu/ schrader Motivation I will talk about some
More informationarxiv: v2 [math.co] 24 Mar 2015
CLUSTER SUPERALGEBRAS arxiv:1503.01894v2 [math.co] 24 Mar 2015 VALENTIN OVSIENKO Abstract. We introduce cluster superalgebras, a class of Z 2 -graded commutative algebras generalizing cluster algebras
More information1. Quivers and their representations: Basic definitions and examples.
1 Quivers and their representations: Basic definitions and examples 11 Quivers A quiver Q (sometimes also called a directed graph) consists of vertices and oriented edges (arrows): loops and multiple arrows
More informationCLUSTER-TILTED ALGEBRAS OF FINITE REPRESENTATION TYPE
CLUSTER-TILTED ALGEBRAS OF FINITE REPRESENTATION TYPE ASLAK BAKKE BUAN, ROBERT J. MARSH, AND IDUN REITEN Abstract. We investigate the cluster-tilted algebras of finite representation type over an algebraically
More informationAn introduction to Hodge algebras
An introduction to Hodge algebras Federico Galetto May 28, 2009 The Grassmannian Let k be a field, E a k-vector space of dimension m Define Grass(n, E) := {V E dim V = n} If {e 1,, e m } is a basis of
More informationA COMPENDIUM ON THE CLUSTER ALGEBRA AND QUIVER PACKAGE IN Sage
Séminaire Lotharingien de Combinatoire 65 (2011), Article B65d A COMPENDIUM ON THE CLUSTER ALGEBRA AND QUIVER PACKAGE IN Sage GREGG MUSIKER AND CHRISTIAN STUMP Abstract. This is the compendium of the cluster
More informationArithmetics of 2-friezes
DOI 0007/s080-02-0348-2 Arithmetics of 2-friezes Sophie Morier-Genoud Received: 0 September 20 / Accepted: 9 January 202 Springer Science+Business Media, LLC 202 Abstract We consider the variant of Coxeter
More informationQuadratic differentials of exponential type and stability
Quadratic differentials of exponential type and stability Fabian Haiden (University of Vienna), joint with L. Katzarkov and M. Kontsevich January 28, 2013 Simplest Example Q orientation of A n Dynkin diagram,
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationBrane Tilings and Cluster Algebras
Brane Tilings and Cluster Algebras Gregg Musiker (U. Minnesota), In-Jee Jeong (Brown Univ.), and Sicong Zhang (Columbia Univ.) Algebra, Geometry and Combinatorics Day Loyola University Chicago November
More information5 Quiver Representations
5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )
More informationParameterizing orbits in flag varieties
Parameterizing orbits in flag varieties W. Ethan Duckworth April 2008 Abstract In this document we parameterize the orbits of certain groups acting on partial flag varieties with finitely many orbits.
More informationCluster algebras: Notes for the CDM-03 conference
Current Developments in Mathematics, 2003 Pages 1-34 Cluster algebras: Notes for the CDM-03 conference Sergey Fomin and Andrei Zelevinsky Abstract. This is an expanded version of the notes of our lectures
More informationarxiv: v5 [math.qa] 4 Aug 2011
Periodicities in cluster algebras and dilogarithm identities Tomoki Nakanishi arxiv:1006.0632v5 [math.qa] 4 Aug 2011 Abstract. We consider two kinds of periodicities of mutations in cluster algebras. For
More informationarxiv: v2 [math.co] 14 Sep 2009
Q-SYSTEM CLUSTER ALGEBRAS, PATHS AND TOTAL POSITIVITY arxiv:090.v [math.co] Sep 009 PHILIPPE DI FRANCESCO AND RINAT KEDEM Abstract. We review the solution of the A r Q-systems in terms of the partition
More informationRings. EE 387, Notes 7, Handout #10
Rings EE 387, Notes 7, Handout #10 Definition: A ring is a set R with binary operations, + and, that satisfy the following axioms: 1. (R, +) is a commutative group (five axioms) 2. Associative law for
More informationH. Schubert, Kalkül der abzählenden Geometrie, 1879.
H. Schubert, Kalkül der abzählenden Geometrie, 879. Problem: Given mp subspaces of dimension p in general position in a vector space of dimension m + p, how many subspaces of dimension m intersect all
More informationENDOMORPHISM ALGEBRAS OF MAXIMAL RIGID OBJECTS IN CLUSTER TUBES
ENDOMORPHISM ALGEBRAS OF MAXIMAL RIGID OBJECTS IN CLUSTER TUBES DONG YANG Abstract. Given a maximal rigid object T of the cluster tube, we determine the objects finitely presented by T. We then use the
More informationarxiv:math/ v4 [math.co] 22 Sep 2005
CLUSTER ALGEBRAS OF FINITE TYPE AND POSITIVE SYMMETRIZABLE MATRICES arxiv:math/0411341v4 [math.co] Sep 005 MICHAEL BAROT, CHRISTOF GEISS, AND ANDREI ZELEVINSKY Abstract. The paper is motivated by an analogy
More informationDerived equivalence classification of the cluster-tilted algebras of Dynkin type E
Derived equivalence classification of the cluster-tilted algebras of Dynkin type E Janine Bastian, Thorsten Holm, and Sefi Ladkani 2 Leibniz Universität Hannover, Institut für Algebra, Zahlentheorie und
More informationThe Geometry of Cluster Varieties arxiv: v2 [math.ag] 26 Dec from Surfaces
The Geometry of Cluster Varieties arxiv:1606.07788v2 [math.ag] 26 Dec 2018 from Surfaces A Dissertation Presented to the Faculty of the Graduate School of Yale University in Candidacy for the Degree of
More information