arXiv subjects
Publications and source records attributed to Andrei Zelevinsky.
This is a preliminary draft of Chapter 7 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. Chapter 6 has been posted as arXiv:2008.09189. This installment contains: Chapter 7. Plabic graphs
This is a preliminary draft of Chapter 6 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. Chapters 4-5 have been posted as arXiv:1707.07190. This installment contains: Chapter 6. Cluster structures in commutative rings
This is a preliminary draft of Chapters 4-5 of our forthcoming textbook "Introduction to Cluster Algebras." Chapters 1-3 have been posted as arXiv:1608.05735. This installment contains: Chapter 4. New patterns from old Chapter 5. Finite type classification
This is a preliminary draft of Chapters 1-3 of our forthcoming textbook "Introduction to Cluster Algebras." This installment contains: Chapter 1. Total positivity Chapter 2. Mutations of quivers and matrices Chapter 3. Clusters and seeds
We establish the existence of a quantum lift of the greedy basis.
We identify a quantum lift of the greedy basis for rank 2 coefficient-free cluster algebras. Our main result is that our construction does not depend on the choice of initial cluster, that it builds all cluster monomials, and that it produces bar-invariant elements. We also present several conjectures related to this quantum greedy basis and the triangular basis of Berenstein and Zelevinsky.
Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.
We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.
A lot of recent activity in the theory of cluster algebras has been directed towards various constructions of "natural" bases in them. One of the approaches to this problem was developed several years ago by P.Sherman - A.Zelevinsky who have shown that the indecomposable positive elements form an integer basis in any rank 2 cluster algebra of finite or affine type. It is strongly suspected (but not proved) that this property does not extend beyond affine types. Here we go around this difficulty by constructing a new basis in any rank 2 cluster algebra that we call the greedy basis. It consists of a special family of indecomposable positive elements that we call greedy elements. Inspired by a recent work of K.Lee - R.Schiffler and D.Rupel, we give explicit combinatorial expressions for greedy elements using the language of Dyck paths.
A lot of recent activity has been directed towards various constructions of "natural" bases in cluster algebras. We develop a new approach to this problem which is close in spirit to Lusztig's construction of a canonical basis, and the pioneering construction of the Kazhdan-Lusztig basis in a Hecke algebra. The key ingredient of our approach is a new version of Lusztig's Lemma that we apply to all acyclic quantum cluster algebras. As a result, we construct the "canonical" basis in every such algebra that we call the canonical triangular basis.
We study two families of integer vectors playing a crucial part in the structural theory of cluster algebras: the $\gg$-vectors parameterizing cluster variables, and the $\cc$-vectors parameterizing the coefficients. We prove two identities relating these vectors to each other. The proofs depend on the sign-coherence assumption for $\cc$-vectors that still remains unproved in full generality.
This is an extended abstract of my talk at the Oberwolfach Workshop "Interactions between Algebraic Geometry and Noncommutative Algebra" (May 10 - 14, 2010). We present some properties of quiver Grassmannians and examples of explicit computations of their Euler characteristics.
We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the "Cluster algebras IV" paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F-polynomials. In the case of skew-symmetric exchange matrices we find an interpretation of these g-vectors and F-polynomials in terms of (decorated) representations of quivers with potentials. Using this interpretation, we prove most of the conjectures about g-vectors and F-polynomials made in loc. cit.
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in the simply connected semisimple algebraic group of the same Cartan-Killing type. In this realization, the cluster variables appear as certain (generalized) principal minors.
We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.
This is an extended abstract of my talk at the Oberwolfach Workshop "Algebraic Groups" (April 22 - 28, 2007). It is based on a joint work with H.Derksen and J.Weyman (arXiv:0704.0649v2 [math.RA]).
We study the dependence of a cluster algebra on the choice of coefficients. We write general formulas expressing the cluster variables in any cluster algebra in terms of the initial data; these formulas involve a family of polynomials associated with a particular choice of "principal" coefficients. We show that the exchange graph of a cluster algebra with principal coefficients covers the exchange graph of any cluster algebra with the same exchange matrix. We investigate two families of parametrizations of cluster monomials by lattice points, determined, respectively, by the denominators of their Laurent expansions and by certain multi-gradings in cluster algebras with principal coefficients. The properties of these parametrizations, some proven and some conjectural, suggest links to duality conjectures of V.Fock and A.Goncharov [math.AG/0311245]. The coefficient dynamics leads to a natural generalization of Al.Zamolodchikov's Y-systems. We establish a Laurent phenomenon for such Y-systems, previously known in finite type only, and sharpen the periodicity result from [hep-th/0111053]. For cluster algebras of finite type, we identify a canonical "universal" choice of coefficients such that an arbitrary cluster algebra can be obtained from the universal one (of the same type) by an appropriate specialization of coefficients.
We study the cluster variables and "imaginary" elements of the semicanonical basis for the coefficient-free cluster algebra of affine type $A_1^{(1)}$. A closed formula for the Laurent expansions of these elements was obtained by P.Caldero and the author in math.RT/0604054. As a by-product, there was given a combinatorial interpretation of the Laurent polynomials in question, equivalent to the one obtained by G.Musiker and J.Propp in math.CO/0602408. The arguments in math.RT/0604054 used a geometric interpretation of the Laurent polynomials due to P.Caldero and F.Chapoton (math.RT/0410184). This note provides a quick, self-contained and completely elementary alternative proof of the same results.