Quantum Cluster Variables via Serre Polynomials

For skew-symmetric acyclic quantum cluster algebras, we express the quantum $F$-polynomials and the quantum cluster monomials in terms of Serre polynomials of quiver Grassmannians of rigid modules. As byproducts, we obtain the existence of counting polynomials for these varieties and the positivity conjecture with respect to acyclic seeds. These results complete previous work by Caldero and Reineke and confirm a recent conjecture by Rupel.Comment: minor corrections, reference added, example 4.3 added, 38 page

Cluster algebras of type A2(1)A_2^{(1)}

In this paper we study cluster algebras \myAA of type $A_2^{(1)}$. We solve the recurrence relations among the cluster variables (which form a T--system of type $A_2^{(1)}$). We solve the recurrence relations among the coefficients of \myAA (which form a Y--system of type $A_2^{(1)}$). In \myAA there is a natural notion of positivity. We find linear bases \BB of \myAA such that positive linear combinations of elements of \BB coincide with the cone of positive elements. We call these bases \emph{atomic bases} of \myAA. These are the analogue of the "canonical bases" found by Sherman and Zelevinsky in type $A_{1}^{(1)}$. Every atomic basis consists of cluster monomials together with extra elements. We provide explicit expressions for the elements of such bases in every cluster. We prove that the elements of \BB are parameterized by \ZZ^3 via their $\mathbf{g}$--vectors in every cluster. We prove that the denominator vector map in every acyclic seed of \myAA restricts to a bijection between \BB and \ZZ^3. In particular this gives an explicit algorithm to determine the "virtual" canonical decomposition of every element of the root lattice of type $A_2^{(1)}$. We find explicit recurrence relations to express every element of \myAA as linear combinations of elements of \BB.Comment: Latex, 40 pages; Published online in Algebras and Representation Theory, springer, 201

Categorification of skew-symmetrizable cluster algebras

We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with the action of a finite group G, we construct a G-equivariant mutation on the set of maximal rigid G-invariant objects of C. Using an appropriate cluster character, we can then attach to these data an explicit skew-symmetrizable cluster algebra. As an application we prove the linear independence of the cluster monomials in this setting. Finally, we illustrate our construction with examples associated with partial flag varieties and unipotent subgroups of Kac-Moody groups, generalizing to the non simply-laced case several results of Gei\ss-Leclerc-Schr\"oer.Comment: 64 page

A homological interpretation of the transverse quiver Grassmannians

In recent articles, the investigation of atomic bases in cluster algebras associated to affine quivers led the second-named author to introduce a variety called transverse quiver Grassmannian and the first-named and third-named authors to consider the smooth loci of quiver Grassmannians. In this paper, we prove that, for any affine quiver Q, the transverse quiver Grassmannian of an indecomposable representation M is the set of points N in the quiver Grassmannian of M such that Ext^1(N,M/N)=0. As a corollary we prove that the transverse quiver Grassmannian coincides with the smooth locus of the irreducible components of minimal dimension in the quiver Grassmannian.Comment: final version, 7 pages, corollary 1.2 has been modifie

Generalised Moore spectra in a triangulated category

In this paper we consider a construction in an arbitrary triangulated category T which resembles the notion of a Moore spectrum in algebraic topology. Namely, given a compact object C of T satisfying some finite tilting assumptions, we obtain a functor which "approximates" objects of the module category of the endomorphism algebra of C in T. This generalises and extends a construction of Jorgensen in connection with lifts of certain homological functors of derived categories. We show that this new functor is well-behaved with respect to short exact sequences and distinguished triangles, and as a consequence we obtain a new way of embedding the module category in a triangulated category. As an example of the theory, we recover Keller's canonical embedding of the module category of a path algebra of a quiver with no oriented cycles into its u-cluster category for u>1.Comment: 26 pages, improvement to exposition of the proof of Theorem 3.

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix $A$, for any index $k$, one can define an automorphism associated with $A,$ of the field $\mathbf{Q}(u_1, >..., u_n)$ of rational functions of $n$ independent indeterminates $u_1,..., u_n.$ It is an isomorphism between two cluster algebras associated to the matrix $A$ (see section 4 for precise meaning). When $A$ is of finite type, these isomorphisms behave nicely, they are compatible with the BGP-reflection functors of cluster categories defined in [Z1, Z2] if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined in [FZ2, FZ3]. Using the construction of preprojective or preinjective modules of hereditary algebras by Dlab-Ringel [DR] and the Coxeter automorphisms (i.e., a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.Comment: revised versio

Discrete integrable systems, positivity, and continued fraction rearrangements

In this review article, we present a unified approach to solving discrete, integrable, possibly non-commutative, dynamical systems, including the $Q$- and $T$-systems based on $A_r$. The initial data of the systems are seen as cluster variables in a suitable cluster algebra, and may evolve by local mutations. We show that the solutions are always expressed as Laurent polynomials of the initial data with non-negative integer coefficients. This is done by reformulating the mutations of initial data as local rearrangements of continued fractions generating some particular solutions, that preserve manifest positivity. We also show how these techniques apply as well to non-commutative settings.Comment: 24 pages, 2 figure

Q-systems, Heaps, Paths and Cluster Positivity

We consider the cluster algebra associated to the $Q$-system for $A_r$ as a tool for relating $Q$-system solutions to all possible sets of initial data. We show that the conserved quantities of the $Q$-system are partition functions for hard particles on particular target graphs with weights, which are determined by the choice of initial data. This allows us to interpret the simplest solutions of the Q-system as generating functions for Viennot's heaps on these target graphs, and equivalently as generating functions of weighted paths on suitable dual target graphs. The generating functions take the form of finite continued fractions. In this setting, the cluster mutations correspond to local rearrangements of the fractions which leave their final value unchanged. Finally, the general solutions of the $Q$-system are interpreted as partition functions for strongly non-intersecting families of lattice paths on target lattices. This expresses all cluster variables as manifestly positive Laurent polynomials of any initial data, thus proving the cluster positivity conjecture for the $A_r$ $Q$-system. We also give an alternative formulation in terms of domino tilings of deformed Aztec diamonds with defects.Comment: 106 pages, 38 figure

The solution of the quantum A1A_1 T-system for arbitrary boundary

We solve the quantum version of the $A_1$ $T$-system by use of quantum networks. The system is interpreted as a particular set of mutations of a suitable (infinite-rank) quantum cluster algebra, and Laurent positivity follows from our solution. As an application we re-derive the corresponding quantum network solution to the quantum $A_1$ $Q$-system and generalize it to the fully non-commutative case. We give the relation between the quantum $T$-system and the quantum lattice Liouville equation, which is the quantized $Y$-system.Comment: 24 pages, 18 figure

Quantum symmetric pairs and representations of double affine Hecke algebras of type C∨CnC^\vee C_n

We build representations of the affine and double affine braid groups and Hecke algebras of type $C^\vee C_n$, based upon the theory of quantum symmetric pairs $(U,B)$. In the case $U=U_q(gl_N)$, our constructions provide a quantization of the representations constructed by Etingof, Freund and Ma in arXiv:0801.1530, and also a type $BC$ generalization of the results in arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic