127 research outputs found
Quantum Cluster Variables via Serre Polynomials
For skew-symmetric acyclic quantum cluster algebras, we express the quantum
-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
In this paper we study cluster algebras \myAA of type . We solve
the recurrence relations among the cluster variables (which form a T--system of
type ). We solve the recurrence relations among the coefficients of
\myAA (which form a Y--system of type ). 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 . 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 --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 . 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 , for any index , one
can define an automorphism associated with of the field of rational functions of independent indeterminates It is an isomorphism between two cluster algebras associated to the
matrix (see section 4 for precise meaning). When 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 - and
-systems based on . 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 -system for as a
tool for relating -system solutions to all possible sets of initial data. We
show that the conserved quantities of the -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 -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 -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 T-system for arbitrary boundary
We solve the quantum version of the -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 -system and generalize it to
the fully non-commutative case. We give the relation between the quantum
-system and the quantum lattice Liouville equation, which is the quantized
-system.Comment: 24 pages, 18 figure
Quantum symmetric pairs and representations of double affine Hecke algebras of type
We build representations of the affine and double affine braid groups and
Hecke algebras of type , based upon the theory of quantum symmetric
pairs . In the case , our constructions provide a
quantization of the representations constructed by Etingof, Freund and Ma in
arXiv:0801.1530, and also a type generalization of the results in
arXiv:0805.2766.Comment: Final version, to appear in Selecta Mathematic
- …