84 research outputs found
Categorification of a frieze pattern determinant
Broline, Crowe and Isaacs have computed the determinant of a matrix
associated to a Conway-Coxeter frieze pattern. We generalise their result to
the corresponding frieze pattern of cluster variables arising from the
Fomin-Zelevinsky cluster algebra of type A. We give a representation-theoretic
interpretation of this result in terms of certain configurations of
indecomposable objects in the root category of type A.Comment: 14 pages; 8 figures. Sections 1-3 rewritten, postponing the cluster
interpretation until Section 3. Minor correction
Denominators in cluster algebras of affine type
The Fomin-Zelevinsky Laurent phenomenon states that every cluster variable in
a cluster algebra can be expressed as a Laurent polynomial in the variables
lying in an arbitrary initial cluster. We give representation-theoretic
formulas for the denominators of cluster variables in cluster algebras of
affine type. The formulas are in terms of the dimensions of spaces of
homomorphisms in the corresponding cluster category, and hold for any choice of
initial cluster.Comment: 22 pages, no figures. Correction to Defn 1.2. Minor correction
A geometric model of tube categories
We give a geometric model for a tube category in terms of homotopy classes of
oriented arcs in an annulus with marked points on its boundary. In particular,
we interpret the dimensions of extension groups of degree 1 between
indecomposable objects in terms of negative geometric intersection numbers
between corresponding arcs, giving a geometric interpretation of the
description of an extension group in the cluster category of a tube as a
symmetrized version of the extension group in the tube. We show that a similar
result holds for finite dimensional representations of the linearly oriented
quiver of type A-double-infinity.Comment: 15 pages, 7 figures. Discussion of maximal rigid objects and
triangulations at end of Section 3. Minor correction
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
Degenerate flag varieties: moment graphs and Schr\"oder numbers
We study geometric and combinatorial properties of the degenerate flag
varieties of type A. These varieties are acted upon by the automorphism group
of a certain representation of a type A quiver, containing a maximal torus T.
Using the group action, we describe the moment graphs, encoding the zero- and
one-dimensional T-orbits. We also study the smooth and singular loci of the
degenerate flag varieties. We show that the Euler characteristic of the smooth
locus is equal to the large Schr\"oder number and the Poincar\'e polynomial is
given by a natural statistics counting the number of diagonal steps in a
Schr\"oder path. As an application we obtain a new combinatorial description of
the large and small Schr\"oder numbers and their q-analogues.Comment: 25 page
Mutation in triangulated categories and rigid Cohen-Macaulay modules
We introduce the notion of mutation of -cluster tilting subcategories in a
triangulated category with Auslander-Reiten-Serre duality. Using this idea, we
are able to obtain the complete classifications of rigid Cohen-Macaulay modules
over certain Veronese subrings.Comment: 52 pages. To appear in Invent. Mat
The geometry of Brauer graph algebras and cluster mutations.
In this paper we establish a connection between ribbon graphs and Brauer graphs. As a result, we show that a compact oriented surface with marked points gives rise to a unique Brauer graph algebra up to derived equivalence. In the case of a disc with marked points we show that a dual construction in terms of dual graphs exists. The rotation of a diagonal in an m-angulation gives rise to a Whitehead move in the dual graph, and we explicitly construct a tilting complex on the related Brauer graph algebras reflecting this geometrical move
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