Frieze patterns for punctured discs

We construct frieze patterns of type D _N with entries which are numbers of matchings between vertices and triangles of corresponding triangulations of a punctured disc. For triangulations corresponding to orientations of the Dynkin diagram of type D _N , we show that the numbers in the pattern can be interpreted as specialisations of cluster variables in the corresponding Fomin-Zelevinsky cluster algebra. This is generalised to arbitrary triangulations in an appendix by Hugh Thomas.     

Inclusion results on statistical cluster points

We study the concepts of statistical cluster points and statistical core of a sequence for A _λ methods defined by deleting some rows from a nonnegative regular matrix A. We also relate A _λ-statistical convergence to A _μ-statistical convergence. Finally we give a consistency theorem for A-statistical convergence and deduce a core equality result.     

Mutation of Torsion Pairs in Triangulated Categories and its Geometric Realization

We introduce and study mutation of torsion pairs, as a generalization of mutation of cluster tilting objects, rigid objects and maximal rigid objects. It is proved that any mutation of a torsion pair is again a torsion pair. A geometric realization of mutation of torsion pairs in the cluster category of type A _n or A _∞ is given via rotation of Ptolemy diagrams.     

On the quiver Grassmannian in the acyclic case

Let A be the path algebra of a quiver Q with no oriented cycle. We study geometric properties of the Grassmannians of submodules of a given A-module M. In particular, we obtain some sufficient conditions for smoothness, polynomial cardinality and we give different approaches to Euler characteristics. Our main result is the positivity of Euler characteristics when M is an exceptional module. This solves a conjecture of Fomin and Zelevinsky for acyclic cluster algebras.     

On cluster algebras arising from unpunctured surfaces II

We study cluster algebras with principal and arbitrary coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of certain paths on a triangulation of the surface. As an immediate consequence, we prove the positivity conjecture of Fomin and Zelevinsky for these cluster algebras.Furthermore, we obtain direct formulas for F-polynomials and g-vectors and show that F-polynomials have constant term equal to 1. As an application, we compute the Euler-Poincaré characteristic of quiver Grassmannians in Dynkin type A and affine Dynkin type .     

Igusa–Todorov functions for Artin algebras

In this paper we study the behavior of the Igusa–Todorov functions for Artin algebras A with finite injective dimension, and Gorenstein algebras as a particular case. We show that the ?-dimension and ψ-dimension are finite in both cases. Also we prove that monomial, gentle and cluster tilted algebras have finite ?-dimension and finite ψ-dimension.     

Derived representation type and Gorenstein projective modules of an algebra under crossed product

Let H and its dual H* be finite dimensional semisimple Hopf algebras. In this paper, we firstly prove that the derived representation types of an algebra A and the crossed product algebra A#σH are coincident. This is an improvement of the conclusion about representation type of an algebra in Li and Zhang [Sci China Ser A, 2006, 50: 1-13]. Secondly, we give the relationship between Gorenstein projective modules over A and that over A#σH. Then, using this result, it is proven that A is a finite dimensional CM-finite Gorenstein algebra if and only if so is A#σH.     

Subword complexes, cluster complexes, and generalized multi-associahedra

In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k=1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes of types A and B coincide with known simplicial complexes, namely with the simplicial complexes of multi-triangulations and centrally symmetric multi-triangulations, respectively. Furthermore, we show that the multi-cluster complex is universal in the sense that every spherical subword complex can be realized as a link of a face of the multi-cluster complex.     

A Simple Proof of the Sharp Weighted Estimate for Calderón–Zygmund Operators on Homogeneous Spaces

Here we show that Lerner’s method of local mean oscillation gives a simple proof of the A ₂ conjecture for spaces of homogeneous type, that is, the linear dependence on the A ₂ norm for weighted L ² Calderón–Zygmund operator estimates. In the Euclidean case, the result is due to Hytönen, and for geometrically doubling spaces, Nazarov, Reznikov, and Volberg obtained the linear bound.     

Construction of pure states in mean field models for spin glasses

If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda–Guerra identities, and if the law of the overlaps has a point mass at the largest point q* of its support, we prove that one can decompose the configuration space into a sequence of sets (A _k ) such that, generically, the overlap of two configurations is equal to q* if and only if they belong to the same set A _k . For the study of the overlaps each set A _k can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda–Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.     

Hamming graphs in Nomura algebras

Let A be an association scheme on q≥3 vertices. We show that the Bose-Mesner algebra of the generalized Hamming scheme H(n,A), for n?2, is not the Nomura algebra of any type II matrix.This result gives examples of formally self-dual Bose-Mesner algebras that are not the Nomura algebras of type II matrices.     

Determinantal formulae for matrices with sparse inverses,II: asymmetric zero patterns

In an earlier paper, formulae for det A as a ratio of products of principal minors of A were exhibited, for any given symmetric zero-pattern of A⁻¹. These formulae may be presented in terms of a spanning tree of the intersection graph of certain index sets associated with the zero pattern of A⁻¹. However, just as the determinant of a diagonal and of a triangular matrix are both the product of the diagonal entries, the symmetry of the zero pattern is not essential for these formulae. We describe here how analogous formulae for det A may be obtained in the asymmetric-zero-pattern case by introducing a directed spanning tree. We also examine the converse question of determining all possible zero patterns of A⁻¹ which guarantee that a certain determinantal formula holds.     

Expressions for certain minors and permanents

We provide explicit formulas for the minors of I–A and for per(I–A) where A is a row stochastic matrix.     

Reflection groupoids of rank two and cluster algebras of type A

We extend the classification of finite Weyl groupoids of rank two. Then we generalize these Weyl groupoids to ‘reflection groupoids’ by admitting non-integral entries of the Cartan matrices. This leads to the unexpected observation that the spectrum of the cluster algebra of type An−3 completely describes the set of finite reflection groupoids of rank two with 2n objects.     

A two-step problem of hedging a European call option under a random duration of transactions

We improve previous sum–product estimates in ?; namely, we prove the inequality max{|A + A|, |AA|} ? |A|^4/3+c, where c is any number less than 5/9813. New lower bounds for sums of sets with small product set are found. We also obtain results on the additive and multiplicative energies; in particular, we improve a result of Balog and Wooley.     

Some reducible Specht modules for Iwahori–Hecke algebras of type A with q=−1

The reducibility of the Specht modules for the Iwahori–Hecke algebras in type A is still open in the case where the defining parameter q equals ?1. We prove the reducibility of a large class of Specht modules for these algebras.     

Crystals and total positivity on orientable surfaces

We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated. We describe in terms of our model the crystal structure and R-matrix of the affine geometric crystal of products of symmetric and dual symmetric powers of type A. Local realizations of the R-matrix and crystal actions are used to construct a double affine geometric crystal on a torus, generalizing the commutation result of Kajiwara et al. (Lett Math Phys, 60(3):211–219, 2002) and an observation of Berenstein and Kazhdan (MSJ Mem, 17:1–9, 2007). We show that our model on a cylinder gives a decomposition and parametrization of the totally non-negative part of the rational unipotent loop group of GL _n .