arXiv subjects
Publications and source records attributed to Frederic Chapoton.
We use dualities of quiver moduli induced by reflection functors to describe generating series of motives of Kronecker moduli spaces of central slope as solutions of algebraic and q-difference equations.
We study two constructions related to the intervals of finite posets. The first one is a poset. The second one is more complicated. Loosely speaking it can be seen as a poset with some extra zero-relations. As main result, we show that these two constructions are equivalent at the level of derived categories.
Recent articles have shown the connection between representation theory of quivers and the theory of cluster algebras. In this article, we prove that some cluster algebras of type ADE can be recovered from the data of the corresponding quiver representation category. This also provides some explicit formulas for cluster variables.
The theory of cluster algebras of S. Fomin and A. Zelevinsky has assigned a fan to each Dynkin diagram. Then A. Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov have generalized this construction using arbitrary quivers on Dynkin diagrams. In the special case of the unioriented quiver of type A, we describe the cohomology ring of the toric variety associated to this fan. A natural base is obtained and an explicit rule is given for the product of any two generators.
On obtient une relation entre la transformation de Coxeter agissant sur le groupe de Grothendieck de la categorie derivee des modules sur un poset de Tamari d'une part et la structure de l'operade anticyclique dendriforme d'autre part. A relation between the anticyclic structure of the dendriform operad and the Coxeter transformations in the Grothendieck groups of the derived categories of modules over the Tamari posets is obtained.
Some binary quadratic operads are endowed with anticyclic structures and their characteristic functions as anticyclic operads are determined, or conjectured in one case.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds to a tilting object in the cluster category. The cluster tilted algebra is the algebra of endomorphisms of that tilting object. Viewing the cluster tilted algebra as a path algebra of a quiver with relations, we prove in this paper that the quiver of the cluster tilted algebra is equal to the cluster diagram. We study also the relations. As an application of these results, we answer several conjectures on the connection between cluster algebras and quiver representations.
The aim of this paper is to define and study pointed and multi-pointed partition posets of type A and B (in the classification of Coxeter groups). We compute their characteristic polynomials, incidence Hopf algebras and homology groups. As a corollary, we get that some operads are Koszul over Z. Cet article introduit et etudie les posets des partitions pointees et multi-pointees en type A et B (dans la classification des groupes de Coxeter finis). Leurs polynomes caracteristiques, leurs algebres de Hopf d'incidence et leurs groupes d'homologie sont calcules. Comme corollaire, on obtient que certaines operades sont de Koszul sur Z.
On etudie divers aspects d'une formule qui compte les reflexions pleines dans les groupes de Coxeter finis. ***** We study several points about a formula which counts reflexions in a finite Coxeter group whose reduced decompositions involve all simple reflexions.
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.
Some enumerative aspects of the fans, called generalized associahedra, introduced by S. Fomin and A. Zelevinsky in their theory of cluster algebras are considered, in relation with a bicomplex and its two spectral sequences. A precise enumerative relation with the lattices of generalized noncrossing partitions is conjectured and some evidence is given.
A new Hopf operad Ram is introduced, which contains both the well-known Poisson operad and the Bessel operad introduced previously by the author. Besides, a structure of cooperad R is introduced on a collection of algebras given by generators and relations which have some similarity with the Arnold relations for the cohomology of the type A hyperplane arrangement. A map from the operad Ram to the dual operad of R is defined which we conjecture to be a isomorphism.
Some posets of binary leaf-labeled trees are shown to be supersolvable lattices and explicit EL-labelings are given. Their characteristic polynomials are computed, recovering their known factorization in a different way.
Let G be a chordal graph, X(G) the complement of the associated complex arrangement and Gamma(G) the fundamental group of X(G). We show that Gamma(G) is a limit of colored braid groups over the poset of simplices of G. When G = G_T is the comparability graph associated with a rooted tree T, a case recently investigated by the first author, the result takes the following very simple form: Gamma(G_T) is a limit over T of colored braid groups.
The ring of locally-constant integer-valued functions on the dominant chamber of the Shi arrangement is endowed with a filtration and a new basis, compatible with this filtration, is found. This basis is compared to the trivial basis. The ring is given a presentation by generators and relations.
Each labeled rooted tree is associated with a hyperplane arrangement, which is free with exponents given by the depths of the vertices of this tree. The intersection lattices of these arrangements are described through posets of forests. These posets are used to define coalgebras, whose dual algebras are shown to have a simple presentation by generators and relations.
A nice factorization is given for the characteristic polynomials of intervals in some posets of leaf-labeled forests of rooted binary trees.
This paper deals with a group of generalized power series associated to any augmented operad, focusing on the case of the PreLie operad. The solution of flow equations using the pre-Lie structure on vector fields on an affine space gives rise to an interesting element of this group, which deserves to be called the PreLie exponential.