From m-clusters to m-noncrossing partitions via exceptional sequences

Abstract

Abstract. Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W , and with the number of noncrossing partitions for W . Natural bijections between these two sets are known. For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss-Catalan number Cm(W ). We give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category D b (H) for any finite-dimensional hereditary algebra H.

Hom-configurations and noncrossing partitions

“…Note that the full subcategory D [10], is just a different fundamental domain for the action of τ [2] in D b (Q).…”

“…In [10], Buan, Reiten and Thomas define a new object in the bounded derived category D b (Q) of an arbitrary hereditary Artin algebra, called a Hom ≤0 -configuration. Definition 3.3 [10] An object X ∈ D b (Q) is a Hom ≤0 -configuration if it satisfies the following axioms:…”

“…We will study the relation between the bijection ρ and the bijection θ between 1-noncrossing partitions and Hom ≤0 -configurations (contained in KQ-mod ∪ KQ-mod [1]), given by A. Buan, I. Reiten and H. Thomas in [10,Theorem 7.3]. Definition 5.9 A 1-noncrossing partition associated to W Q is a pair (u 1 , u 2 ) of elements of W Q such that c = u 1 u 2 and l T (u 1 ) + l T (u 2 ) = n.…”

“…Definition 3.3 [10] An object X ∈ D b (Q) is a Hom ≤0 -configuration if it satisfies the following axioms:…”

“…Our work is closely related to [10]. In this paper the authors give a natural bijection between m-clusters and m-noncrossing partitions, for m ≥ 1, using special classes of exceptional sequences in the bounded derived category.…”