$m$-noncrossing partitions and $m$-clusters
Abstrak
Let $W$ be a finite crystallographic reflection group, with root system $\Phi$. Associated to $W$ there is a positive integer, the generalized Catalan number, which counts the clusters in the associated cluster algebra, the noncrossing partitions for $W$, and several other interesting sets. Bijections have been found between the clusters and the noncrossing partitions by Reading and Athanasiadis et al. There is a further generalization of the generalized Catalan number, sometimes called the Fuss-Catalan number for $W$, which we will denote $C_m(W)$. Here $m$ is a positive integer, and $C_1(W)$ is the usual generalized Catalan number. $C_m(W)$ counts the $m$-noncrossing partitions for $W$ and the $m$-clusters for $\Phi$. In this abstract, we will give an explicit description of a bijection between these two sets. The proof depends on a representation-theoretic reinterpretation of the problem, in terms of exceptional sequences of representations of quivers.
Topik & Kata Kunci
Penulis (3)
Aslak Bakke Buan
Idun Reiten
Hugh Thomas
Akses Cepat
- Tahun Terbit
- 2009
- Sumber Database
- DOAJ
- DOI
- 10.46298/dmtcs.2719
- Akses
- Open Access ✓