The number of complete exceptional sequences for a Dynkin algebra

TL;DR

This paper counts the number of complete exceptional sequences in Dynkin algebras, linking representation theory with non-crossing partitions, and provides a categorification of related combinatorial results.

Contribution

It determines the exact number of complete exceptional sequences for any Dynkin algebra, establishing a connection with non-crossing partitions and maximal chains in their lattice.

Findings

Number of complete exceptional sequences for Dynkin algebras is explicitly determined.

Establishes a bijection between exceptional sequences and maximal chains in non-crossing partition lattices.

Provides a categorification linking representation theory and combinatorial structures.

Abstract

We consider Dynkin algebras, these are the hereditary artin algebras of finite representation type. The indecomposable modules for a Dynkin algebra correspond bijectively to the positive roots of a Dynkin diagram. Given a Dynkin algebra with n simple modules, a complete exceptional sequence is a sequence M_1,..., M_n of indecomposable modules such that Hom(M_i,M_j) = 0 = Ext(M_i,M_j) for i > j. The aim of this paper is to determine the number of complete exceptional sequences for any Dynkin algebra. There are direct connections between the representation theory of a Dynkin algebra A and the lattice L of non-crossing partitions of the same Dynkin type: As Ingalls and Thomas have shown, the lattice of the thick subcategories of mod A can be identified with L. Hubery and Krause have pointed out that this identification provides a bijection between the complete exceptional sequences for A and the maximal chains in L. Thus, our calculations may also be considered as a categorification of results concerning non-crossing partitions.